Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 9 Resources: SpringBoardGeometry

Online Resources:

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Student Focus

Springboard Geometry Main Ideas for success in lessons 9-1, 9-2, 9-3, and 9-4:

Unit 2 Vocabulary:

 Perform transformations, translations, rotations, and reflections on and off the coordinate plane.

Transformations Pre-image Image Rigid motion Translation Rhombus Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Composition of transformations Congruent Corresponding parts SSS SAS ASA AAS AAA SSA Triangle Congruence Criteria Criterion Flowchart Proof

Page 1 of 53

 Identify characteristics of transformations that are rigid motions and characteristics of transformations that are non-rigid motions.  Represent a transformation as a function using coordinates, and show how a figure is transformed by a function.  Predict the effect of a translation on a figure.  Identify reflection symmetry in plane figures.  Identify and distinguish between reflectional and rotational symmetry.

Example: 9-1

Complete the table to show the coordinates of the image and preimage for the four corners of the rectangle.

Is the transformation of the figure above a rigid motion or non-rigid motion? Explain how you know.  Rigid motion. The image has the same size and shape as the pre-image. Page 2 of 53

Example: 9-2 The figure shows hexagon ABCDEF undergoing a translation to the right.

Which part of the pre-image is translated onto CD?  is translated onto .

Express the translation as a function of two points of the hexagon.



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or

Example: 9-3

Draw the reflection of the arrow described by each of these functions, and identify the line of reflection. a. line of reflection: x = 4

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b. line of reflection: x = -1

c. line of reflection: y = 0 (x-axis)

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Example: 9-4 Describe the rotation that would move the arrow to these positions.

a. pointing up, with the tip at (4,7) 90⁰ clockwise about (4,3) b. Pointing down, with the tip at (0,3) 90⁰ counterclockwise about (0,3) c. Pointing up, with the tip at (3,0) 90⁰ clockwise about (0,0)

Page 6 of 53

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 10 Resources: SpringBoard- Geometry

Online Resources:

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Springboard Geometry

Student Focus

Unit 2 Vocabulary:

Main Ideas for success in lessons 10-1 and 10-2:

Transformations Pre-image Image Rigid motion Translation Rhombus Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Composition of transformations Congruent Corresponding parts SSS SAS ASA AAS AAA SSA Triangle Congruence Criteria Criterion Flowchart Proof

Page 7 of 53

 Finding the image of a figure under a composition of rigid motions.  Finding the pre-image of a figure under a composition of rigid motions.  Determining whether given figures are congruent.  Specifying a sequence of rigid motions that will carry a given figure to a congruent figure.

Example 10-1: An isosceles triangle has vertices at (-3,0), (0,1) , and (3,0).

Example 10-2: An arrow is placed with its base at the origin and its tip at the point (0,2). For each arrow listed below, find the rigid motion or composition that shows the two arrows are congruent.

Page 8 of 53

a. Base at (3,3), tip at (3,5)

T(3,3)

b. Base at (-2,2), tip at (-2,0)

T(-20) (r(y=1))

c. Base at (4,0), tip at (6,0)

T(4,0) R(0,90⁰)

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 11 Resources: SpringBoardGeometry

Online Resources: Springboard Geometry

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Student Focus Main Ideas for success in lessons 11-1, 11-2, 11-3, and 11-4:

Unit 2 Vocabulary Transformations Pre-image Image Rigid motion Translation Rhombus Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Composition of transformations Congruent Corresponding parts SSS SAS ASA AAS AAA SSA Triangle Congruence Criteria Criterion Flowchart Proof

Page 9 of 53

 Use the fact that congruent triangles have congruent corresponding parts.  Determine unknown angle measures or side lengths in congruent triangles.  Develop criteria for proving triangle congruence.  Determine which congruence criteria can be used to show that two triangles are congruent  Prove that congruence criteria follow from the definition of congruence.  Use the congruence criteria in simple proofs.  Apply congruence criteria to figures on the coordinate plane.  Prove the AAS criterion and develop the HL criterion.

Example 11-1:

In the figure, ∆ABC

∆DEF, find the length of

.

 18 in.

Find the measure of all angles in ∆DEF that is possible to find. 

What is the perimeter of ∆DEF?  73 in.

Page 10 of 53

Example 11-2: For each pair of triangles, write the congruence criterion, if any, that can be used to show the triangles are congruent.

none

SSS

ASA

Page 11 of 53

Example 11-3:

Example 11-4: On a coordinate plane, plot triangles ABC and DEF with vertices A(-3,-1), B(-1,2), C(1,1), D(3,-4), E(1,-1), and F(-1,-2). Then prove ∆ABC ∆DEF. Using the Distance Formula

Since

Page 12 of 53

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 12 Resources

Unit Overview

Spring BoardGeometry

In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Online Resources: Springboard Geometry

Student Focus

Unit 2 Vocabulary:

Main Ideas for success in lessons 12-1 and 12-2:

Transformations Pre-image Image Rigid motion Translation Rhombus Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Composition of transformations Congruent Corresponding parts SSS SAS ASA AAS AAA SSA Triangle Congruence Criteria Criterion Flowchart Proof

Page 13 of 53

 Write a flowchart proof.  Write a simple flowchart proof as a two-column proof.  Write a proof in three different formats.  Write proofs using the fact that corresponding parts of congruent triangles are congruent.

Example 12-1: Write a flowchart proof for the following. Given: ∆ACD ≡ ∆BDE;  CAD Prove: ∆ACD

∆ABD

See flowchart proof below.

Page 14 of 53

BDA

Example 12-2:

Page 15 of 53

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 13 Resources: SpringBoard- Geometry

Online Resources:

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Springboard Geometry

Student Focus Unit 2 Vocabulary Transformations Pre-image Image Rigid motion Translation Directed line segment Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Composition of transformations Congruent Corresponding parts SSS, SAS, ASA, AAS, AAA,SSA Triangle Congruence Criteria Criterion Flowchart Proof Auxiliary line Interior angle Exterior angle Remote Interior Angle Exterior Angle Theorem Isosceles Triangle Theorem

Page 16 of 53

Main Ideas for success in lessons 13-1 and 13-2:  Prove theorems about angle measures in triangles.  Apply theorems about angle measures in triangles.  Develop theorems about Isosceles triangles.  Prove theorems about isosceles triangles.

Example 13-1: Use the figure below to find each measure.

A. 

B.

A = 89⁰

C = ______ 

C.

ABC = _____ 

D.

ABC = 43.5⁰

ABD = _____ 

Page 17 of 53

C = 47.5⁰

ABD = 136.5⁰

Example 13-2: Given

, =

= ⁰.

⁰,

=

⁰, and

A. Find the value of .

B. Determine the measure of each of the three angles.

m A = 20, m B = 30, m C = 130

C. Classify

by side length and angle measure.

Scalene, obtuse

Page 18 of 53

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 14 Resources: SpringBoard- Geometry

Online Resources: Springboard Geometry

Unit 2 Vocabulary Transformations Pre-image Image Rigid motion Translation Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Congruent Corresponding parts SSS, SAS, ASA, AAS, AAA,SSA Triangle Congruence Criteria Criterion Flowchart Proof Auxiliary line Interior angle Exterior angle Remote Interior angle Exterior angle theorem Isosceles Triangle theorem Altitudes Point of Concurrency Orthocenter Medians Centroid Circumcenter Incenter Circumscribed Circle Inscribed Circle

Page 19 of 53

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Student Focus Main Ideas for success in lessons 14-1, 14-2, and 14-3:  Determine the point of concurrency of the altitudes of a triangle.  Use the point of concurrency of the altitudes of a triangle to solve problems.  Determine the point of concurrency of the medians of a triangle.  Use the point of concurrency of the medians of a triangle to solve problems.  Determine the points of concurrency of the perpendicular bisectors and the angle bisectors of a triangle to solve problems.

Example 14-1: Find the coordinates of the orthocenter of the triangle with the given vertices: 

Example 14-2: Given:

A. If

with centroid . Complete the following.

= 8cm, then

= ______ and

= ______

16cm, 24cm

B. If

= 12cm, then 8cm, 4cm

Page 20 of 53

= _____ and

= _____

Example 14-3: Given:

If

with circumcenter .

= 18mm, then  18mm, 18mm

Page 21 of 53

= _____ and

= _____

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 15 Resources: SpringBoard- Geometry

Online Resources: Springboard Geometry

Unit 2 Vocabulary Transformations Pre-image Image Rigid motion Translation Directed line segment Reflection Line of reflection Reflectional symmetry Line of symmetry Rotation Rotational symmetry Angle of rotational symmetry Congruent Corresponding parts SSS, SAS, ASA, AAS, AAA,SSA Flowchart Proof Interior angle Exterior angle Remote Interior angle Exterior angle theorem Isosceles Triangle theorem Altitudes Point of Concurrency Orthocenter Medians Centroid Circumcenter Incenter Circumscribed Circle Inscribed Circle Page 22 of 53

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Student Focus Main Ideas for success in lessons 15-1, 15-2, 15-3, and 15-4:  Develop properties of kites.  Prove the triangle Midsegment Theorem.  Develop properties of trapezoids.  Prove properties of trapezoids.  Develop properties of parallelograms.  Prove properties of parallelograms.  Develop properties of rectangles, rhombuses, and squares.  Prove properties of rectangles, rhombuses, and squares.

Example 15-1: is a midsegment of

x = _____

. Find each measure.

x=6

= _____

= 19

= ______

= 9.5

Example 15-2: is a midsegment of trapezoid QRST. Find each measure.

Page 23 of 53

QU = _____

QU = 3

VS = _____

VS = 4

UV = _____

UV = 13

Example 15-3:

and are diagonals of parallelogram ABCD. Find each measure.

= _____

Page 24 of 53

= 10

= _____

= 10

= _____

=8

= _____

=8

Example 15-4: and measure.

m

Page 25 of 53

are diagonals of rectangle ABCD. Find each

= _____

90⁰

m ACD = _____

90⁰

m BDC = _____

25⁰

m BDA = _____

65⁰

m AEB = _____

130⁰

m BEC = _____

50⁰

m BCE = _____

65⁰

Geometry Honors: Transformations, Triangles, and Quadrilaterals Semester 1, Unit 2: Activity 16 Resources: SpringBoard- Geometry

Online Resources: Springboard Geometry

Unit Overview In this unit, students study how transformations are connected to congruence. They will use idea of congruence to write proofs involving triangles and quadrilaterals.

Student Focus Main Ideas for success in lessons 16-1, 16-2, 16-3, and 16-4:

Unit 2 Vocabulary Transformations Pre-image Image Rigid motion Translation Directed line segment Reflection Line of reflection Reflectional symmetry Line of symmetry symmetry Composition of Transformations Congruent Corresponding parts SSS, SAS, ASA, AAS, AAA Flowchart Proof Auxiliary line Interior angle Exterior angle Remote Interior angle Exterior angle theorem Isosceles Triangle theorem Altitudes Point of Concurrency Orthocenter Medians Centroid Circumcenter Incenter Circumscribed Circle Inscribed Circle Page 26 of 53

 Develop criteria for showing that a quadrilateral is a parallelogram.  Prove that a quadrilateral is a parallelogram.  Develop criteria for showing that a quadrilateral is a rectangle.  Prove that a quadrilateral is a rectangle.  Develop criteria for showing that a quadrilateral is a rhombus.  Prove that a quadrilateral is a rhombus.  Develop criteria for showing that a quadrilateral is a square.  Prove that a quadrilateral is a square.

Example 16-1: What theorem can be used to prove the quadrilateral below is a parallelogram? If there is not enough information to prove it is a parallelogram, write “not enough information”.

 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

 Not enough information

Example 16 – 2: Find the value of x that makes the parallelogram a rectangle.

 x = 15

Page 27 of 53

Example 16 – 3: Three vertices of a rhombus are given. Find the coordinates of the fourth vertex.



Example 16 – 4: The coordinates of a parallelogram are given. Determine whether the figure is a square.

 No

Page 28 of 53

Name

class

date

Geometry Unit 2 Practice Lesson 9-1 1. Which of the following shows the rectangle on the grid under the transformation (x, y) → (x 2 1, y 1 3)? y 5

10

5

25

x

25

A.

C.

y 5

y 5

5

25

x

25

10

5

25

x

25

210

B.

D.

y

y 5

5

5

25

10

x

5

25

x

25

25

210



© 2015 College Board. All rights reserved.

Page 29 of 53

1

SpringBoard Geometry, Unit 2 Practice

Name

class

date

2. What function describes the transformation shown? Write your answer as (x, y) → (?, ?). y

y 5

5

5

25

x

5

25

25

x

25

5. A rectangle on a coordinate plane has side lengths of 5 units and 4 units, so its diagonal is 41 units. The position of the rectangle is changed by a rigid transformation. a. What is the length of the diagonal of the transformed rectangle?

3. Express regularity with repeated reasoning. One of the vertices of a rectangle is (23, 24). What is the image of that vertex after each transformation? a. (x, y) → (x, y 1 7) b. (x, y) → (x 2 3, y 1 2)

b. Explain your answer to Part a.

c. (x, y) → (2x, y 2 5)

Lesson 9-2

d. (x, y) → (2y, 2x)

6. Model with mathematics. Use the diagram shown.

4. Reason quantitatively. Label each transformation as rigid or nonrigid. II

a. (x, y) → (x 1 7, y 1 3)

I

b. (x, y) → (x 1 1, 2y) IV

III

c. (x, y) → (2x, 2y)

d. (x, y) → (2x, 2y 21) Which pair of figures can represent the pre-image and image in a translation? A. I and IV B.  II and III

e. (x, y) → (x 1 5, 0)

C. III and IV © 2015 College Board. All rights reserved.

Page 30 of 53

2

D.  I and III SpringBoard Geometry, Unit 2 Practice

Name

class

7. Use the diagram shown.

10. Make sense of problems. In the diagram shown, two of the triangles represent the translation (x, y) → (x 2 4, y 1 5).

y

A

date

5 B

y 5

5

25

x

C

B

C

5

25

x

A

25 25

a. What is the image of point A under the translation (x, y) → (x 2 3, y 2 2)?

a. Which figure is the pre-image in the translation?

b. What translation maps point B to (0, 0)? Write your answer as a function (x, y) → (?, ?).

b. Which figure is the image in the translation?

8. Line segment PQ has endpoints P(3, 22) and Q(2, 4). The translation (x, y) → (x 2 3, y 1 5) maps PQ to RS .

c. Explain your answers to Parts a and b. d. What translation will take the image back to the pre-image? Write your answer as a function (x, y) → (?, ?).

a. What is the relationship between PQ and RS ?

b. What are the coordinates of the endpoints of RS ?

Lesson 9-3 11. A point is located at (24, 3). Which reflection maps that point to (10, 3)?

9. Consider the point (26, 3). a. What is the image of (26, 3) under the translation (x, y) → (x 1 5, y 2 3)?

A. a reflection across the line x 5 3 B. a reflection across the line x 5 7 C. a reflection across the line x 5 10

b. What translation takes (26, 3) to (0, 22)? Write your answer as a function (x, y) → (?, ?).

D. a reflection across the line y 5 3 12. How is the point (3, 25) related to its reflection across the line y 5 2? A. The point (3, 25) and its image are 2 units apart. B. The point (3, 25) is on the line of reflection. C. The point and its image are the same distance from the line y 5 2. D. The line of reflection is perpendicular to the line y 5 2.

© 2015 College Board. All rights reserved.

Page 31 of 53

3

SpringBoard Geometry, Unit 2 Practice

Name

class

date

15. Construct viable arguments. Consider the capital letters from H through P.

13. The point (25, 22) is reflected over a line of reflection. Find the equation of the line of reflection if the image of (25, 22) is each of the following.

a. List the letters with no lines of symmetry.

a. (5, 22)

b. List the letters with exactly one line of symmetry.

b. (25, 2)

c. List the letters with exactly two lines of symmetry.

c. (25, 0)

d. Consider the four operation symbols 1, 2, 3, and 4. How many lines of symmetry does each one have?

d. (0, 22) e. (25, 22)

Lesson 9-4

14. Model with mathematics. Use the square shown.

16. Use the diagram of a T with an arrow, as shown.

y

y

5 5 A

B

5

25 D

x 5

25

x

C

25 25

a. What are the coordinates of the image of vertex A if the square is reflected across the line x 5 5?

After each rotation of the figure, indicate the direction of the arrow. a. 90° counterclockwise

b. What are the coordinates of the image of vertex B if the square is reflected across the line y 5 0? b. 90° clockwise c. What are the coordinates of the image of vertex C under the reflection (x, y) → (x, 2y)?

c. 180°

d. Under a reflection, vertex D maps to (0, 6). What is the line of reflection?

© 2015 College Board. All rights reserved.

Page 32 of 53

d. 270° clockwise

4

SpringBoard Geometry, Unit 2 Practice

Name

class

date

19. A rotation maps figure ABCDE to figure PQRST.

17. Model with mathematics. Use the diagram shown. Which rotation maps Figure II onto Figure I?

a. What angle of PQRST must have the same measure as angle D?

y 5

b. What angle of ABCDE must have the same measure as angle T? 25

I

5

x

c. What side of PQRST must have the same length as BC ?

II 25

d. What general property can you use to answer Parts a through c?

A. a rotation of 180° around (5, 25) B. a rotation of 90° clockwise around (3, 23)

20. Attend to precision. Describe the rotation that would move the arrow from (22, 1) to each position.

C. a rotation of 90° counterclockwise around (5, 23) D. a rotation of 90° clockwise around (5, 23)

y 5

18. In the diagram, figure IV is the rotated image of figure III. y

5

25

x

5 III

25 5

25

x

a. pointing up, with the tip at (22, 7)

IV 25

b. pointing left, with the tip at (24, 3) a. What is the center of rotation? c. pointing right, with the tip at (5, 22)

b. What are the angle and direction of rotation?

d. pointing up, with the tip at (2, 1)

© 2015 College Board. All rights reserved.

Page 33 of 53

5

SpringBoard Geometry, Unit 2 Practice

Name

class

date

25. Which of the following statements is NOT true?

Lesson 10-1

A. The composition of two translations can be represented as a single translation.

21. Write the notations for these compositions of transformations. a. a reflection about the line x 5 2, followed by a rotation of 180° about the origin

B. The composition of two rotations can be represented as a single rotation. C. The composition of two reflections can be represented as a single reflection. D. The composition of a translation followed by a reflection can be represented as the composition of a reflection followed by a translation.

b. a translation of the directed line segment from the origin to (0, 5), followed by a reflection around the line y 5 5

Lesson 10-2 26. Consider the figures in the diagram shown. Complete each transformation or composition to show the two rectangles are congruent. A

22. Reason quantitatively. An arrow is placed with its base at (21, 1) and its tip at (4, 1). Identify the positions of the image’s base and tip under the composition T(3, 21) ((R0, 90°) T(2, 3)).

B

m

G

D C

23. Identify the inverses of these transformations and compositions.

F

a. T(5, 21)

a. reflection of ABCD over ?

b. R0, 180° (T(2, 3))

b. translation of DEFG by directed line segment DC , followed by rotation ?

24. Attend to precision. For each of these compositions, identify the single rigid motion that performs the same mapping.

c. rotation of DEFG 90° clockwise around D, followed by translation ?

a. R(3, 4), 90° (R(3, 4), 90°)

d. rotation of DEFG 90° clockwise around D, followed by reflection ?

b. 2T(2, 23) (T(25, 5))

© 2015 College Board. All rights reserved.

Page 34 of 53

E

6

SpringBoard Geometry, Unit 2 Practice

Name

class

date

30. In the diagram shown, ACFD and CHGB are rectangles, CF and CB have the same length, and AC and CH have the same length. Explain why the following composition shows that ACFD and CHGB are congruent.

27. Reason abstractly. Explain why it is possible that the diameter of one circle can be congruent to the radius of another circle. 28. Which combination shows that rectangles A and B are congruent?

RC, 90°(TCA (CHGB))

y

A

B

C

5 E

A

25

5

B

F

D

10

x

G

H

Lesson 11-1

25

31. Suppose you have a sequence of rigid motions to map XYZ to PQR. Fill in the blank for each transformation.

A. R(0, 3), 90° (T(23, 0) (A)) B. T(23, 21), 90° (R(23, 2), 90° (A))

a. ∠Y →  ? 

C. T(23, 21), 90° (R(23, 2), 90° (B)) D. T(23, 27), 90° (R(23, 27), 90° (A))

b.  ?  → ∠P

29. Express regularity in repeated reasoning. Arrow A is placed with its base at (1, 3) and its tip at (6, 3). For each arrow, complete the rigid motion or composition that shows the new arrow is congruent to the original one.

c. YZ →  ?  d.  ?  → PR e.  ?  → RPQ

a. base (6, 23), tip (1, 23); translation  ?  followed by reflection  ? 

32. ABC is divided into two congruent triangles by BP . Fill in the blanks to show the congruent sides and angles. b. base (6, 23), tip (1, 23); rotation  ? 

B

A

P

C

a. AP   ? 

c. base (6, 23), tip (1, 23); reflection  ?  followed by another reflection  ? 

b. ∠APB   ?  c.  ?   ∠PCB

d. base (0, 2), tip (26, 2); rotation  ?  followed by translation  ? 

© 2015 College Board. All rights reserved.

Page 35 of 53

d. PBA   ?  7

SpringBoard Geometry, Unit 2 Practice

Name

class

Lesson 11-2

33. MNT  RPQ. Complete the following. a. Name three pairs of corresponding sides.

36. Write the triangle congruence criterion illustrated by each pair of triangles. a.

b. Name three pairs of corresponding angles. 34. Critique the reasoning of others. In the figure, DGF  EFG. Which statement is NOT correct? F

D

date

b.

E

c.

G

A. DG  EF B. FD  GE

d.

C. ∠DFG  ∠EGF D. ∠DGF  ∠EGF 37. Make use of structure. Which additional sides or angles must be congruent in order to use the given triangle congruence criterion?

35. Make use of structure. For each diagram, write the congruence criterion that can be used to show the triangles are congruent.

a. ASA A

a.



D

B

E

C

F

b. b. AAS E

A

c.

A

F

B



C

B

C

D

D

DB bisects ∠ADC and ∠ABC.

c. SSS A

d. B

© 2015 College Board. All rights reserved.

Page 36 of 53

8

D

C

F

E

SpringBoard Geometry, Unit 2 Practice

Name

class

d. SAS

date

Lesson 11-3 A

C

D

B

41. Suppose you are given that ABC and XYZ satisfy the ASA triangle congruence criterion. How can you show that ABC is congruent to XYZ using the definition of congruence?

F

E

42. Construct viable arguments. You are given that ∠ABC  ∠DCB, BD bisects ∠ABC, and CA bisects ∠BCD.

38. Which statements are NOT enough to show that AED  BEC?

B

C

1

2

C

E A

B

E

D

A

a. How do you know that ∠1  ∠2?

D

A. ∠D  ∠C, AE  BE , ∠AED  ∠BEC

b. What triangle seems to be congruent to ABC?

B. ∠A  ∠B, AE  BE , ∠AED  ∠BEC C. AE  BE , ∠AED  ∠BEC, DE  CE

c. What side is common in the two triangles?

D. AE  BE , ∠AED  ∠BEC, AD  BC d. What triangle congruence criterion can you use to prove the two triangles are congruent?

39. Construct viable arguments. Can an obtuse triangle be congruent to an isosceles triangle? Explain.

43. In the diagram shown, ABD is a reflection image of ABC over AB . Which relationship CANNOT be established using the properties of reflection?

40. Use the diagram shown. Find two possible ordered pairs for point F so that ABC  EDF.

A

y 10

5

(2, 5) A

C

210



B. ∠CAD is a right angle.

B (2, 3)

25

D

A. AC  AD

(24, 3) C

B

D (4, 21)

5

E (6, 21)

10

C. ∠C  ∠D

x

D. BC  BD

25

© 2015 College Board. All rights reserved.

Page 37 of 53

9

SpringBoard Geometry, Unit 2 Practice

Name

class

47. Reason quantitatively. In the diagram shown, ∠A and ∠X are right angles.

44. Make use of structure. In the diagram shown, ∠PQR and ∠SRQ are right angles and PR  SQ . P

date

y

Q 20

S

R

10

a. What other congruence statement about segments can you make about the diagram?

X(24, 21)

A(3, 4)

C(15, 4) 10

220 Z (216, 21)

b. What triangle congruence criterion can you use to conclude that the two triangles are congruent?

B (3, 9)

20

x

Y (24, 26) 210

c. Complete this statement: RSQ   ? .

220

a. Are ABC and XYZ right triangles? Explain.

d. Once the triangles are shown to be congruent, what other relationships must be true about pairs of sides and angles?

b. Use the Distance Formula to find BC and YZ. c. Use the Distance Formula to find AB and XY.

45. Suppose MNP  QRS. If m∠M 5 47°, m∠N 5 52°, and m∠S 5 7x 1 4, find x.

d. Do you have enough information to prove that XYZ  ABC? Explain.

Lesson 11-4 46. Make use of structure. In triangles PQR and STV, what information satisfies the HL congruence criterion? P

48. In the diagram shown, YW bisects ∠XYZ and ∠XWZ. Which triangle congruence criterion lets you conclude that XYW  ZYW?

S

X R

Q

T

V Y

A. ∠Q and ∠T are right angles, PQ  ST, and PR  SV.

Z

A. HL

B. ∠Q and ∠T are right angles, PQ  ST, and QR  TV.

B. ASA C. SAS

C. ∠Q  ∠T, PQ  ST, PR  SV

D. SSS

D. ∠Q and ∠T are right angles, ∠P  ∠S, and ∠R  ∠V. © 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

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class

49. In an isosceles triangle, a segment joins the vertex of the triangle to the midpoint of the third side.

date

50. In this diagram, PQ  PR and PS bisects ∠QPR. P

a. How is that segment related to the vertex angle of the triangle? Q

b. Describe the angle formed by that segment and the side it intersects.

S

R

a. Complete this statement: SQP   ? . b. How is PS related to QR?

Lesson 12-1 51. Use appropriate tools strategically. Rewrite this flowchart proof as a two-column proof. Given: ∠1  ∠4, MN  PT Prove: MNT  TPM M

1

N 2

3 P

T

4

1. /1 > /4 3. /2 > /3

Given

Transitive prop. 2. /1 > /2 /3 > /4

4. MN > PT

6. nMNT > nTPM

Vertical /s are >.

Given

SAS

5. MT > TM Reflexive Property

Statements

Reasons

1. a.

1. Given

2. ∠1  ∠2, ∠3  ∠4

2. b.

3. c.

3. Transitive Property

4. d.

4. Reflexive Property

5. e.

5. SAS

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

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class

date

52. Use appropriate tools strategically. A plan for this proof appears below the diagram. Given: m∠QPS 5 m∠TPR PR  PS, ∠QRP  ∠TSP



Prove: PQR  PTS P

Q

R

T

S

Follow the plan to complete the flowchart proof. Plan: Subtract m∠RPS from m∠QPS and from m∠TPR to show that ∠QPR  ∠TPS. Then use the other given angles and sides to show that PQR  PTS by ASA. 1. a. Angle Addition Postulate

2. m/QPR 5 m/QPS 2 m/RPS m/TPS 5 m/TPR 2 m/RPS b. 3. m/QPS 5 m/TPR

5. d.

Given

Subst. and Trans. Prop.

4. m/RPS 5 m/RPS

6. PR > PS, /QRP > /TPS

c.

Given

7. nPQR > nPTS e.

Which could be the last two statements of the proof?

53. In a flowchart proof, suppose Statement 5 is that two triangles are congruent by SAS. Describe the statements that should precede Step 5.

A. ∠3  ∠4, ∠1  ∠2 B. PQ  SR, ∠1  ∠2 C. SRV  QPT, ∠3  ∠4 D. SRV  QPT, ∠1  ∠2

54. A student wrote a correct proof for this problem.

55. In a flowchart proof, suppose a statement in the middle of the proof is that two triangles are congruent. What will be the reason for the NEXT statement in the proof?

Given: SV  TQ, VR  TP, ∠3  ∠4 Prove: ∠1  ∠2 P V

2

Q

3

4 S

1

T

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

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class

date

Lesson 12-2

Lesson 13-1

56. How are the Statements and Reasons in a twocolumn proof related to the boxes and the lines below them, in a flowchart proof?

61. Use the figure to find the missing angle measures. 1 140°

80°

57. What do the arrows in a flowchart proof represent?

2 3

58. Use appropriate tools strategically. The first step in a flowchart proof is that AB intersects BC at point M, the midpoint of BC . If an arrow leads to the next step, which of the following could be the next step in the proof?

a. m∠1 1 m∠2 5 b. m∠1 5

A. BM 5 MC

c. m∠3 5

B. BM 1 MC 5 BC

d. m∠2 5

C. ∠BMA  ∠CMA

62. Use the figure to find each measure.

D. AM 5 MB

P (3x 1 4)° (2x 1 1)°

59. A student is writing a flowchart proof. A part of the flowchart proof is shown below.

R

S

nPQR > nWXY AAS

Q

(6x 2 7)°

a. m∠P 5

CPCTC

b. m∠Q 5

Which statement could appear in the empty box? A. ∠P  ∠W

c. m∠PRQ 5

B. PQ  WX

d. m∠PRS 5 63. Model with mathematics. In the diagram shown, lines m and n are parallel.

C. PR  WY D. ∠P and ∠R are complementary angles.

Find the measure of ∠ABC. A

60. Reason abstractly. It is given that FJ  HG and FG  HJ . Is ∠1  ∠2? Explain your answer in paragraph form. F

2

Page 41 of 53

B

? 30° C

a. Redraw the diagram in Item 63, and add a line through B that is parallel to m and n. Explain how to use your diagram to find m∠ABC.  b. Redraw the original diagram and extend CB through line m. Explain how to use your diagram to find m∠ABC.

H

© 2015 College Board. All rights reserved.

50°

n

G

1

J

m

13

SpringBoard Geometry, Unit 2 Practice

Name

class

Use ∠1, ∠2, or ∠3 to complete the statements.

64. a. Persevere in solving problems. In the diagram, M is the midpoint of AC. ABC is rotated 180° about M. A

date

∠DCP    ?  ∠CDP    ? 

D

∠P    ? 

1

c. List the three angles that have their vertex at point C.

M 2

3

B

C

d. How is your result in Part c related to the Triangle Sum Theorem?

Use ∠1, ∠2, or ∠3 to complete the following statements: ∠ADC    ?  ,

65. Which statement is NOT correct?

∠ACD    ?  ,

A. An exterior angle of a triangle can equal one of the interior angles of the triangle.

∠CAD    ?  . b. The diagram below extends the diagram in Part a by translating ABC along the directed segment BC. A

B. An exterior angle of a triangle can be acute, right, or obtuse. C. A triangle can have two exterior angles that are obtuse angles.

D

1

D. A triangle can have two exterior angles that are acute angles.

M 2

3

B

C

P

Lesson 13-2 66. Use appropriate tools strategically. Use the lines below the flowchart proof to complete the proof of the Isosceles Triangle Theorem, using the altitude from the vertex angle. Given: TM 5 TN MN TP

1. TP ' MN Given 3. nTMP and nTNP are right triangles. 2. /TPM and /TPN are right angles.

a.

Perpendicular lines form right angles.



Prove: ∠M  ∠N

4. b.

6. nTPM > nTPN

Reflexive Property

c.

M 5. TM 5 TN T

Given

P

N

7. d. e.

a. b. c. d. e.

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

Name

class

date

Lesson 14-1

67. Reason quantitatively. In an isosceles triangle, the bisector of the vertex angle forms an angle of 27° with each leg. What are the base angles of the triangle?

71. Consider the diagram shown. y 10

68. a. State the Converse of the Isosceles Triangle Theorem.

C (11, 5)

5 A (1, 1)

b. Use this information to prove the Converse of the Isosceles Triangle Theorem. Write your proof as a paragraph proof. X

W

B (7, 1) 0

5

10

15

x

a. Use the diagram to show the altitudes of ABC.

Y

1 2



Z

Given: m∠X 5 m∠Y

ZW bisects ∠XZY.

b. For ABC, do the altitudes intersect inside, on, or outside the triangle?

Prove: XZ  YZ 69. One angle of an isosceles triangle is 40°. Which of the following CANNOT describe the triangle? A. The triangle can be obtuse. B. The triangle can be right. C. Another angle of the triangle can be 70°.

c. Is your answer to Part b related to the shape of ABC? Explain.

D. Another angle of the triangle can be 100°. 70. In an isosceles triangle, the measure of a base angle is (2x 1 5)°. At the vertex, the measure of an exterior angle is (5x 2 3)°. a. Write and solve an equation to find x. Explain what properties you used to write the equation.

d. Find the point of intersection of the altitudes. b. Find the measures of the angles of the triangle.

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

Name

class

72. Attend to precision. Find an ordered pair for the orthocenter of the triangle with vertices M(26, 22), N(2, 6), and P(4, 0).

75. Express regularity in repeated reasoning. The algebraic process of finding the orthocenter of a triangle uses equations for the altitudes of the triangle. If you know the coordinates of the vertices of a triangle, explain the steps you take to find the equation of any altitude.

73. In the diagram of QRS and its three altitudes, m∠RQS 5 86° and m∠QSR 5 40°. Find the measure of each angle.

Lesson 14-2

Q V

date

T

76. Consider the diagram shown.

Y

y 5 P(0, 3)

R Q(6, 0)

W S

5

10

x

R(0, 23)

a. m∠QRT 5   ? 

25



a. Use the diagram to show the medians of PQR.

b. m∠VSR 5   ? 

c. m∠WQS 5   ?  b. For PQR, do the medians intersect inside, on, or outside the triangle? d. m∠VYQ 5   ? 

e. m∠QYS 5   ?  c. Is your answer to Part b related to the shape of PQR? Explain. 74. Suppose RPQ is either an acute triangle or an obtuse triangle. Which of the following can be true? A. The orthocenter can be on the triangle.

d. Find the point of intersection of the medians.

B. The orthocenter must be outside the triangle. C. The orthocenter must be inside the triangle. D. The orthocenter cannot be on the triangle. © 2015 College Board. All rights reserved.

Page 44 of 53

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SpringBoard Geometry, Unit 2 Practice

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class

Lesson 14-3

77. Express regularity in repeated reasoning. Find the centroid for the triangle whose vertices are F(23, 6), G(3, 6), and H(9, 26).

81. Consider the diagram shown. y 10 9 8 7 6 V (0, 5) 5 4 3 2 1 X (7, 0) W (0, 0) x 1 2 3 4 5 6 7 8 9 10

78. Attend to precision. In the diagram of CDE and its medians, CJ 5 3, HJ 5 4.5, and CD 5 12. Find the following lengths. C F H

J D G

E

date

a. Use the diagram to show the perpendicular bisectors of the sides of VWX.

a. JG 5   ?  b. HD 5   ?  b. For VWX, is the intersection of the perpendicular bisectors inside, on, or outside the triangle?

c. FD 5   ?  d. CG 5   ?  79. PQR is a scalene, obtuse triangle with obtuse ∠PRQ. Also, RT  PQ. Which statement is NOT true?

c. Is your answer to Part b related to the shape of VWX? Explain.

R

P

T

Q

d. Find the point of intersection of the perpendicular bisectors.

A. The centroid is inside PRQ. B. The centroid is inside PRT. C. The centroid is inside RTQ. e. Make another copy of VWX. Use your diagram to find the approximate coordinates for the point of intersection of the angle bisectors.

D. The centroid is on the median from Q to PR. 80. Describe the process of finding the centroid of a triangle.

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

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82. In ABC, the lines m, n, and p are the perpendicular bisectors of the sides and intersect at point T. AT 5 2x 2 5, CT 5 3(x 2 1), and BT 5 5y 1 1.

date

Lesson 15-1 86. In isosceles ABC, AC is the base. Points D, E, and F are the midpoints of AB , AC , and BC , respectively. The length of BD is 13 inches and m∠BFD 5 65°. Find each measure.

T

A

A

D

E

m C

B

p

B

n

F

C

a. AB 5   ? 

a. What is the value of x?

b. DE 5   ? 

b. What is the length of BT  ?

c. AD 5   ?  d. m∠EDF 5   ? 

c. What is the value of y?

87. Model with mathematics. Two segments PQ and TS intersect at point R, and TS is the perpendicular bisector of PQ.

d. A circle has center T and contains point B. What is the diameter of the circle?

a. What kind of figure is PTQS?

83. Which of the following sentences is true? A. The circumcenter of a right triangle is outside the triangle.

b. Name a pair of congruent triangles.

B. The incenter of an obtuse triangle is outside the triangle.

c. Prove that PTR  QTR.

C. The centroid of a right triangle is on the triangle.

d. Prove that PTQS is a kite.

D. The orthocenter of an obtuse triangle is outside the triangle.

88. In the diagram, ABC is isosceles with base BC and BMC is equilateral. Prove that AM is the perpendicular bisector of BC. 

84. Express regularity in repeated reasoning. Describe the process of finding the incenter of a triangle.

B

A

85. Express regularity in repeated reasoning. Describe the process of finding the circumcenter of a triangle. © 2015 College Board. All rights reserved.

Page 46 of 53

18

M

C

SpringBoard Geometry, Unit 2 Practice

Name

class

89. XYZ is an obtuse triangle, and RST is the triangle that is formed by joining the midpoints of the three sides of XYZ. Which of the following statements is NOT true?

date

92. Attend to precision. Figures ABCD and DCSR are both isosceles trapezoids and MN and XY are the medians of the trapezoids. Use the measurements in the diagram to find each measure. 33

A

A. The orthocenter of XYZ is inside RST.

B

62°

B. The centroid of XYZ is inside RST.

28

N

C. The perimeter of RST is half the perimeter of XYZ.

C

D

D. The angle measures of XYZ are the same as the angle measures of RST.

X

27 135°

R

90. Reason quantitatively. In kite MNPQ, MN is the perpendicular bisector of NQ. If m∠NMP 5 40° and m∠MQP 5 110°, find each measure.

M

Y S

a. DC b. RS c. CS

N

d. m∠BMN M

e. m∠CSR

P

Q

a. m∠MQN

93. Model with mathematics. Complete the steps to prove that if the base angles of a trapezoid are congruent, then the trapezoid is isosceles.

b. m∠MNP c. m∠NPM

Given: Trapezoid ABCD with AD  BC

d. m∠NPQ

∠A  ∠D

Lesson 15-2

A

91. In trapezoid TVME, TV  EM, m∠V 5 125°, m∠TME 5 25°, and m∠E 5 55°. Find each measure. T

E

Prove: AB  DC

B

V

C

M

D

a. m∠ETM

  Proof: AD  BC, so AB is not parallel to DC because a trapezoid has exactly  one pair of parallel sides. So AB and DC must intersect.

b. m∠VTM c. m∠VMT d. m∠ETV e. m∠VME © 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

Name

class A

date

95. Figure ABCD is a trapezoid with AB  DC . Also, AC bisects ∠DAB, m∠CAB 5 55°, and m∠EBC 5 140°. Find each measure.

B

B

A

1

55°

Y

E

140°

2 C

D

D

∠A  ∠1 and ∠D  ∠2 because they are a.  . Using ∠1 and ∠2 in BCY, BY  CY because if two angles in a triangle are congruent, then the sides opposite those angles are congruent.

a. m∠DAC b. m∠ADC c. m∠DCA d. m∠ACB

Using the same reasoning in APD, ∠A  ∠D so b.  . Using the Segment Addition Postulate, AP 5 AB 1 BY and DY 5 c.

C

e. m∠DCB

 .

Lesson 15-3

Using subtraction, AB 5 AP 2 BY and DC 5 DY 2 CY. So d. by substitution and AB  DC by e.  .

96. In the diagram, ABCD is a parallelogram. The diagonals of the parallelogram intersect at point T. A

B T

D

94. Which of the following statements is NOT true?

C

a. If AT 5 2x 2 1 and AC 5 3x 1 5, what is TC?

A. In an isosceles trapezoid, the diagonals are congruent. B. In an isosceles trapezoid, opposite angles are supplementary.

b. If m∠ABC 5 3y 1 5 and m∠ADC 5 5y 2 45, what is m∠BCD?

C. In an isosceles trapezoid, a diagonal forms two congruent triangles. c. If AB 5 3z 1 1, DC 5 z 1 7, and AD 5 2z, what is BC?

D. In an isosceles trapezoid, the diagonals form four small triangles, and two of the triangles are congruent.

d. If m∠ATD 5 6n 1 2, m∠BTC 5 5n 1 8, and m∠ADC 5 13n 1 2, what is m∠DAB?

© 2015 College Board. All rights reserved.

Page 48 of 53

20

SpringBoard Geometry, Unit 2 Practice

Name

class

99. Attend to precision. Points T(23, 1), A(2, 5), and P(2, 22) are three vertices of a parallelogram. Which of the following ordered pairs CANNOT be the vertex of the parallelogram?

97. In parallelogram TPQR, side TP is 8 units longer than side PQ. If the perimeter of the figure is 56, find the lengths of the sides.

y

98. Use appropriate tools strategically. Follow these steps for a different way to construct the median of a triangle.

date

5

A (2, 5)

T (23, 1)

Given: ABC 5

25

T ask: Construct the median from vertex B.

x

P (2, 22)

A 25

A. (23, 26) B

B. (21, 10)

C

C. (7, 2)

Step 1: Use a compass to find point D by drawing two arcs, first with center at A and then with center at C, so that AD 5 BC and CD 5 BA. A

D. (23, 8) 100. Complete the steps in this proof.

D

Given: Quadrilateral ABCD

         X, Y, Z, and W are midpoints. Prove: WXYZ is a(n) a. A B

C

D

A

X

W

Step 2: Draw BD. Use F to label the intersection of BD and AC.

B

D

F

Z

Y C

B

C

Statements

1. b.

a. What kind of figure is ADCB? Explain.

2. WX  DB , ZY  DB 3. c.

b. What do BD and AC represent in figure ADCB? c. Why is F the midpoint of AC?

4. WX  ZY  , WZ  XY

d. Suppose you want to construct the median to BC. What would be the first step?

5. d. © 2015 College Board. All rights reserved.

Page 49 of 53

21

Reasons

1. Given 2. Midsegment Theorem 3. Midsegment Theorem 4. If 2 lines are  to the same line, they are  to each other. 5. Def. of parallelogram SpringBoard Geometry, Unit 2 Practice

Name

class

Lesson 15-4

date

103. In this figure, MNPQ is a rhombus, QN 5 12, and m∠MNQ 5 37°. Find each measure.

1 01.  Model with mathematics. Use this diagram of rectangle RECT.

M

N

y R (0, 5)

T

4

212 28

4

24

8

E (12, 0) x 12

Q

P

24 28

a. TN

P 212 T (210, 219)

b. TP

216

c. MP

220 224

d. m∠TMN C (2, 224)

e. m∠QPN 104.  Persevere in solving problems. In rhombus TRUE, ER 5 24 and TU 5 18.

a. Find the coordinates of point P, the intersection of the diagonals.

T 1

2 3

b. Find the lengths of the sides of RECT.

R 4

B 8 E

c. Find the lengths of PR , PE , PC, and PT .

7 6

5 U

d. Find the slopes of RE and EC. What is the result when you find the product of the two slopes?

a. What is TR? b. List three angles that are congruent to ∠1. c. List three angles that are congruent to ∠TBR.

102. Which statement is NOT true? A. All squares are rhombuses.

d. Express the area of TRUE in terms of TU and ER.

B. All rhombuses are parallelograms. C. The diagonals bisect each other in squares, rhombuses, and rectangles.

e. What is the distance between TR and EU ? Explain how you found your answer.

D. The diagonals are perpendicular in squares, rhombuses, and rectangles.

© 2015 College Board. All rights reserved.

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SpringBoard Geometry, Unit 2 Practice

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1 05. In the diagram, ABCD is a square with diagonal BD, and BDEC is a rhombus with diagonals BE and DC. Each side of the square is 6 units. A

Lesson 16-1 1 06. Construct viable arguments. The vertices of a quadrilateral are A(1, 5), B(5, 23), C(2, 25), and D(22, 3). Use the diagonals to show that ABCD is a parallelogram.

B

F

D

date

107. In figure ABCD, m∠A 5 12x 2 7, m∠B 5 4x 1 11, m∠C 5 11x 1 9, and m∠D 5 4x 1 6.

C

A

B 12x 2 7

4x 1 11

E 4x 1 6

11x 1 9

D

C

a. What is BD? Explain your answer. a. Use the fact that the sum of the interior angles of a quadrilateral is 360° to find x. b. What are the measures of the angles of ABCD?

b. What is m∠ABD? Explain your answer.

c. Are any sides of ABCD parallel to other sides? Explain. d. Is ABCD a parallelogram? Explain.

c. What is m∠BCE? Explain your answer.

108. Use the diagram shown. R

M

d. What is the area of ABD? Explain your answer. C

P

a. What information is indicated in the diagram? e. What is the area of BCE? Explain your answer.

© 2015 College Board. All rights reserved.

Page 51 of 53

b. Is the given information enough to conclude that RMPC is a parallelogram? Explain.

23

SpringBoard Geometry, Unit 2 Practice

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114. Which of the following sets of information is NOT enough to prove that parallelogram GRAM is a rectangle?

1 09. Reason quantitatively. Find the values of m and n that make the quadrilateral a parallelogram. 3n –2

date

7m

G

R Y

10n M

1 10. Which of the following can be used to prove that quadrilateral ABCD is a parallelogram?

A

A. ∠GMR  ∠AMR, ∠GRM  ∠ARM B. GY 5 AY 5 RY 5 MY

A. AB  CD and ∠D  ∠C

C. ∠GRM and ∠ARM are complementary.

B. AC forms two obtuse triangles.

D. GR2 1 RA2 5 GA2

C. The midpoint of diagonal BD is equidistant from sides AB and DC.

1 15. Three vertices of a rectangle are A(0, 2), B(2, 0), and C(24, 26).

D. The midpoint of diagonal AC is equidistant from vertices B and D.

a. Find the coordinates for D, the other vertex.

Lesson 16-2 b. Verify that the diagonals of ABCD are congruent.

1 11. Construct viable arguments. The vertices of quadrilateral MNPQ are M(4, 8), N(7, 2), P(3, 0), and Q(0, 6). Use the diagonals to show that MNPQ is a rectangle.

c. Find the lengths of sides AB and BC. d. Is ABCD a square? Explain your answer.

112. Three vertices of a rectangle are (3, 6), (0, 21), and (22, 4). Find the coordinates of the fourth vertex.

Lesson 16-3 1 16. Construct viable arguments. The vertices of a quadrilateral are Q(23, 3), U(2, 10), A(7, 3), and D(2, 24). Use the diagonals to show that QUAD is a rhombus.

113. Make use of structure. Which of the following conditions is NOT enough to conclude that a figure is a rectangle?

117. Given: Triangles RTM, RTQ, RPQ, and RPM are congruent. Prove: MTQP is a rhombus.

A. It is a parallelogram and the diagonals bisect their angles. B. It is a parallelogram with a right angle.

M

T

C. It is a rhombus with congruent diagonals.

R

D. It is a parallelogram with opposite angles that are supplementary.

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P

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SpringBoard Geometry, Unit 2 Practice

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122. Figure FACE is a square. If the coordinates for two vertices are F(1, 6) and C(23, 24), what are the ordered pairs for the other two vertices?

1 18. Which of the following is NOT sufficient to conclude that a figure is a rhombus? A. A figure is a parallelogram with two consecutive congruent sides.

123. In the diagram, ABCD is a rectangle. DF bisects ∠ADC and FG || AD. Complete this proof that AFGD is a square.

B. A figure is a parallelogram with perpendicular diagonals. C. A figure is a parallelogram and one diagonal forms two congruent triangles.

A

F

B

D

G

C

D. A figure is a parallelogram and one diagonal bisects its angles. 119. Find the value of x that makes the parallelogram a rhombus.

It is given that DF bisects a right angle, so m∠ADF 5 a. . We know that ∠A is a right angle because ABCD is a rectangle, so AFD is a right triangle. In a right triangle the acute angles are b. , so m∠AFD 5 45°. That means AFD is an isosceles triangle, and AF 5 AD. We know that AFDG is a c. because AF || DG (opposite sides of a rectangle are parallel) and AD || FG (Given). We have shown that AFGD is a parallelogram with a right angle and two d. , so e. .

10 – 20x

1 20. Construct viable arguments. One student proved that quadrilateral QUAD is a rectangle. Another student proved that QUAD is a rhombus. What else can you prove about QUAD? Explain.

Lesson 16-4

124. Model with mathematics. One vertex of a square is (1, 3) and the common midpoint of the diagonals is (7, 3). Find the other three vertices.

1 21. Make use of structure. Which statement is NOT sufficient to prove that a figure is a square? A. The figure is a rectangle with perpendicular diagonals.

125. A square has vertices A(5, 5), B(5, 25), C(25, 25), and D(25, 5).

B. The figure is a parallelogram with perpendicular diagonals.

a. What is the length of the diagonals of the square?

C. The figure is a rhombus with one right angle. D. The figure is a rhombus with congruent diagonals.

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Page 53 of 53

b. Suppose the square is rotated 90° clockwise around (0, 0). What are the coordinates for the vertices of the image?

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SpringBoard Geometry, Unit 2 Practice