9.2 Parallelograms
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The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. The key properties are that opposite sides are congruent, opposite angles are congruent, and consecutive angles are supplementary. Examples are provided to demonstrate using these properties to solve for unknown variables.
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Prove and apply properties of parallelograms.
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A parallelogram is a quadrilateral with two pairs of parallel sides. The document outlines several properties of parallelograms: opposite sides are parallel and equal in length, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other. There are several ways to prove that a quadrilateral is a parallelogram, including showing that opposite sides are parallel or equal, or that diagonals bisect each other.
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9.2 Parallelograms
- 1. Parallelograms The student is able to (I can): • Prove and apply properties of parallelograms. • Use properties of parallelograms to solve problems.
- 2. parallelogramparallelogramparallelogramparallelogram – a quadrilateral with two pairs of parallel sides. Therefore, a quadrilateral is a parallelogram if and only if it has two pairs of parallel sides. >> >> T I ME ,TI ME TE IM
- 3. Properties ofProperties ofProperties ofProperties of ParallelogramsParallelogramsParallelogramsParallelograms If a quadrilateral is a parallelogram, then opposite sides are congruent. If a quadrilateral is a parallelogram, then opposite angles are congruent. ,KI NG GK IN≅ ≅ K NG I >> >> K NG O ∠K ≅ ∠N, ∠O ≅ ∠G
- 4. Properties ofProperties ofProperties ofProperties of Parallelograms (cont.)Parallelograms (cont.)Parallelograms (cont.)Parallelograms (cont.) If a quadrilateral is a parallelogram, then consecutive angles are supplementary. If a quadrilateral is a parallelogram, then its diagonals bisect each other. 1 2 34 >> >> T U NE S ,TS NS ES US≅ ≅ m 1 m 2 180 m 2 m 3 180 m 3 m 4 180 m 4 m 1 180 ∠ + ∠ = ° ∠ + ∠ = ° ∠ + ∠ = ° ∠ + ∠ = °
- 5. Examples Find the value of the variable: 1. x = 2. x = 3. y = 5x + 3 2x + 15 (3x)° (x + 84)° y°
- 6. Examples Find the value of the variable: 1. x = 2. x = 3. y = 5x + 3 2x + 15 4 (3x)° (x + 84)° y° 5x + 3 = 2x + 15 3x = 12 3x = x + 84 2x = 84 42 3(42) = 126° y = 180 – 126 54
- 7. Examples Find the value of the variable. 4. x = y = 16 23 x° y°
- 8. Examples Find the value of the variables. 4. x = 44 y = 136 16 23 x° y° − = = ° 1 16 sin 23 44 x = − = ° 180 44 136 y
- 9. Examples Find the value of the variable: 5. x = 67° 25 x
- 10. Examples Find the value of the variable: 5. x = 67° 25 x °= = ° = 25 sin67 25 sin67 27.2 x x