CHAPTER 6
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Table of Contents
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Topic:
Description:
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Geometry
Section 6.1 Notes: Angles of Polygons
Date:
Today’s Objectives:
Question, Topics and Vocabulary
1. Students will be able to find and use the sum of the measures of the interior
angles of a polygon.
2. Students will be able to find and use the sum of the measures of the exterior
angles of a polygon.
Problems, Definitions and Work
Diagonal of a Polygon
The vertices of polygon PQRST that are not consecutive with vertex P are vertices R and S.
Therefore, polygon PQRST has two diagonals from vertex P, PR and PS . Notice that the
diagonals from vertex P separate the polygon into three triangles.
Sum of the Angles of a
Polygon
The _______ of the angle measures of a polygon is the __________ of the angle measures of
the triangles formed by drawing all the possible diagonals from one vertex.
Polygon
Number of
Sides
Number of
Triangles
Sum of
Interior Angle
Measures
Triangle
Quadrilateral
Pentagon
Hexagon
n-gon
Polygon Interior Angle
Sum
Example 1: Find the sum of the measures of the interior angles of a convex nonagon.
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You can use the Polygon Interior
Angles Sum Theorem to find the
sum of the interior angles of a
polygon and to find missing
measures in polygons.
Example 2: Find the measures of each interior angle of parallelogram RSTU.
Recall from Lesson 1.6 that in a regular polygon, all of the interior angles are congruent. You
can use this fact and the Polygon Interior Angle Sum Theorem to find the interior angle
measure of any regular polygon.
Example 3: A mall is designed so that five walkways meet at a food court that is in the shape of
a regular pentagon. Find the measure of one of the interior angles of the pentagon.
Given the interior angle measure of a regular polygon, you can also use the Polygon Interior
Angles Sum Theorem to find a polygon’s number of sides.
Example 4: a) The measure of an interior angle of a regular polygon is 150. Find the number
of sides in the polygon.
b) The measure of an interior angle of a regular polygon is 144. Find the number of sides in the
polygon.
Using the polygons below, does a relationship exist between the number of sides and sum of its
exterior angles?
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Polygon Exterior
Angles Sum
Example 4: Find the value of x in the diagram.
Example 5: Find the measure of each exterior angle of a regular decagon.
Interior Angle Sum vs. Exterior
Angle Sum Post-It Activity
Summary:
1.
Find the sum of the measures of the interior angles of a convex octagon.
2. Find the value of x.
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3. The measure of an interior angle of a regular polygon is given. Find the number of
sides.
a. 150
b. 160
c. 175
4. Find the sum of the measures of the interior angles of each convex polygon.
a. octagon
b. 12-gon
c. 35-gon
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Geometry
Section 6.1 Worksheet
Name: _____________________________________
For numbers 1 – 3, find the sum of the measures of the interior angles of each convex polygon.
1. 11-gon
2. 14-gon
3. 17-gon
For numbers 4 – 6, the measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
4. 144
5. 156
6. 160
For numbers 7 & 8, find the measure of each interior angle.
7.
8.
For numbers 9 – 14, find the measures of an exterior angle and an interior angle given the number of sides of each regular polygon.
Round to the nearest tenth, if necessary.
9. 16
10. 24
11. 30
12. 14
13. 22
14. 40
Find the value of x.
15.
16.
9
17.
18.
10
Geometry
Section 6.2 Notes: Parallelograms
Date:
1. Students will recognize and apply properties of the sides and angles of
parallelograms.
2. Students will recognize and apply properties of the diagonals of
parallelograms.
Today’s Objectives:
Question, Topics and Vocabulary
Problems, Definitions and Work
Parallelogram
*To name a parallelogram, use the symbol
definition.
. In
ABCD, BC || AD and AB || DC by
Other Properties of Parallelograms
Property:
Property 6.3
Ex:
Property:
Property 6.4
Ex:
Property:
Property 6.5
Ex:
Property:
Property 6.6
Ex:
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Example 1: In parallelogram ABCD, suppose mB = 32, CD = 80 inches, and BC = 15 inches.
a) Find AD.
b) Find mC.
c) Find mD.
Example 2: ABCD is a parallelogram.
a) Find AB.
b) 𝑚∠𝐶
c) 𝑚∠𝐷
Diagonals of a Parallelogram
Theorem 6.7
Ex:
Theorem 6.8
Ex:
Example 3: If WXYZ is a parallelogram…
a) Find the value of r.
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b) Find the value of s.
c) Find the value of t.
Example 3:
a) What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with
vertices M(–3, 0), N(–1, 3), P(5, 4) and R(3, 1)?
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b) What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with
vertices L(0, –3), M(–2, 1), N(1, 5) and O(3, 1)?
1) Find the values of x and y given that GFED is a parallelogram.
Summary:
2) Find the value of each variable.
3. Find the coordinates of the intersection of the diagonals of
H(–1, 4), J(3, 3), K(3, –2), L(–1, –1)
HJKL with the given vertices.
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Geometry
Section 6.2 Worksheet
Name: _____________________________________
For numbers 1 – 4, find the value of each variable.
1.
2.
3.
4.
For numbers 5 – 8, use parallelogram RSTU to find the measure or value.
5. mRST
6. mSTU
7. mTUR
8. b
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For numbers 9 & 10, find the coordinates of the intersection of the diagonals of parallelogram PRYZ with the given vertices.
9. P(2, 5), R(3, 3), Y(–2, –3), Z(–3, –1)
10. P(2, 3), R(1, –2), Y(–5, –7), Z(–4, –2)
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Geometry
Section 6.3 Notes: Tests for Parallelograms
Date:
1. Students will recognize the conditions that ensure a quadrilateral is a
parallelogram.
2. Students will prove that a set of points forms a parallelogram in the coordinate
plane.
Today’s Objectives:
Question, Topics and Vocabulary
Problems, Definitions and Work
We already learned that if a quadrilateral has opposite sides parallel, it is a parallelogram by definition.
However there are more tests to determine if a quadrilateral is a parallelogram.
Conditions for Parallelograms
Condition 6.9
Ex:
Condition 6.10
Ex:
Condition 6.11
Ex:
Condition 6.12
Ex:
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Example 1:
a) Determine whether the quadrilateral is a parallelogram. Justify your answer.
b) Which theorem would prove the quadrilateral is a parallelogram?
Example 2: Scissor lifts, like the platform lift shown, are commonly applied to
tools intended to lift heavy items. In the diagram, A  C and B  D. Explain
why the consecutive angles will always be supplementary, regardless of the height
of the platform.
Example 3:
a) Find x and y so that the quadrilateral is a parallelogram.
b) Find m so that the quadrilateral is a parallelogram.
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How to Prove that a Quadrilateral is a Parallelogram:
Definition:
Theorem 6.9:
Concept Summary
Theorem 6.10:
Theorem 6.11:
Theorem 6.12:
Example 4: Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1).
Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope
Formula.
Coordinate Geometry
We can use the Distance, Slope,
and Midpoint Formulas to
determine whether a quadrilateral
in the coordinate plane is a
parallelogram.
Example 5: Graph quadrilateral EFGH with vertices E(-2, 2), F(2, 0), G(1, 5), and H(-3, -2).
Determine whether the quadrilateral is a parallelogram.
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Summary
Determine whether each quadrilateral is a parallelogram. Justify your answer.
a)
b)
Find x and y so that the quadrilateral is a parallelogram.
a)
b)
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Geometry
Section 6.3 Worksheet
Name: _____________________________________
For numbers 1 – 4, determine whether each quadrilateral is a parallelogram. Justify your answer.
1.
2.
3.
4
For numbers 5 & 6, graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your
answer with the method indicated.
5. P(–5, 1), S(–2, 2), F(–1, –3), T(2, –2); Slope Formula
6. R(–2, 5), O(1, 3), M(–3, –4), Y(–6, –2); Distance and Slope Formulas
For numbers 7 – 10, solve for x and y so that the quadrilateral is a parallelogram.
7.
8.
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9.
10.
11. Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets are located at the vertices of a
parallelogram, what are the three possible locations of the missing jet?
12. When a coordinate plane is placed over the Harrisville town map, the four street lamps in the center are located as shown. Do the
four lamps form the vertices of a parallelogram? Explain.
13. Aaron is making a wooden picture frame in the shape of a parallelogram. He has two pieces of wood that are 3 feet long and two
that are 4 feet long.
a) If he connects the pieces of wood at their ends to each other, in what order must he connect them to make a parallelogram?
b) How many different parallelograms could he make with these four lengths of wood?
c)Explain something Aaron might do to specify precisely the shape of the parallelogram.
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Geometry
Section 6.4 Notes: Rectangles
Date:
Today’s Objectives:
Question, Topics and Vocabulary
Rectangle
1. Students will be able to recognize and apply properties of rectangles.
2. Students will be able to determine whether parallelograms are rectangles.
Problems, Definitions and Work
By definition, a rectangle has the following properties:





All four angles are right angles.
Opposite sides are parallel and congruent.
Opposite angles are congruent
Consecutive angles are supplementary.
Diagonals bisect each other.
Diagonals of a
Rectangle
(Theorem 6.13)
Example 1: A rectangular garden gate is reinforced with diagonal braces to prevent it from
sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.
Questions:
Example 2: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.
Questions:
Example 3: Quadrilateral RSTU is a rectangle. If mRTU = (8x + 4) and mSUR = (3x – 2),
solve for x.
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Example 4: Quadrilateral EFGH is a rectangle. If mFGE = (6x – 5) and mHFE = (4x –5),
solve for x.
Diagonals of a
Rectangle
(Theorem 6.14)
*This is the converse of
theorem 6.13
Example 5: Some artists stretch their own canvas over wooden frames.
This allows them to customize the size of canvas. In order to ensure that the
frame is rectangular before stretching the canvas, an artist measures the
sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches,
CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches,
explain how an artist can be sure that the frame is rectangular.
Example 6: Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3).
Determine whether JKLM is a rectangle using the Distance Formula.
Coordinate Geometry
You can also use the properties of
rectangles to prove that a
quadrilateral positioned on a
coordinate plane is a rectangle
given the coordinates of the
vertices.
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1. In rectangle ABCD, AC = 24 and DE = 2x – 8. Find the value of x.
A. 10
B. 16
C. 7
D. 13
Summary:
2. In rectangle RSTU, which pair of segments is not necessarily congruent?
̅̅̅̅, 𝑇𝑈
̅̅̅̅
A. 𝑅𝑆
̅̅̅ , 𝑇𝑈
̅̅̅̅
B. ̅𝑆𝑇
̅̅̅̅ , ̅̅̅̅
C. 𝑆𝑉
𝑉𝑈
̅̅̅̅, ̅̅̅̅
D. 𝑅𝑇
𝑆𝑈
3. Quadrilateral ABCD is a rectangle.
a. If AE = 36 and CE = 2x – 4, find x.
b. If BE = 6y + 2 and CE = 4y + 6, find y.
4. Graph the quadrilateral with the given vertices. Determine whether the figure is a rectangle.
Justify your answer using the indicated formula.
A(–3, 1), B(–3, 3), C(3, 3), D(3, 1); Distance Formula
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Geometry
Section 6.4 Worksheet
Name: _____________________________________
For numbers 1 – 6, quadrilateral RSTU is a rectangle.
1. If UZ = x + 21 and ZS = 3x – 15, find US.
2. If RZ = 3x + 8 and ZS = 6x – 28, find UZ.
3. If RT = 5x + 8 and RZ = 4x + 1, find ZT.
4. If mSUT = (3x + 6)° and mRUS = (5x – 4)°, find mSUT.
5. If mSRT = (x + 9)° and mUTR = (2x – 44)°, find mUTR.
6. If m∠RSU = (x + 41)° and m∠TUS = (3x + 9)°, find mRSU.
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For numbers 7 – 12, quadrilateral GHJK is a rectangle. Find each measure if m1 = 37°.
7. m2
8. m3
9. m4
10. m5
11. m6
12. m7
For numbers 13 – 15, graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer
using the indicated formula.
13. B(–4, 3), G(–2, 4), H(1, –2), L(–1, –3); Slope Formula
14. N(–4, 5), O(6, 0), P(3, –6), Q(–7, –1); Distance Formula
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Geometry
Section 6.5 Notes: Rhombi and Squares
Date:
Today’s Objectives:
Question, Topics and Vocabulary
1. Students will be able to recognize and apply the properties of rhombi and
squares.
2. Students will be able to determine whether quadrilaterals are rectangles,
rhombi, or squares.
Problems, Definitions and Work
Rhombus
Diagonals of a Rhombus
Theorem 6.15
Theorem 6.16
Example 1: The diagonals of rhombus WXYZ intersect at V. If mWZX = 39.5, find mZYX.
Questions:
Example 2: The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3,
solve for x.
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Example 3: ABCD is a rhombus. Find 𝑚∠𝐶𝐵𝐷 if 𝑚∠𝐴𝐵𝐶 = 126.
Square
Parallelograms (Opp. Sides are Parallel)
Relationship between
Parallelograms,
Rhombi, Rectangles,
and Squares
Properties to Recognize

Recall that a parallelogram with four right angles is a rectangle, and a parallelogram
with four congruent sides is a rhombus. Therefore, a parallelogram that is both a
rectangle and a rhombus is also a square.

All of the properties of parallelograms, rectangles, and rhombi apply to squares. For
example, the diagonals of a square bisect each other (parallelogram), are congruent
(rectangle), and are perpendicular (rhombus).
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Conditions for Rhombi and Squares
Theorem 6.17
Theorem 6.18
Theorem 6.19
Theorem 6.20
Questions:
Example 4: Write a two-column proof
Given: LMNP is a parallelogram
1  2
2  6
Prove: LMNP is a rhombus.
Statements
Reason
1.
1.Given
2. 𝐿𝑀 ∥ 𝑃𝑁
2.
3.
3. Alternate Interior Angles
4. 1  2
4.
5.
5. Given
6. 5  6
6.
7.
7.
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Example 5: Hector is measuring the boundary of a new garden. He wants the garden to be
square. He has set each of the corner stakes 6 feet apart. What does Hector need to know to
make sure that the garden is square?
Example 6: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for
the given vertices: A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.
Coordinate Geometry
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Summary:
1. Determine whether parallelogram EFGH is a rhombus, a rectangle, or a square for E(0,
–2), F(–3, 0), G(–1, 3), and H(2, 1). List all that apply.
2. Quadrilateral DKLM is a rhombus.
a. If DK = 8, find KL.
b. If m∠DML = 82 find m∠DKM.
c. If m∠KAL = 2x – 8, find x.
d.If DA = 4x and AL = 5x – 3, find DL.
e.If DA = 4x and AL = 5x – 3, find AD.
f. If DM = 5y + 2 and DK = 3y + 6, find KL.
3. PROOF Write a two-column proof.
Given: RSTU is a parallelogram.
̅̅̅̅ ≅ 𝑈𝑋
̅̅̅̅≅ 𝑇𝑋
̅̅̅̅ ≅ 𝑆𝑋
̅̅̅̅
𝑅𝑋
Prove: RSTU is a rectangle.
Statements
Reasons
33
34
Geometry
Section 6.5 Worksheet
Name: _____________________________________
For numbers 1 – 4, PRYZ is a rhombus. If RK = 5, RY = 13 and mYRZ = 67°, find each measure.
1. KY
2. PK
3. mYKZ
4. mPZR
For numbers 5 – 8, MNPQ is a rhombus. If PQ = 3√2 and AP = 3, find each measure.
5. AQ
6. mAPQ
7. mMNP
8. PM
For numbers 9 – 11, use the given set of vertices to determine whether
apply. Explain.
BEFG is a rhombus, a rectangle, or a square. List all that
9. B(–9, 1), E(2, 3), F(12, –2), G(1, –4)
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10. B(1, 3), E(7, –3), F(1, –9), G(–5, –3)
11. B(–4, –5), E(1, –5), F(–2, –1), G(–7, –1)
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Geometry
Section 6.6 Trapezoids
Date:
Today’s Objectives:
Question, Topics and Vocabulary
1. Students will be able to apply properties of trapezoids.
2. Students will be able to apply properties of kites.
Problems, Definitions and Work
Trapezoid
Bases
Legs
Base Angles
Isosceles Trapezoid
Theorem 6.21
Theorem 6.22
Theorem 6.23
37
Example 1: Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN =
6.7 feet, and LN = 3.6 feet. Find mMJK.
Questions:
Example 2: Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show
that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.
Midsegment
Trapezoid Midsegment
Theorem
(6.24)
Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x?
38
Example 4: WXYZ is an isosceles trapezoid with median. Find XY if JK = 18 and WZ = 25.
Kite
Theorem 6.25
Theorem 6.26
Example 5: If WXYZ is a kite, find mXYZ.
Example 6: If MNPQ is a kite, find NP.
39
Example 7: If BCDE is a kite, find mCDE.
Summary:
1. If JKLM is a kite, find KL.
2. Find m∠S
3. For trapezoid HJKL, T and S are midpoints of the legs.
a. If HJ = 14 and LK = 42, find TS.
b. If LK = 19 and TS = 15, find HJ.
c. If HJ = 7 and TS = 10, find LK.
d. If KL = 17 and JH = 9, find ST.
COORDINATE GEOMETRY
4. EFGH is a quadrilateral with vertices E(1, 3), F(5, 0), G(8, –5), H(–4, 4).
a. Verify that EFGH is a trapezoid.
b. Determine whether EFGH is an isosceles
trapezoid. Explain.
40
Geometry
Section 6.6 Worksheet
Name: _____________________________________
For numbers 1 – 4, find each measure.
1. mT
3. mQ
2. mY
4. BC
For numbers 5 & 6, use trapezoid FEDC, where V and Y are midpoints of the legs.
5. If FE = 18 and VY = 28, find CD.
6. If mF = 140 and mE = 125°, find mD.
41
For numbers 7 & 8, RSTU is a quadrilateral with vertices R(–3, –3), S(5, 1), T(10, –2), U(–4, –9).
7. Verify that RSTU is a trapezoid.
8. Determine whether RSTU is an isosceles trapezoid. Explain.
42
Geometry
Systems of Equations
Name___________________________________Date_______________
1. QUAD is a parallelogram. Determine the values of x and y.
U
A
18
Q
D
2. ABCD is a parallelogram. Determine the values of x and y
B
C
y
A
D
3. MNOP is a parallelogram. Determine the values of x and y.
N
M
O
P
4. Draw the parallelogram ABCD with sides AB = y, DC = 3x, BC = x, and AD = 4 – y. Then, find the values of x and y.
43
5. Given: parallelogram PSTM
mP = (2x + y)°
mM = (3x + 5y)°
mT = (4x – 3y + 8)°
Find the values of x, y, mP, and mS.
S
P
T
M
M
6. Given: Parallelogram KMOP,
O
mM  ( x  3 y ),
mO  ( x  4),
mP  (4 y  8)
Find:
mK.
K
P
44