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Five-Minute Check (over Lesson 6–1) CCSS 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 Lesson Menu

Find the measure of an interior angle of a regular polygon that has 10 sides. B. 162 C. 144 D. 126 5-Minute Check 1

Find the measure of an interior angle of a regular polygon that has 10 sides. B. 162 C. 144 D. 126 5-Minute Check 1

Find the measure of an interior angle of a regular polygon that has 12 sides. B. 150 C. 165 D. 180 5-Minute Check 2

Find the measure of an interior angle of a regular polygon that has 12 sides. B. 150 C. 165 D. 180 5-Minute Check 2

What is the sum of the measures of the interior angles of a 20-gon? B. 3420 C. 3240 D. 3060 5-Minute Check 3

What is the sum of the measures of the interior angles of a 20-gon? B. 3420 C. 3240 D. 3060 5-Minute Check 3

What is the sum of the measures of the interior angles of a 16-gon? B. 2880 C. 2700 D. 2520 5-Minute Check 4

What is the sum of the measures of the interior angles of a 16-gon? B. 2880 C. 2700 D. 2520 5-Minute Check 4

Find x if QRSTU is a regular pentagon. B. 15.25 C. 12 D. 10 5-Minute Check 5

Find x if QRSTU is a regular pentagon. B. 15.25 C. 12 D. 10 5-Minute Check 5

What type of regular polygon has interior angles with a measure of 135°? A. pentagon B. hexagon C. octagon D. decagon 5-Minute Check 6

What type of regular polygon has interior angles with a measure of 135°? A. pentagon B. hexagon C. octagon D. decagon 5-Minute Check 6

G.CO.11 Prove theorems about parallelograms. Content Standards G.CO.11 Prove theorems about parallelograms. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. Mathematical Practices 4 Model with mathematics. 3 Construct viable arguments and critique the reasoning of others. CCSS

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. Then/Now

parallelogram Vocabulary

Concept 1

Concept 2

Use Properties of Parallelograms A. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find AD. Example 1A

AD = BC Opposite sides of a are . Use Properties of Parallelograms AD = BC Opposite sides of a are . = 15 Substitution Answer: Example 1

AD = BC Opposite sides of a are . Use Properties of Parallelograms AD = BC Opposite sides of a are . = 15 Substitution Answer: AD = 15 inches Example 1

Use Properties of Parallelograms B. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mC. Example 1B

mC + mB = 180 Cons. s in a are supplementary. Use Properties of Parallelograms mC + mB = 180 Cons. s in a are supplementary. mC + 32 = 180 Substitution mC = 148 Subtract 32 from each side. Answer: Example 1

mC + mB = 180 Cons. s in a are supplementary. Use Properties of Parallelograms mC + mB = 180 Cons. s in a are supplementary. mC + 32 = 180 Substitution mC = 148 Subtract 32 from each side. Answer: mC = 148 Example 1

Use Properties of Parallelograms C. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mD. Example 1C

mD = mB Opp. s of a are . = 32 Substitution Answer: Use Properties of Parallelograms mD = mB Opp. s of a are . = 32 Substitution Answer: Example 1

mD = mB Opp. s of a are . = 32 Substitution Answer: mD = 32 Use Properties of Parallelograms mD = mB Opp. s of a are . = 32 Substitution Answer: mD = 32 Example 1

A. ABCD is a parallelogram. Find AB. Example 1A

A. ABCD is a parallelogram. Find AB. Example 1A

B. ABCD is a parallelogram. Find mC. Example 1B

B. ABCD is a parallelogram. Find mC. Example 1B

C. ABCD is a parallelogram. Find mD. Example 1C

C. ABCD is a parallelogram. Find mD. Example 1C

Concept 3

A. If WXYZ is a parallelogram, find the value of r. 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: Example 2A

A. If WXYZ is a parallelogram, find the value of r. 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 Example 2A

B. If WXYZ is a parallelogram, find the value of s. Use Properties of Parallelograms and Algebra B. If WXYZ is a parallelogram, find the value of s. 8s = 7s + 3 Diagonals of a bisect each other. s = 3 Subtract 7s from each side. Answer: Example 2B

B. If WXYZ is a parallelogram, find the value of s. Use Properties of Parallelograms and Algebra B. If WXYZ is a parallelogram, find the value of s. 8s = 7s + 3 Diagonals of a bisect each other. s = 3 Subtract 7s from each side. Answer: s = 3 Example 2B

C. If WXYZ is a parallelogram, find the value of t. 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 mYWX = mWYZ Definition of congruence Example 2C

2t = 18 Substitution t = 9 Divide each side by 2. Answer: Use Properties of Parallelograms and Algebra 2t = 18 Substitution t = 9 Divide each side by 2. Answer: Example 2C

2t = 18 Substitution t = 9 Divide each side by 2. Answer: t = 9 Use Properties of Parallelograms and Algebra 2t = 18 Substitution t = 9 Divide each side by 2. Answer: t = 9 Example 2C

A. If ABCD is a parallelogram, find the value of x. Example 2A

A. If ABCD is a parallelogram, find the value of x. Example 2A

B. If ABCD is a parallelogram, find the value of p. Example 2B

B. If ABCD is a parallelogram, find the value of p. Example 2B

C. If ABCD is a parallelogram, find the value of k. Example 2C

C. If ABCD is a parallelogram, find the value of k. Example 2C

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 Example 3

Parallelograms and Coordinate Geometry Answer: Example 3

Parallelograms and Coordinate Geometry Answer: The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2). Example 3

What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5), O(3, 1)? A. B. C. D. Example 3

What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5), O(3, 1)? A. B. C. D. Example 3

Write a paragraph proof. Proofs Using the Properties of Parallelograms Write a paragraph proof. Given: are diagonals, and point P is the intersection of 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. Example 4

To complete the proof below, which of the following is relevant information? Given: LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO. Prove: LNO  NLM A. LO  MN B. LM║NO C. OQ  QM D. Q is the midpoint of LN. Example 4

To complete the proof below, which of the following is relevant information? Given: LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO. Prove: LNO  NLM A. LO  MN B. LM║NO C. OQ  QM D. Q is the midpoint of LN. Example 4

End of the Lesson