1. 8.3 Tests for Parallelograms

2. Objectives
Recognize the conditions that ensure a quadrilateral is a parallelogram.
Prove that a set of points forms a parallelogram in the coordinate plane.

3. Conditions for a Parallelogram
Obviously, if the opposite sides of a quadrilateral are parallel, then it is a parallelogram; but there are other tests we can also apply to a quadrilateral to test whether it is a parallelogram or not.

4. Conditions for a Theorems
Theorem 8.9 – If both pairs of opposite sides are ≅, then the quad. is a
Theorem 8.10 – If both pairs of opposite s are ≅, then the quad. is a
Theorem 8.11 – If diagonals bisect each other, then the quad. is
Theorem 8.12 – If one pair of opposite sides is ║ and ≅, then the quad. is a

5. Example 1:
Write a paragraph proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram.
Given:
Prove: ABCD is a parallelogram.
Proof: CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, ABCD is a parallelogram.

6. Your Turn:
Write a paragraph proof of the statement: If two diagonals of a quadrilateral divide the quadrilateral into four triangles where opposite triangles are congruent, then the quadrilateral is a parallelogram.
Given:
Prove: WXYZ is a parallelogram.

7. Your Turn:
Proof: by CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, WXYZ is a parallelogram.

8. Example 2:
Some of the shapes in this Bavarian crest appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms.
Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is a congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

9. Your Turn:
The shapes in the vest pictured here appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms.
Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

10. Example 3:
Determine whether the quadrilateral is a parallelogram. Justify your answer.
Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

11. Your Turn:
Determine whether the quadrilateral is a parallelogram. Justify your answer.
Answer: One pair of opposite sides is parallel and has the same measure, which means these sides are congruent. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

12. Tests for Parallelograms
Both pairs of opposite sides are parallel.
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
The diagonals bisect each other.
A pair of opposite sides is both parallel and congruent.

13. Example 4a: Find x so that the quadrilateral is a parallelogram. A B C
Opposite sides of a parallelogram are congruent.

14. Example 4a: Substitution Distributive Property
Subtract 3x from each side.
Add 1 to each side.
Answer: When x is 7, ABCD is a parallelogram.

15. Example 4b: Find y so that the quadrilateral is a parallelogram. F D E
Opposite angles of a parallelogram are congruent.

16. Example 4b: Substitution Subtract 6y from each side.
Subtract 28 from each side.
Divide each side by –1.
Answer: DEFG is a parallelogram when y is 14.

17. Your Turn: Find m and n so that each quadrilateral is a parallelogram.b.
Answer:
Answer:

18. Parallelograms on the Coordinate Plane
We can use the Distance Formula and the Slope Formula to determine if a quadrilateral is a parallelogram on the coordinate plane.
Just pick one of the tests… and apply either or both of the formulas.

19. Example 5a:
COORDINATE GEOMETRY Determine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and
D(1, –1) is a parallelogram. Use the Slope Formula.

20. Example 5a:
If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition.

21. Example 5b:
COORDINATE GEOMETRY Determine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3) is a parallelogram. Use the Distance and Slope Formulas.

22. Example 5b:
First use the Distance Formula to determine whether the opposite sides are congruent.

23. Example 5b: Next, use the Slope Formula to determine whether
and have the same slope, so they are parallel.
Answer: Since one pair of opposite sides is congruent and parallel, PQRS is a parallelogram.

24. Your Turn:
Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.
a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1);
Slope Formula

25. Your Turn:
Answer: The slopes of and the slopes of Therefore, Since opposite sides are parallel, ABCD is a parallelogram.

26. Your Turn:
Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.
Distance and Slope Formulas
b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);

27. Your Turn:
Answer: Since the slopes of Since one pair of opposite sides is congruent and parallel, LMNO is a parallelogram.

28. Assignment Pre-AP Geometry: Pg. 421 #13 – 32
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