6-2 Properties of Parallelograms page 294 Objective: To use relationships among sides, angles, diagonals or transversals of parallelograms.
Mar 26, 2015
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
CD
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
CD
Use KMOQ to find m O.
Q and O are consecutive angles of KMOQ, so they are supplementary.
Definition of supplementary anglesm O + m Q = 180
Substitute 35 for m Q.m O + 35 = 180
Subtract 35 from each side.m O = 145
Properties of Parallelograms
6-2
Theorem 6-2
Opposite angle of a parallelogram are congruent.
<A = <C
<B = <D
A B
CD
Find the value of x in ABCD. Then find m A.
2x + 15 = 135 Add x to each side.
2x = 120 Subtract 15 from each side.
x = 60 Divide each side by 2.
x + 15 = 135 – x Opposite angles of a are congruent.
Substitute 60 for x. m B = 60 + 15 = 75
Consecutive angles of a parallelogram are supplementary.
m A + m B = 180
Subtract 75 from each side.m A = 105
m A + 75 = 180 Substitute 75 for m B.
6-2
Theorem 6-3
The diagonals of a parallelogram bisect each other.
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.
Properties of Parallelograms
–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 = DFA 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