Name

5-1

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Parallel Lines (Pages 188–192)

MG3.6 S T A N D A R D S

Parallel lines are lines in a plane that will never intersect. If line p is parallel to line q, then write p || q. A line that intersects two or more other lines is called a transversal. Congruent angles formed by parallel lines and a transversal have special names. Angles formed by parallel lines and a transversal also have certain special relationships.

Congruent Angles With Parallel Lines

If a pair of parallel lines is intersected by a transversal, these pairs of angles are congruent. alternate interior angles: 4  6, 3  5 alternate exterior angles: 1  7, 2  8 corresponding angles: 1  5, 2  6, 3  7, 4  8

Vertical Angles and Supplementary Angles

Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent. (For example, 1  3 above.) Supplementary angles are two angles whose measures have a sum of 180°. (For example, 1 is supplementary to 2 above.)

12 43 56 87

EXAMPLES Use the figure above for these examples. A Find m1 if m5  60°. B Find m6 if m7  75°. 1 and 5 are corresponding angles. Corresponding angles are congruent. Since m5  60°, m1  60°.

6 and 7 are supplementary angles. So, m6  m7  180°. m6  75°  180° Substitute 75° for m7. m6  105° Subtract 75° from each side.

Try These Together Use the figure at the right for Exercises 1–4. The two lines are parallel. 1. Find m2 if m8  110°. 2. Find m4 if m6  122°.

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HINT: Identify the type of angles first.

PRACTICE 3. Find m3 if m2  98°. 4. Find m7 if m3  45°. 5. p and q are congruent. Solve for x if mp  (2x 5)° and mq  75°. 6. Hobbies Alexis is making a quilt with a pattern that uses parallel lines and transversals. The pattern is shown at the right. If m1 is 68°, what should m2 be? B

4.

C C

A

B

5.

C B

6.

A

7. 8.

1

B A

7. Standardized Test Practice a and b are alternate exterior angles of parallel lines. If ma is 138°, what is mb? A 180° B 138° C 42° D 48° Answers: 1. 110° 2. 122° 3. 82° 4. 45° 5. 40 6. 68° 7. B

3.

2

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Date

Classifying Triangles (Pages 196–199)

MR2.5, S T MR2.8, A MR3.1 N D A R D S

A polygon is a simple closed figure in a plane formed by three or more line segments. A polygon formed by three line segments is a triangle. Triangles can be classified by their angles and their sides. Triangles Classified by Angle

• Acute triangles have three acute angles. • Right triangles have one right angle. • Obtuse triangles have one obtuse angle.

Triangles Classified by Sides

• Scalene triangles have no two sides that are congruent. • Isosceles triangles have at least two sides congruent. • Equilateral triangles have three sides congruent.

EXAMPLES Classify each triangle. A ABC has one angle that measures 136°, and no sides that are the same length.

B EFG has one angle that measures 90°. Since it has one right angle, you know that EFG is a right triangle. You cannot determine whether it is scalene or isosceles without knowing the lengths of the sides of the triangle.

Because the angle is greater than 90°, this is an obtuse triangle. Because none of the sides are the same length, it is also a scalene triangle. ABC is an obtuse, scalene triangle.

PRACTICE Classify each triangle by its angles and by its sides. 1. 2. 3. 6.2 in. 110 45 5 cm

8 in.

7.1 cm

30

45 5 cm

40 11.7 in.

60 5m

5m

60

60 5m

4. Gift Wrapping Classify the triangles used in the pattern on the wrapping paper shown at the right.

B

C C B

C B

6.

A

7. 8.

B A

5. Standardized Test Practice How would you classify a triangle that has one right angle and two congruent sides? A right, isosceles B acute, scalene C obtuse, isosceles D right, equilateral 5. A

A

5.

4. acute, equilateral

4.

Answers: 1. right, isosceles 2. obtuse, scalene 3. acute, equilateral

3.

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

Name

5-3

Date MR1.2 S T A N D A R D S

Classifying Quadrilaterals (Pages 201–204)

A quadrilateral is a polygon with four sides and four angles. The sum of the measures of the angles of a quadrilateral is 360°.

Types of Quadrilaterals

• A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel. • A rectangle is a parallelogram with four right angles. • A rhombus is a parallelogram with all sides congruent. • A square is a parallelogram with all sides congruent and four right angles. • A trapezoid is a quadrilateral with exactly one pair of opposite sides that are parallel.

parallelgram

rectangle

rhombus

square

trapezoid

EXAMPLES Classify each quadrilateral. A Quadrilateral ABCD has only one pair of parallel sides.

B Quadrilateral HIJK has all sides congruent, with four right angles.

The only quadrilateral with only one pair of sides parallel is a trapezoid. Quadrilateral ABCD is a trapezoid.

A quadrilateral with four sides congruent and four right angles is a square. A square is also a parallelogram, a rectangle, and a rhombus. Therefore, quadrilateral HIJK is a parallelogram, a rectangle, a rhombus, and a square.

PRACTICE Let Q  quadrilateral, P  parallelogram, R  rectangle, S  square, RH  rhombus, and T  trapezoid. Write all of the letters that describe each figure. 1. 2. 3. 4.

5. Architecture An architect is designing a rhombus-shaped window for a new house. A sketch of the window is shown at the right. Find the value of x so the architect will know the measures of all four angles. B

C C

A

B

5.

C B

6.

A

7. 8.

45 x

B A

6. Standardized Test Practice What is the best way to classify a quadrilateral that is also a parallelogram with 4 right angles? A trapezoid B rhombus C square

D rectangle

2. Q, P, RH 3. Q, T 4. Q, P, R 5. 135 6. D

4.

45

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Answers: 1. Q

3.

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

Name

5-4

Date

Symmetry (Pages 206–209)

Many geometric and other figures have one or more of the types of symmetry described below.

Types of Symmetry

• A figure has line symmetry if it can be folded so that one half of the figure coincides with the other half. • In figures with line symmetry, one half of the figure is a reflection of the other half. The line that divides the two halves is the line of reflection, or line of symmetry. Some figures have more than one line of symmetry. • Often, an object will have symmetry if you spin it. This kind of movement is called a rotation. • If you can rotate an object less than 360° and it still looks like the original, the figure has rotational symmetry.

EXAMPLES Identify the type of symmetry. A A drawing that looks the same if you turn the paper so that the bottom is now at the top.

B The brand for Lee’s family cattle ranch looks like it could be folded in half and the two sides would match.

Since the drawing looks the same if you turn it 180°, the drawing has rotational symmetry.

Figures that can be folded in half to make matching sides have line symmetry.

PRACTICE Determine whether each figure has line symmetry. If so, draw the lines of reflection. 1. 2. 3. 4.

5. Which of the figures in Exercises 1–4 have rotational symmetry? 6. Sports Sailing is a popular sport in areas near lakes and oceans. Draw a line of symmetry on the sail of the boat at the right.

B

C C B

C B

6.

A

7. 8.

B A

7. Standardized Test Practice Which of the following figures shows correct lines of symmetry? A B C

4. See Answer Key. 5. the star in Exercise 1

A

5.

3. no lines of symmetry

4.

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D

Answers: 1. See Answer Key. 2. no lines of symmetry 6. See Answer Key. 7. B

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

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Congruent Triangles (Pages 210–212)

MG3.4 S T A N D A R D S

Triangles that have the same size and shape are called congruent triangles. When two triangles are congruent, the parts that “match” are called corresponding parts. Two triangles are congruent when all of their corresponding parts are congruent. However, you do not need to know that all six parts are congruent to know that two triangles are congruent. The table below explains three such shortcuts. Two triangles are congruent if the following corresponding parts of the triangles are congruent. • three sides (side-side-side, or SSS) • two angles and the included side (angle-side-angle, or ASA) • two sides and the included angle (side-angle-side, or SAS)

Congruent Triangles

SSS

ASA

SAS

PRACTICE Determine whether each pair of triangles is congruent. If so, write a congruence statement and tell why the triangles are congruent. 1.

Z

2.

Y

K

3.

L

A

N X

B

S T

Q

U

J M

C

P

R

V

Find the value of x in each pair of congruent triangles. E G 4. 5. (5x – 5) m 10 m D

3x

F

J

45

H

6. Flags International code flags are used at sea to signal distress or give warnings. The flag that corresponds to the letter O, shown at the right, warns there is a person overboard. How many congruent triangles are on the flag? B

4.

C C

A

B

5.

C B

6.

A

7. 8.

B A

7. Standardized Test Practice Which of the following sets of corresponding parts will not tell you if two triangles are congruent? A SSA B SAS C ASA

D SSS

Answers: 1. yes; ABC  XYZ; SAS 2. no 3. yes; QRS  VUT; SSS 4. 15 5. 3 6. 2 7. A

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

Name

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Date MG3.4 S T A N D A R D S

Similar Triangles (Pages 215–218)

Triangles that have the same shape but differ in size are called similar triangles. Two triangles are similar if their corresponding angles are congruent. So, if ABC is similar to DEF, you know that A  D, B  E, and C  F. Use the symbol  to indicate similar triangles. For example, ABC  DEF.

EXAMPLES Use the figure below. B

Q C R

A

P

A List the congruent angles in ABC and PQR.

B Is ABC similar to PQR?

Use the arcs marked on the angles as your guide to which angles are congruent. According to the arcs, A  P, B  Q, and C  R.

Since the corresponding angles of ABC and PQR are congruent, then ABC  PQR.

PRACTICE Tell whether each pair of triangles is congruent, similar, or neither. 1. 2. 3.

95

80

40

Find the value of x in each pair of similar triangles. 4. 5. 25 125

68 87

(5x + 25)

6. Design Janey is a wallpaper designer. In her new wallpaper design, she wants to include similar triangles. Are the two triangles at the right similar? Explain. B

A

7. 8.

110 30

40 30 110 40

C B

C B

6.

25

C

A

5.

B A

7. Standardized Test Practice What is the value of x in the pair of similar triangles at the right? A 8 B 9 C 5 D 6

4. 20 5. 4 6. Yes, corresponding angles are congruent. 7. C

4.

87 17x

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9x

45

Answers: 1. similar 2. congruent 3. neither

3.

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

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Date

Transformations and M.C. Escher (Pages 220–223)

A tessellation is a tiling made up of copies of the same shape or shapes that fit together without gaps and without overlapping. Unique, interlocking, puzzle-like patterns can be made using transformations to change tessellations. One common transformation is a slide, or translation, of a figure. Some tessellations can be modified by using changes with a rotation.

EXAMPLE The square at the right has had the left side changed. To make sure the pieces, or pattern units, will tessellate, slide or translate the change to the opposite side and copy it. Now change all of the squares in a tessellation the same way. The tessellation takes on qualities like a painting by M. C. Escher when you add different colors or designs.

PRACTICE Make an Escher-like drawing for each pattern described. Use a tessellation of two rows of three squares as the base. 1. 2. 3.

4. Remodeling The Louden family is remodeling the entrance to their home. They would like to put in a new tile floor pattern using the transformation shown at the right. Draw the tessellation.

B

4.

C C

A

B

5.

C B

6.

A

7. 8.

B A

5. Standardized Test Practice Name the polygon and the transformation that were used to produce the tessellation at the right. A square, rotation B square, translation C rectangle, translation D triangle, rotation

Answers: 1–4. See Answer Key. 5. A

3.

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3

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Date

Chapter 5 Review

MG3.4 S T A N D A R D S

Triangle Treasure Hunt Search this figure to find the triangles described. A

D

C

G

E

B

F

1. Find two triangles that the markings indicate are congruent by SSS and write the congruence statement.

2. Find a different pair of triangles that are congruent by SAS and write the congruence statement.

3. Find two triangles that are similar but not congruent and write the similarity statement.

4. Draw and mark a figure that has two triangles that are congruent by ASA. Then ask your parent to find and name the two congruent triangles.

Answers are located on page 121. © Glencoe/McGraw-Hill

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California Parent and Student Study Guide Mathematics: Applications and Connections, Course 3