Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals...
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Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals to solve a real-world problem Objectives: • Determine whether a quadrilateral is a parallelogram Recall Theorem 6-3 The diagonals of a parallelogram bisect each other.
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Transcript of Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals...
Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals to solve a real-world problem
Objectives:• Determine whether a quadrilateral is a parallelogram
Recall Theorem 6-3The diagonals of a parallelogram bisect each other.
Theorem 6-5Converse of Theorem 6-3
If the diagonals of a quadrilateral bisect each other,then
the quadrilateral is a parallelogram.
Theorem 6-6
If one pair of opposite sides of a quadrilateral is both congruent and parallel, then
the quadrilateral is a parallelogram.
Find values of x and y for which ABCD must be a parallelogram.
If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other.
If x = 18 and y = 89, then ABCD is a parallelogram.
10x – 24 = 8x + 12 Diagonals of parallelograms 2y – 80 = y + 9bisect each other.
x = 182x = 36
y = 89 Solve.
2x – 24 = 12 y – 80 = 9Collect the variable terms on one side.
Use Theorems to Solve Problems
Theorem 6-7
If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Theorem 6-8
If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Determine whether the quadrilateral is a parallelogram. Explain.
a.
b.
a. All you know about the quadrilateral is that only one pair of opposite sides is congruent.
b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360.
Therefore, you cannot conclude that the quadrilateral is a parallelogram.
If x represents the measure of the unmarked angle, x + 75 + 105 + 75 = 360, so x = 105.
Theorem 6-8 states If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-8.
Use Theorems to Solve Problems
Parallelograms and Real-world
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