CCSS

Content StandardsG.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Mathematical Practices4 Model with mathematics.3 Construct viable arguments and critique the reasoning of others.

Then/Now

You named and classified polygons.

• Find and use the sum of the measures of the interior angles of a polygon.

• Find and use the sum of the measures of the exterior angles of a polygon.

Vocabulary

• diagonal

Concept 1

Example 1A

Find the Interior Angles Sum of a Polygon

A. Find the sum of the measures of the interior angles of a convex nonagon.

Example 1B

Find the Interior Angles Sum of a Polygon

B. Find the measure of each interior angle of parallelogram RSTU.

Example 1A

A. 900

B. 1080

C. 1260

D. 1440

A. Find the sum of the measures of the interior angles of a convex octagon.

Example 1B

A. x = 7.8

B. x = 22.2

C. x = 15

D. x = 10

B. Find the value of x.

Example 2

Interior Angle Measure of Regular Polygon

ARCHITECTURE 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.

Example 2

A. 130°

B. 128.57°

C. 140°

D. 125.5°

A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl.

Example 3

Find Number of Sides Given Interior Angle Measure

The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon.

Example 3

A. 12

B. 9

C. 11

D. 10

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Concept 2

Example 4A

Find Exterior Angle Measures of a Polygon

A. Find the value of x in the diagram.

Example 4B

Find Exterior Angle Measures of a Polygon

B. Find the measure of each exterior angle of a regular decagon.

Example 4A

A. 10

B. 12

C. 14

D. 15

A. Find the value of x in the diagram.

Example 4B

A. 72

B. 60

C. 45

D. 90

B. Find the measure of each exterior angle of a regular pentagon.

CCSS

Content StandardsG.CO.11 Prove theorems about parallelograms.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.Mathematical Practices4 Model with mathematics.3 Construct viable arguments and critique the

reasoning of others.

Then/Now

You classified polygons with four sides as quadrilaterals.

• Recognize and apply properties of the sides and angles of parallelograms.

• Recognize and apply properties of the diagonals of parallelograms.

Vocabulary

• parallelogram

Concept 1

Concept 2

Example 1A

Use Properties of Parallelograms

A. CONSTRUCTION In suppose m∠B = 32, CD = 80 inches, BC = 15 inches. Find AD.

Example 1B

Use Properties of Parallelograms

B. CONSTRUCTION In suppose m∠B = 32, CD = 80 inches, BC = 15 inches. Find m∠C.

Example 1C

Use Properties of Parallelograms

C. CONSTRUCTION In suppose m∠B = 32, CD = 80 inches, BC = 15 inches. Find m∠D.

Example 1A

A. 10

B. 20

C. 30

D. 50

A. ABCD is a parallelogram. Find AB.

Example 1B

A. 36

B. 54

C. 144

D. 154

B. ABCD is a parallelogram. Find m∠C.

Example 1C

A. 36

B. 54

C. 144

D. 154

C. ABCD is a parallelogram. Find m∠D.

Concept 3

Example 2A

Use Properties of Parallelograms and Algebra

A. If WXYZ is a parallelogram, find the value of r.

Example 2B

Use Properties of Parallelograms and Algebra

B. If WXYZ is a parallelogram, find the value of s.

Example 2C

Use Properties of Parallelograms and Algebra

C. If WXYZ is a parallelogram, find the value of t.

Example 2A

A. 2

B. 3

C. 5

D. 7

A. If ABCD is a parallelogram, find the value of x.

Example 2B

A. 4

B. 8

C. 10

D. 11

B. If ABCD is a parallelogram, find the value of p.

Example 2C

A. 4

B. 5

C. 6

D. 7

C. If ABCD is a parallelogram, find the value of k.

Example 3

Parallelograms and Coordinate Geometry

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)?

Example 3

What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with verticesL(0, –3), M(–2, 1), N(1, 5), O(3, 1)?

A.

B.

C.

D.

Example 4

Proofs Using the Properties of Parallelograms

Write a paragraph proof.

Prove: AC and BD bisect each other.

Given: are diagonals, and point P is the intersection of

Example 4

To complete the proof below, which of the following is relevant information?

Prove: ∠LNO ≅ ∠NLM

Given: LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO.

A. LO ≅ MN

B. LM║NO

C. OQ ≅ QM

D. Q is the midpoint of LN.

CCSS

Content StandardsG.CO.11 Prove theorems about parallelograms.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.Mathematical Practices3 Construct viable arguments and critique the reasoning of others.2 Reason abstractly and quantitatively.

Then/Now

You recognized and applied properties of parallelograms.

• Recognize the conditions that ensure a quadrilateral is a parallelogram.

• Prove that a set of points forms a parallelogram in the coordinate plane.

Concept 1

Concept 2

Example 1

Identify Parallelograms

Determine whether the quadrilateral is a parallelogram. Justify your answer.

Example 1

A. Both pairs of opp. sides ||.

B. Both pairs of opp. sides ≅.

C. Both pairs of opp. ∠s ≅.

D. One pair of opp. sides both || and ≅.

Which method would prove the quadrilateral is a parallelogram?

Example 2

Use Parallelograms to Prove Relationships

MECHANICS 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 2

The diagram shows a car jack used to raise a car from the ground. In the diagram, AD ≅ BC and AB ≅ DC. Based on this information, which statement will be true, regardless of the height of the car jack.A. ∠A ≅ ∠B

B. ∠A ≅ ∠C

C. AB ≅ BC

D. m∠A + m∠C = 180

Example 3

Use Parallelograms and Algebra to Find Values

Find x and y so that the quadrilateral is a parallelogram.

Example 3

A. m = 2

B. m = 3

C. m = 6

D. m = 8

Find m so that the quadrilateral is a parallelogram.

Concept 3

Example 4

Parallelograms and Coordinate Geometry

COORDINATE GEOMETRY 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.

Example 4

A. yes

B. no

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.

Example 5

Parallelograms and Coordinate Proofs

Write a coordinate proof for the following statement.

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

Example 5

Parallelograms and Coordinate Proofs

Example 5

Which of the following can be used to prove the statement below?If a quadrilateral is a parallelogram, then one pair of opposite sides is both parallel and congruent.A. AB = a units and DC = a units; slope

of AB = 0 and slope of DC = 0

B. AD = c units and BC = c units;

slope of and slope of

CCSS

Content StandardsG.CO.11 Prove theorems about parallelograms.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.Mathematical Practices3 Construct viable arguments and critique the reasoning of others.5 Use appropriate tools strategically.

Then/Now

You used properties of parallelograms and determined whether quadrilaterals were parallelograms.

• Recognize and apply properties of rectangles.

• Determine whether parallelograms are rectangles.

Vocabulary

• rectangle

Concept 1

Example 1

Use Properties of Rectangles

CONSTRUCTION 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 1

A. 3 feet

B. 7.5 feet

C. 9 feet

D. 12 feet

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Example 2

Use Properties of Rectangles and Algebra

Quadrilateral RSTU is a rectangle. If m∠RTU = 8x + 4 and m∠SUR = 3x – 2, find x.

Example 2

A. x = 1

B. x = 3

C. x = 5

D. x = 10

Quadrilateral EFGH is a rectangle. If m∠FGE = 6x – 5 and m∠HFE = 4x – 5, find x.

Concept 2

Example 3

Proving Rectangle Relationships

ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a 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 3

Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90?

A. Since opp. sides are ||, STUR must be a rectangle.

B. Since opp. sides are ≅, STUR must be a rectangle.

C. Since diagonals of the are ≅, STUR must be a rectangle.

D. STUR is not a rectangle.

Example 4

Rectangles and Coordinate Geometry

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.

Example 4

A. yes

B. no

C. cannot be determined

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). Determine whether WXYZ is a rectangle by using the Distance Formula.

Example 4

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). What are the lengths of diagonals WY and XZ?

A.

B. 4

C. 5

D. 25

CCSS

Content StandardsG.CO.11 Prove theorems about parallelograms.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.Mathematical Practices3 Construct viable arguments and critique the reasoning of others.2 Reason abstractly and quantitatively.

Then/Now

You determined whether quadrilaterals were parallelograms and/or rectangles.

• Recognize and apply the properties of rhombi and squares.

• Determine whether quadrilaterals are rectangles, rhombi, or squares.

Vocabulary

• rhombus

• square

Concept 1

Concept 2

Example 1A

Use Properties of a Rhombus

A. The diagonals of rhombus WXYZ intersect at V.If m∠WZX = 39.5, find m∠ZYX.

Example 1B

Use Properties of a Rhombus

B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

Example 1A

A. m∠CDB = 126

B. m∠CDB = 63

C. m∠CDB = 54

D. m∠CDB = 27

A. ABCD is a rhombus. Find m∠CDB if m∠ABC = 126.

Example 1B

A. x = 1

B. x = 3

C. x = 4

D. x = 6

B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

Concept 3

Concept

Example 2

Proofs Using Properties of Rhombi and Squares

Write a paragraph proof.Given: LMNP is a parallelogram.

∠1 ≅ ∠2 and ∠2 ≅ ∠6Prove: LMNP is a rhombus.

Example 2

Proofs Using Properties of Rhombi and Squares

Example 2

Is there enough information given to prove that ABCD is a rhombus?

Given: ABCD is a parallelogram.AD ≅ DC

Prove: ADCD is a rhombus

Example 2

A. Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

B. No, you need more information.

Example 3

Use Conditions for Rhombi and Squares

GARDENING 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 3

A. The diagonal bisects a pair of opposite angles.

B. The diagonals bisect each other.

C. The diagonals are perpendicular.

D. The diagonals are congruent.

Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square?

Example 4

Classify Quadrilaterals Using Coordinate Geometry

Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

Example 4

Classify Quadrilaterals Using Coordinate Geometry

Example 4

A. rhombus only

B. rectangle only

C. rhombus, rectangle, and square

D. none of these

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.

CCSS

Content StandardsG.GPE.4 Use coordinates to prove simple geometric theorems algebraically. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices1 Make sense of problems and persevere in solving

them.2 Reason abstractly and quantitatively.

Then/Now

You used properties of special parallelograms.

• Apply properties of trapezoids.

• Apply properties of kites.

Vocabulary

• trapezoid

• bases

• legs of a trapezoid

• base angles

• isosceles trapezoid

• midsegment of a trapezoid

• kite

Concept 1

Concept 2

Example 1A

Use Properties of Isosceles Trapezoids

A. BASKET 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.

Example 1B

Use Properties of Isosceles Trapezoids

B. BASKET Each side of the basket shown is an isosceles trapezoid. If m∠JML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.

Example 1A

A. 124

B. 62

C. 56

D. 112

A. Each side of the basket shown is an isosceles trapezoid. If m∠FGH = 124, FI = 9.8 feet, and IG = 4.3 feet, find m∠EFG.

Example 1B

A. 4.3 ft

B. 8.6 ft

C. 9.8 ft

D. 14.1 ft

B. Each side of the basket shown is an isosceles trapezoid. If m∠FGH = 124, FI = 9.8 feet, and EG = 14.1 feet, find IH.

Example 2

Isosceles Trapezoids and Coordinate Geometry

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.

Example 2

Isosceles Trapezoids and Coordinate Geometry

Example 2

Quadrilateral QRST has vertices Q(–1, 0), R(2, 2), S(5, 0), and T(–1, –4). Determine whether QRST is a trapezoid and if so, determine whether it is an isosceles trapezoid.

A. trapezoid; not isosceles

B. trapezoid; isosceles

C. not a trapezoid

D. cannot be determined

Concept 3

Example 3

In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x?

Midsegment of a Trapezoid

Example 3

Midsegment of a Trapezoid

Example 3

A. XY = 32

B. XY = 25

C. XY = 21.5

D. XY = 11

WXYZ is an isosceles trapezoid with medianFind XY if JK = 18 and WZ = 25.

Concept 4

Example 4A

Use Properties of Kites

A. If WXYZ is a kite, find m∠XYZ.

Example 4B

Use Properties of Kites

B. If MNPQ is a kite, find NP.

Example 4A

A. 28°

B. 36°

C. 42°

D. 55°

A. If BCDE is a kite, find m∠CDE.

Example 4B

A. 5

B. 6

C. 7

D. 8

B. If JKLM is a kite, find KL.