1. Section 6.2: Parallelograms
EQ: What are the properties of parallelograms?

2. Vocab!
Parallelogram
Parallelogram Opposite Sides Theorem
Parallelogram Opposite Angles Theorem
A quadrilateral with both pairs of opposite sides parallel
If JKLM is a parallelogram then 𝐽𝐾 ≅ 𝑀𝐿 & 𝐽𝑀 ≅ 𝐾𝐿 J K M L
If JKLM is a parallelogram, then ∠J ≅ ∠L and ∠M ≅ ∠K J K M L

3. Example 1
In parallelogram ABCD, suppose m∠B = 32, CD = 80 inches, and BC = 15 inches.
Find AD
AD = BC
AD = 15
b) Find m∠C
= 64
360 – 64 = 296
296/2 = 148
148°
c) Find m∠D
m∠D = m∠B
m∠D = 32°

4. Example 2
Find the values of x and y.
𝐴𝐵=𝐷𝐶
𝑥+4=12
𝑥=8
𝑚∠𝐴=𝑚∠𝐶
𝑦=65°

5. You Try! 1. Find FG and the measure of angle G.
2. Find the values of x and y.
m∠E = m∠G
m∠G = 60°
m∠𝐽=𝑚∠𝐿
2x = 50
x = 25
ML = JK
y + 3 = 18
y = 15

6. Consecutive interior angles theorem…
x + y = 180°

7. Vocab! Parallelogram Consecutive Angles Theorem
If JKLM is a parallelogram, then x + y = 180°
Parallelogram Diagonals Theorem
If ABCD is a parallelogram then AP = PC and DP = PB J K y x y x M L A B P D C

8. Example 2 Find AB m∠C m∠D AB = DC AB = 30 m∠C = m∠A m∠C = 36°
180 – 36 = 144°

9. Example 3 If WXYZ is a parallelogram… Find the value of r.
b) Find the value of s.
c) Find the value of t.
WX = ZY
4r = 18
r = 4.5
7s + 3 = 8s
s = 3
Alternate Interior Angles
2t = 18
t = 9

10. You Try!
Find the indicated measure in parallelogram LMNQ. Explain your reasoning. 1. LM 2. LP 3. LQ 4. MQ
LM = QN 13
LP = NP 7
LQ = MN 8
MP = QP
QP = 8.2
QP + MP = MQ
16.4

11. You Try Cont. 5. 𝑚∠𝐿𝑀𝑁 6. 𝑚∠𝑁𝑄𝐿 7. 𝑚∠𝑀𝑁𝑄 8. 𝑚∠𝐿𝑀𝑄 180 – 100 = ∠LMN
m∠LMN = 80°
m∠NQL = m∠LMN
80°
m∠MNQ = m∠MLQ
100°
Alternate interior angles
m∠LMQ = m∠NQM
29°

12. Coordinate Geometry Coordinate Geometry
Using a coordinate grid to prove certain relationships about polygons.
Using coordinate geometry, how can you prove a shape is a parallelogram?
Use the theorems to prove the shape is a parallelogram.

13. Example 4
a) 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)?
Need to find the midpoint of 𝑁𝑅 & 𝑀𝑃
Midpoint formula: ( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 )
𝑁𝑅 =( −1+3 2 , ) = (1, 2)
𝑀𝑃 =( −3+5 2 , ) = (1, 2)
Diagonal intersection= (1, 2) P N R M

14. Example 4 cont.
b) What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5) and O(3, 1)?
Need to find the midpoint of 𝑀𝑂 & 𝑁𝐿
Midpoint formula: ( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 )
𝑀𝑂 =( −2+3 2 , ) = ( 1 2 ,1)
𝑁𝐿 =( , 5−3 2 ) = ( 1 2 ,1)
Diagonal intersection: ( 1 2 , 1) N M O L