1. DRILL 1. Name 6 Different types of Quadrilaterals.
2. Are all Squares considered Rectangles?
3. Are all Parallelograms considered Rectangles?
4. How would you find the third side of a right triangle given two of the sides?

2. 9.1 Properties of Parallelograms
Geometry
Mr. Calise

3. Objectives: Use some properties of parallelograms.
Use properties of parallelograms in real-life situations.

4. In this lesson . . .
And the rest of the chapter, you will study special quadrilaterals. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
When you mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel. For example, in the diagram to the right, PQ║RS and QR║SP. The symbol PQRS is read “parallelogram PQRS.”

5. Theorems about parallelogramsQ
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
► PQ ≅ RS and SP ≅ QR R P S

6. Theorems about parallelograms
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
P ≅ R and
Q ≅ S Q R P S

7. Properties of a Parallelogram
1. Opposite Sides are Congruent.
2. Opposite Angles are Congruent.
3. Consecutive Angles are Supplementary.
4. Diagonals Bisect each other.

8. Parallel Lines Cut By A Transversal
If three or more parallel lines are cut by a transversal and the parts of the transversal are congruent, then the parts of all other transversals are also congruent.

9. Theorems about parallelograms
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary (add up to 180°).
mP + mQ = 180°,
mQ + mR = 180°,
mR + mS = 180°,
mS + mP = 180° Q R P S

10. Theorems about parallelogramsQ R
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
QM ≅ SM and
PM ≅ RM P S

11. Ex. 1: Using properties of Parallelograms
FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK 5 F G K 3 H J b.

12. Ex. 1: Using properties of Parallelograms
FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK
a. JH = FG Opposite sides of a are ≅.
JH = 5 Substitute 5 for FG. 5 F G 3 K H J b.

13. Ex. 1: Using properties of Parallelograms
FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK 5 F G 3 K H J
JK = GK Diagonals of a bisect each other.
JK = 3 Substitute 3 for GK b.

14. Ex. 2: Using properties of parallelogramsQ R
PQRS is a parallelogram.
Find the angle measure.
mR
mQ
70° P S

15. Ex. 2: Using properties of parallelograms
PQRS is a parallelogram.
Find the angle measure.
mR
mQ
a. mR = mP Opposite angles of a are ≅.
mR = 70° Substitute 70° for mP. Q R
70° S

16. Ex. 2: Using properties of parallelogramsQ R
PQRS is a parallelogram.
Find the angle measure.
mR
mQ
mQ + mP = 180° Consecutive s of a are supplementary.
mQ + 70° = 180° Substitute 70° for mP.
mQ = 110° Subtract 70° from each side.
70° S P S

17. Ex. 3: Using Algebra with ParallelogramsQ
PQRS is a parallelogram. Find the value of x.
mS + mR = 180°
3x = 180
3x = 60
x = 20
120°
3x° S R
Consecutive s of a □ are supplementary.
Substitute 3x for mS and 120 for mR.
Subtract 120 from each side.
Divide each side by 3.

18. Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are parallelograms.
Prove 1 ≅ 3.
ABCD is a □ AEFG is a ▭.
1 ≅ 2, 2 ≅ 3
1 ≅ 3
Given

19. Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are parallelograms.
Prove 1 ≅ 3.
1. ABCD is a □ AEFG is a ▭.
2. 1 ≅ 2, 2 ≅ 3
3. 1 ≅ 3
Given
Opposite s of a ▭ are ≅

20. Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are parallelograms.
Prove 1 ≅ 3.
1. ABCD is a □ AEFG is a ▭.
2. 1 ≅ 2, 2 ≅ 3
3. 1 ≅ 3
Given
2. Opposite s of a ▭ are ≅
3. Transitive prop. of congruence.

21. Proofs
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.

22. Ex. 6: Using parallelograms in real life
FURNITURE DESIGN. A drafting table is made so that the legs can be joined in different ways to change the slope of the drawing surface. In the arrangement below, the legs AC and BD do not bisect each other. Is ABCD a parallelogram?

23. ANSWER:
NO. If ABCD were a parallelogram, then by definition of a parallelogram, AC would bisect BD and BD would bisect AC. They do not, so it cannot be a parallelogram.

24. Homework
Textbook Page 451
#’s 1 – 10, 14 – 16