1. Splash Screen

2. Five-Minute Check (over Lesson 6–2) CCSS Then/Now
Theorems: Conditions for Parallelograms
Proof: Theorem 6.9
Example 1: Identify Parallelograms
Example 2: Real-World Example: Use Parallelograms to Prove Relationships
Example 3: Use Parallelograms and Algebra to Find Values
Concept Summary: Prove that a Quadrilateral Is a Parallelogram
Example 4: Parallelograms and Coordinate Geometry
Example 5: Parallelograms and Coordinate Proofs
Lesson Menu

3. ____ ? A. B. C.
5-Minute Check 1

4. ____ ? A. B. C.
5-Minute Check 1

5. ? A. B. C.
5-Minute Check 2

6. ? A. B. C.
5-Minute Check 2

7. ?
A. A
B. B
C. C
5-Minute Check 3

8. ?
A. A
B. B
C. C
5-Minute Check 3

9. An expandable gate is made of parallelograms that have angles that change measure as the gate is adjusted. Which of the following statements is always true?
A. A  C and B  D
B. A  B and C  D C. D.
5-Minute Check 4

10. An expandable gate is made of parallelograms that have angles that change measure as the gate is adjusted. Which of the following statements is always true?
A. A  C and B  D
B. A  B and C  D C. D.
5-Minute Check 4

11. G.CO.11 Prove theorems about parallelograms.
Content Standards
G.CO.11 Prove theorems about parallelograms.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.
Mathematical Practices
3 Construct viable arguments and critique the reasoning of others.
2 Reason abstractly and quantitatively.
CCSS

12. You recognized and applied properties of parallelograms.
Recognize the conditions that ensure a quadrilateral is a parallelogram.
Prove that a set of points forms a parallelogram in the coordinate plane.
Then/Now

13. Concept 1

14. Concept 1

15. Concept 1

16. Concept 1

17. Identify Parallelograms
Determine whether the quadrilateral is a parallelogram. Justify your answer.
Answer: Each pair of opposite sides has the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Example 1

18. Which method would prove the quadrilateral is a parallelogram?
A. Both pairs of opp. sides ||.
B. Both pairs of opp. sides .
C. Both pairs of opp. s .
D. One pair of opp. sides both || and .
Example 1

19. Use Parallelograms to Prove Relationships
MECHANICS Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A  C and B  D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform.
Example 2

20. Use Parallelograms to Prove Relationships
Answer: Since both pairs of opposite angles of quadrilateral ABCD are congruent, ABCD is a parallelogram by Theorem Theorem 6.5 states that consecutive angles of parallelograms are supplementary. Therefore, mA + mB = 180 and mC + mD = 180. By substitution, mA + mD = 180 and mC + mB = 180.
Example 2

21. The diagram shows a car jack used to raise a car from the ground
The diagram shows a car jack used to raise a car from the ground. In the diagram, AD  BC and AB  DC. Based on this information, which statement will be true, regardless of the height of the car jack.
A. A  B
B. A  C
C. AB  BC
D. mA + mC = 180
Example 2

22. Find x and y so that the quadrilateral is a parallelogram.
Use Parallelograms and Algebra to Find Values
Find x and y so that the quadrilateral is a parallelogram.
Opposite sides of a parallelogram are congruent.
Answer: So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram.
Example 3

23. Find m so that the quadrilateral is a parallelogram.
B. m = 3
C. m = 6
D. m = 8
Example 3

24. Concept 3

25. Parallelograms and Coordinate Geometry
COORDINATE GEOMETRY Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.
If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
Example 4

26. Parallelograms and Coordinate Geometry
Answer:
Example 4

27. Parallelograms and Coordinate Geometry
Answer: Since opposite sides have the same slope, QR║ST and RS║TQ. Therefore, QRST is a parallelogram by definition.
Example 4

28. Graph quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram.
A. yes
B. no
Example 4

29. Which of the following can be used to prove the statement below?
If a quadrilateral is a parallelogram, then one pair of opposite sides is both parallel and congruent.
A. AB = a units and DC = a units; slope of AB = 0 and slope of DC = 0
B. AD = c units and BC = c units;
slope of and slope of
Example 5

30. HOMEWORK Pg
4-7,9-14,
18-23,
24,25,
52-54
Concept 3

31. End of the Lesson