6.2 Properties of Parallelograms Geometry Ms. Olifer

Objectives: Use relationships among sides and among angles of parallelograms. Use properties of parallelograms in real-life situations

In this lesson . . . And the rest of the chapter, you will study special quadrilaterals. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. When you mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel. For example, in the diagram to the right, PQ║RS and QR║SP. The symbol PQRS is read “parallelogram PQRS.”

Theorems about parallelograms Q R 6.2—If a quadrilateral is a parallelogram, then its opposite sides are congruent. ►PQ≅RS and SP≅QR P S

Theorems about parallelograms Q R 6.3—If a quadrilateral is a parallelogram, then its opposite angles are congruent. P ≅ R and Q ≅ S P S

Theorems about parallelograms Q R 6.4—If a quadrilateral is a parallelogram, then its consecutive angles are supplementary (add up to 180°). mP +mQ = 180°, mQ +mR = 180°, mR + mS = 180°, mS + mP = 180° P S

Theorems about parallelograms Q R 6.5—If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM ≅ SM and PM ≅ RM P S

Ex. 1: Using properties of Parallelograms 5 F G FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK 3 K H J b.

Ex. 1: Using properties of Parallelograms 5 F G FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. 3 K H J b.

Ex. 1: Using properties of Parallelograms 5 F G FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. 3 K H J b. JK = GK Diagonals of a bisect each other. JK = 3 Substitute 3 for GK

Ex. 2: Using properties of parallelograms Q R PQRS is a parallelogram. Find the angle measure. mR mQ 70° P S

Ex. 2: Using properties of parallelograms Q R PQRS is a parallelogram. Find the angle measure. mR mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. 70° P S

Ex. 2: Using properties of parallelograms Q R PQRS is a parallelogram. Find the angle measure. mR mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. mQ + mP = 180° Consecutive s of a are supplementary. mQ + 70° = 180° Substitute 70° for mP. mQ = 110° Subtract 70° from each side. 70° P S

Ex. 3: Using Algebra with Parallelograms Q PQRS is a parallelogram. Find the value of x. mS + mR = 180° 3x + 120 = 180 3x = 60 x = 20 3x° 120° S R Consecutive s of a □ are supplementary. Substitute 3x for mS and 120 for mR. Subtract 120 from each side. Divide each side by 3.