GE OME TR Y – CH APTER 6 STUDY GUIDE A. Vocabulary Term Parallelogram (pg 306)

Definition & symbols

Picture and/or example

Rhombus (pg 306)

Rectangle (pg 306)

Square (pg 306)

Kite (pg 306)

Trapezoid (pg 306)

Geometry – Ch. 6 Note taking guide

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Isosceles Trapezoid (pg 306)

Consecutive angles (pg 312)

Base angles of a trapezoid (pg 336)

Midsegment of a trapezoid (pg 348)

B.

REVIEW FORMULAS FROM CH. 1

SLOPE OF A LINE:

DISTANCE BETWEEN TWO POINTS:

MIDPOINT:

Geometry – Ch. 6 Note taking guide

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C.

Coordinate Proofs: 1) Determine the most precise name for quadrilateral ABCD with vertices 6 A(-3, 3), B(2, 4), C(3, -1), and D(-2, -2). Prove using appropriate formulas. 4

2

-5

5

10

-2

-4

-6

2) Prove that ABCD form a parallelogram. 6 A(-1, 1), B(-2, -1), C(2, -3), and D(3, -1). 4

2

-5

5

10

-2

-4

-6

Geometry – Ch. 6 Note taking guide

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D.

Rela tionships Among Special Q ua drila tera ls (S ee chart on pg 307 )

Write the name of the quadrilateral and mark the relevant features (congruent sides, parallel sides, right angles, congruent angles)

No pairs of // sides

2 pairs of // sides 1 pair of // sides

E. Draw a Venn diagram to show the relationships among the special quadrilaterals.

Geometry – Ch. 6 Note taking guide

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F. PRO PERTIES OF SPECIAL QUAD RILATERALS : Check each box if the property is ALWAYS TRUE for the given special quadrilaterals. Property

Parallelogram

Rectangle

Rhombus

Square

Kite

Trapezoid

Only one pair of opposite sides are //. Both pairs of opposite sides are //. Only one pair of opposite sides are ". Both pairs of opposite sides are ". All sides are " . Only 1 pair of opposite ! " s are " . Both pairs of opposite " s are " All ! ". ! " s are

!

All ! of ! 4 pairs " s are consecutive ! supplementary. Only 2 pairs of " s are consecutive ! supplementary. Diagonals bisect each ! other. Diagonals are congruent. Diagonals are perpendicular. Each diagonal bisect a pair of opposite angles. Base angles are congruent.

Geometry – Ch. 6 Note taking guide

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Isosceles Trapezoid

G. Ways to prove that a q uadrila teral is a pa rallelogram. 1. Show tha t both pa irs of oppos ite sides are ___________. (Definition of a pa rallelogram) 2. Show tha t both pa irs of oppos ite sides are ____________. (Theorem 6 -5) 3. Show tha t both pa irs of oppos ite angles are ___________. (Theorem 6 -6) 4. Show tha t dia gonals ___________________________. (Theorem 6 -7) 5. Show tha t one pair of opposite sides is both _____ and ___. (Theorem 6 -8)

Examples: 1. Given: " ADB " " DBC, " ABD " " BDC

A

B

Prove: ABCD is a parallelogram.

!

! !

!

! !

Geometry – Ch. 6 Note taking guide

D

C

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2. Use only a compass and a straightedge, construct a parallelogram. Explain why your construction is valid.

3. Given three vertices of a parallelogram A(1, -2), B(2, 2), C(5, 0), determine all possible sets of coordinates of a fourth point that would form a parallelogram with the given vertices. 8

6

4

2

-5

5

10

15

-2

-4

-6

-8

Geometry – Ch. 6 Note taking guide

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H. Ways to prove that a pa rallelogram is a rhom bus. 1. Show tha t one ________________ of the parallelogram bisects two ___________. (Theorem 6 -12) 2. Show tha t the d iagona ls are __ __________________. (Theorem 6 -13 ) 3. Show tha t all four sid es are __ ______________. (Def. of rhom bus ) Ex: 1) Proof of Theorem 6-13. Given: Parallelogram ABCD; AC " BD at E. Prove: ABCD is a rhombus. !

!!

A

B E

D

C

2) Use only a compass and a straightedge, construct a rhombus. Explain why your construction is valid.

Geometry – Ch. 6 Note taking guide

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I. Way to prove that a pa rallelogram is a rectangle. 1. Show tha t the d iagona ls of the para llelogram are ___________ (Theorem 6 -14 ) 2. Show tha t all four a ngles a re _ _____________. (Def. of a rectangle) Examples: 1. Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. Suppose you are on a building team and helping to lay out a rectangular patio. Explain how to use properties of diagonals to locate the four corners of a rectangle.

2. Use only a compass and a straightedge, construct a rectangle. Explain why your construction is valid.

Geometry – Ch. 6 Note taking guide

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I. Trapezoid M idsegment T heorem (Theorem 6 -18, pg 348):

1. The midsegment of a trapezoid is ____________ to the _______.

R

A

M T

2. The length of the midsegment of a trapezoid is _________ the sum of the lengths of the two bases.

MN // ________, MN // ________, ! !

and MN =

Examples: 1) Let RA = 24, TP = 30, MN = ?

2) Let RA = 5, MN = 9, TP = ?

3) Let MN = 12, RA = x + 4, TP = 2x.

Geometry – Ch. 6 Note taking guide

N

Find the values of x, RA, and TP.

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P

J. Coordinate Geometry: When working with a figure in the coordinate plane, it generally is good practice to place a vertex at the origin and one side on an axis. You can also use multiples of 2 to avoid fractions when finding midpoints. 1. Find the most convenient place to draw a rectangle in the coordinate plane with side lengths 2 and 5. 8

6

4

2

-5

5

10

15

-2

-4

-6

2. Give coordinates for unlabeled points without using any new variables. Rectangle Square -8

B(0, b)

D(a, 0)

Parallelogram

D(a, 0)

Isosceles Trapezoid B (b, h)

B (b, h)

D(a, 0)

D(a, 0)

Geometry – Ch. 6 Note taking guide

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3. a. Draw a square whose diagonals of length 2b lie on the x- and y-axes.

b. Give the coordinates of the vertices of the square.

c. Compute the length of a side of the square.

d. Find the slopes of two adjacent sides of the square.

e. Do the slopes show that the sides are perpendicular? Explain.

4. Given an ordinary quadrilateral shown to the right, find the coordinates of the midpoint of each side. Prove that the segments joining the midpoints form a parallelogram. B (b, h)

C (c, e)

A(0, 0)

D(d, 0)

Geometry – Ch. 6 Note taking guide

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