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1. Complete the statement about
and justify your answer. AB ?
2. AD ?
3. D ?
4. In the figure, RSTU is a
parallelogram. Find x.
5. Find y.
• Recognize the conditions that ensure a quadrilateral is a parallelogram.
• Prove that a set of points forms a parallelogram in the coordinate plane.
Write a Proof
Proof: Since ΔABD ΔCDB, CPCTC. By Theorem 6.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, ABCD is a parallelogram.
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.
Prove: ABCD is a parallelogram.
Given: ΔABD ΔCDB
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.
Prove: WXYZ is a parallelogram.
Given: ΔXVY ΔZVW and ΔXVW ΔZVY
A. A
B. B
C. C
D. D
A. Both pairs of opp. sides .
B. Both pairs of opp. ’s .
C. One pair of opp. sides both and ||.
D. Diagonals bisect each other
Proof: Since ΔXVY ΔZVW and ΔXVW ΔZVY, by CPCTC. By which method would you prove WXYZ is a parallelogram?
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 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.
Properties of Parallelograms
Properties of 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.
1. A
2. B
3. C
4. D
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 .
Which method would prove the quadrilateral is a parallelogram?
Find x so that the quadrilateral is a parallelogram.
Opposite sides of a parallelogram are congruent.
Find Measures
Answer: When x is 7, ABCD is a parallelogram.
Find Measures
Substitution
Distributive Property
Add 1 to each side.
Subtract 3x from each side.
AB = DC
A. A
B. B
C. C
D. D
A. m = 2
B. m = 3
C. m = 6
D. m = 8
Find m so that the quadrilateral is a parallelogram.
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.
Use Slope and Distance
If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
Use Slope and Distance
Answer:
1. A
2. B
3. C
Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.
A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula
A. yes
B. no
C. cannot be determined
