FiveMinute Check over Lesson 6 1 CCSS ThenNow

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

Over Lesson 6– 1 Find the measure of an interior angle of a regular polygon that has 10 sides. A. 180 B. 162 C. 144 D. 126

Over Lesson 6– 1 Find the measure of an interior angle of a regular polygon that has 12 sides. A. 135 B. 150 C. 165 D. 180

Over Lesson 6– 1 What is the sum of the measures of the interior angles of a 20 -gon? A. 3600 B. 3420 C. 3240 D. 3060

Over Lesson 6– 1 What is the sum of the measures of the interior angles of a 16 -gon? A. 3060 B. 2880 C. 2700 D. 2520

Over Lesson 6– 1 Find x if QRSTU is a regular pentagon. A. 21 B. 15. 25 C. 12 D. 10

Over Lesson 6– 1 What type of regular polygon has interior angles with a measure of 135°? A. pentagon B. hexagon C. octagon D. decagon

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.

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.

• parallelogram

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

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

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

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

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

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

A. ABCD is a parallelogram. Find AB. A. 10 B. 20 C. 30 D. 50

B. ABCD is a parallelogram. Find m C. A. 36 B. 54 C. 144 D. 154

C. ABCD is a parallelogram. Find m D. A. 36 B. 54 C. 144 D. 154

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. 8 s = 7 s + 3 s=3 Answer: s = 3 Diagonals of a each other. bisect Subtract 7 s 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 m YWX = m WYZ Definition of congruence

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

A. If ABCD is a parallelogram, find the value of x. A. 2 B. 3 C. 5 D. 7

B. If ABCD is a parallelogram, find the value of p. A. 4 B. 8 C. 10 D. 11

C. If ABCD is a parallelogram, find the value of k. A. 4 B. 5 C. 6 D. 7

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).

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.

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.

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.