6-2 Properties of Parallelograms page 294 Objective: To use relationships among sides, angles, diagonals or transversals of parallelograms.

Vocabulary Consecutive angles – angles of a polygon that share a side. NOTE: Consecutive angles of a parallelogram are supplementary. A B C D

You can use what you know about parallel lines & transversals to prove some theorems about parallelograms Theorem 6.1 p. 294---Opposite sides of a parallelogram are congruent

Theorem 6-1 Opposite sides of a parallelogram are congruent. AB = DC AD = BC A B D C

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

Theorem 6-2 Opposite angle of a parallelogram are congruent. <A = <C <B = <D A B D C

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 = 60 + 15 = 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

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

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 – 32 + 5 = 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

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

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.

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?

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