Five-Minute Check (over Lesson 6–1)

NGSSS

Then/Now

New Vocabulary

Theorems: Properties of Parallelograms

Proof: Theorem 6.4

Example 1: Real-World Example: Use Properties of Parallelograms

Theorems: Diagonals of Parallelograms

Example 2: Use Properties of Parallelograms and Algebra

Example 3: Parallelograms and Coordinate Geometry

Example 4: Proofs Using the Properties of Parallelograms

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 180

B. 162

C. 144

D. 126

Find the measure of an interior angle of a regular polygon that has 10 sides.

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 135

B. 150

C. 165

D. 180

Find the measure of an interior angle of a regular polygon that has 12 sides.

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 3600

B. 3420

C. 3240

D. 3060

What is the sum of the measures of the interior angles of a 20-gon?

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 3060

B. 2880

C. 2700

D. 2520

What is the sum of the measures of the interior angles of a 16-gon?

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 21

B. 15.25

C. 12

D. 10

Find x if QRSTU is a regular pentagon.

Over Lesson 6–1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. pentagon

B. hexagon

C. octagon

D. decagon

What type of regular polygon has interior angles with a measure of 135°?

MA.912.G.3.1 Describe, classify, and compare relationships among the quadrilaterals the square, rectangle, rhombus, parallelogram, trapezoid, and kite.

MA.912.G.3.4 Prove theorems involving quadrilaterals.

You classified polygons with four sides as quadrilaterals. (Lesson 1–6)

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

• Recognize and apply properties of the diagonals of parallelograms.

• parallelogram

Use Properties of Parallelograms

A. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find AD.

Use Properties of Parallelograms

AD = BC Opposite sides of a are .

= 15 Substitution

Answer: AD = 15 inches

Use Properties of Parallelograms

B. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mC.

Use Properties of Parallelograms

Answer: mC = 148

mC + mB = 180 Cons. s in a aresupplementary.

mC + 32 = 180 Substitution

mC = 148 Subtract 32 from eachside.

Use Properties of Parallelograms

C. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mD.

Use Properties of Parallelograms

Answer: mD = 32

mD = mB Opp. s of a are .

= 32 Substitution

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 10

B. 20

C. 30

D. 50

A. ABCD is a parallelogram. Find AB.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 36

B. 54

C. 144

D. 154

B. ABCD is a parallelogram. Find mC.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 36

B. 54

C. 144

D. 154

C. ABCD is a parallelogram. Find mD.

Use Properties of Parallelograms and Algebra

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

Opposite sides of a parallelogram are .

Definition of congruence

Substitution

Divide each side by 4.Answer: r = 4.5

Use Properties of Parallelograms and Algebra

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

8s = 7s + 3 Diagonals of a bisecteach other.

Answer: s = 3

s = 3 Subtract 7s from each side.

Use Properties of Parallelograms and Algebra

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

ΔWXY ΔYZW Diagonal separates a parallelogram into 2 triangles.

YWX WYZ CPCTC

mYWX = mWYZ Definition of congruence

Use Properties of Parallelograms and Algebra

2t = 18 Substitution

t = 9 Divide each side by 2.

Answer: t = 9

A. A

B. B

C. C

D. D A B C D

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A. 2

B. 3

C. 5

D. 7

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

A. A

B. B

C. C

D. D A B C D

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A. 4

B. 8

C. 10

D. 11

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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 4

B. 5

C. 6

D. 7

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

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)?

Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of

Find the midpoint of

Midpoint Formula

Parallelograms and Coordinate Geometry

Answer: The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2).

A. A

B. B

C. C

D. D A B C D

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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.

Proofs Using the Properties of Parallelograms

Write a paragraph proof.

Prove: AC and BD bisect each other.

Proof: ABCD is a parallelogram and AC and BD are diagonals; therefore, AB║DC and AC is a transversal. BAC DCA and ABD CDB by Theorem 3.2. ΔAPB ΔCPD by ASA. So, by the properties of congruent triangles BP DP and AP CP. Therefore, AC and BD bisect each other.

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

A. A

B. B

C. C

D. D A B C D

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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.