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Use properties of parallelograms to solve problems.

Prove that a given quadrilateral is a parallelogram.

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- 1. Obj. 26 Parallelograms The student is able to (I can): • Prove and apply properties of parallelograms. • Use properties of parallelograms to solve problems. • Prove that a given quadrilateral is a parallelogram.
- 2. parallelogram A quadrilateral with two pairs of parallel sides. A parallelogram has the following properties: Opposite sides are parallel. (Definition) Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.
- 3. 2 pairs of sides If a quadrilateral has two pairs of parallel sides, then it is a parallelogram T >> E I >> M TI ME, TE IM
- 4. Opp. sides ≅ If a quadrilateral is a parallelogram, then opposite sides are congruent. K G I N KI ≅ NG, GK ≅ IN
- 5. Opp. angles ≅ If a quadrilateral is a parallelogram, then opposite angles are congruent. K >> G O >> N ∠K ≅ ∠N, ∠O ≅ ∠G
- 6. Cons. angles supp. If a quadrilateral is a parallelogram, then consecutive angles are supplementary. 1 4 2 3 m∠1 + m∠2 = 180° m∠2 + m∠3 = 180° m∠3 + m∠4 = 180° m∠4 + m∠1 = 180°
- 7. Diagonals bisect If a quadrilateral is a parallelogram, then its diagonals bisect each other. T U >> S >> E N TS ≅ NS, ES ≅ US
- 8. Examples Find the value of the variable: 1. x = 5x + 3 2x + 15 2. x = (x + 84)º 3. y = yº (3x)º
- 9. Examples Find the value of the variable: 1. x = 4 5x + 3 2x + 15 5x + 3 = 2x + 15 3x = 12 2. x = 42 (x + 84)º 3x = x + 84 2x = 84 3. y = 54 3(42) = 126º y = 180 - 126 yº (3x)º
- 10. Conditions for Parallelograms If one pair of opposite sides of a quadrilateral is congruent and parallel then the quadrilateral is a parallelogram. parallel, We can also use the converses of the theorems from the previous section to prove that quadrilaterals are parallelograms. Parallelogram ⇔ Opposite sides ≅ Opposite angles ≅ Cons. ∠s supp. Diagonals bisect

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