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CHAPTER 6
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Honors Geometry
Section 6.1 Notes: Angles of Polygons
Learning Targets
Question, Topics and
Vocabulary
1. I can find and use the sum of the measures of the interior and 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.
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.
Sum of the Angles of a Polygon
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.
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.
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Example 2: Find the measures of each interior angle of parallelogram RSTU.
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 5: Find the value of x in the diagram.
Example 6: Find the measure of each exterior angle of a regular decagon.
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Honors Geometry
Section 6.2 Notes: Parallelograms
Learning Targets
Question, Topics and
Vocabulary
1. I can recognize and apply properties of the sides, angles, and diagonals of parallelograms.
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:
Theorem 6.3
Ex:
Property:
Theorem 6.4
Ex:
Property:
Theorem 6.5
Ex:
Property:
Theorem 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) Find mC.
c) Find mD.
Example 3: Write a two-column proof.
Given: Parallelogram ABCD
AC and BD are diagonals
P is the intersection of AC and BD
Prove: AC and BD bisect each other
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
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8.
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11.
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Example 4:
Given: WXYZ is a parallelogram
Prove: WXZ ≅ YZX
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Diagonals of a Parallelogram
Theorem 6.7
Ex:
Theorem 6.8
Ex:
Example 5: If WXYZ is a parallelogram…
a) Find the value of r.
b) Find the value of s.
c) Find the value of t.
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Example 6:
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)?
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)?
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Honors Geometry
Section 6.3 Notes: Tests for Parallelograms
Learning Targets
Question, Topics and
Vocabulary
1. I can recognize the conditions that ensure a quadrilateral is a parallelogram.
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
Theorem 6.9
Ex:
Theorem 6.10
Ex:
Theorem 6.11
Ex:
Theorem 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:
Theorem 6.10:
Concept Summary
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.
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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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Honors Geometry
Section 6.4 Notes: Rectangles
Learning Targets
Question, Topics and
Vocabulary
1. I can recognize and apply properties of rectangles.
Problems, Definitions and Work
By definition, a rectangle has the following properties:
Rectangle





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.
Example 2: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.
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.
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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.
Example 7: 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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Honors Geometry
Section 6.5 Notes: Rhombi and Squares
Learning Targets
Question, Topics and
Vocabulary
1. I can recognize and apply the properties of rhombi and 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.
Example 2: The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3,
solve for x.
Example 3: ABCD is a rhombus. Find mCBD if mABC = 126°.
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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).
Conditions for Rhombi and Squares
Theorem 6.17
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Theorem 6.18
Theorem 6.19
Theorem 6.20
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
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6. 5  6
6.
7.
7.
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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Honors Geometry
Section 6.6 Trapezoids
Today’s Objectives:
Question, Topics and
Vocabulary
1. I can apply properties of trapezoids and kites.
2. I can use the trapezoid midsegment theorem.
Problems, Definitions and Work
Trapezoid
Bases
Legs
Base Angles
Isosceles Trapezoid
Theorem 6.21
Theorem 6.22
Theorem 6.23
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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.
a) Find mMJK.
b) Find MN
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
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(6.24)
Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x?
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.
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Example 6: If MNPQ is a kite, find NP.
Example 7: If BCDE is a kite, find mCDE.
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Honors Geometry
Systems of Equations Worksheet
Name: _____________________________________
1. QUAD is a parallelogram. Determine the values of x and y.
U
A
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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.
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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
T
P
M
M
6. Given: Parallelogram KMOP,
mM = (x + 3y)°
mO = (x – 4)°
mP = (4y – 8)°
Find: mK
K
O
P
25
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