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**6-2 Properties of Parallelograms page 294**

Objective: To use relationships among sides, angles, diagonals or transversals of parallelograms.

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**Vocabulary Consecutive angles – angles of a polygon that share a side.**

NOTE: Consecutive angles of a parallelogram are supplementary. A B C D

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You can use what you know about parallel lines & transversals to prove some theorems about parallelograms Theorem 6.1 p Opposite sides of a parallelogram are congruent

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**Theorem 6-1 Opposite sides of a parallelogram are congruent. AB = DC**

AD = BC A B D C

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**Properties of Parallelograms**

Use KMOQ to find m O. Q and O are consecutive angles of KMOQ, so they are supplementary. Definition of supplementary angles m O + m Q = 180 Substitute 35 for m Q. m O + 35 = 180 Subtract 35 from each side. m O = 145 6-2

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**Theorem 6-2 Opposite angle of a parallelogram are congruent.**

<A = <C <B = <D A B D C

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**Find the value of x in ABCD. Then find m A.**

x + 15 = 135 – x Opposite angles of a are congruent. 2x + 15 = 135 Add x to each side. 2x = 120 Subtract 15 from each side. x = 60 Divide each side by 2. Substitute 60 for x. m B = = 75 Consecutive angles of a parallelogram are supplementary. m A + m B = 180 m A + 75 = 180 Substitute 75 for m B. Subtract 75 from each side. m A = 105 6-2

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Theorem 6-3 The diagonals of a parallelogram bisect each other.

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**Properties of Parallelograms**

Find the values of x and y in KLMN. x = 7y – 16 The diagonals of a parallelogram bisect each other. 2x + 5 = 5y 2(7y – 16) + 5 = 5y Substitute 7y – 16 for x in the second equation to solve for y. 14y – = 5y Distribute. 14y – 27 = 5y Simplify. –27 = –9y Subtract 14y from each side. 3 = y Divide each side by –9. x = 7(3) – 16 Substitute 3 for y in the first equation to solve for x. x = 5 Simplify. So x = 5 and y = 3. 6-2

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Theorem 6-4 If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. BD = DF A B C D E F

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Closure Lesson 6-1 defined a rectangle as a parallelogram with four right angles. Explain why you can now define a rectangle as a parallelogram with one right angle.

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**Summary What is true about the opposite sides of a parallelogram?**

What is true about the opposite angles of a parallelogram? What about consecutive angles? What about the diagonals of a parallelogram? When 3 or more parallel lines cut of congruent segments on one transversal, what is true about all other transversals?

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Assignment 6.2 Page 297 #2-32 E, 34, 35, 39-41

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8.2 Parallelograms. Objectives Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals.

8.2 Parallelograms. Objectives Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals.

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