7.2 Properties of Parallelograms Objectives: Use some properties of parallelograms. Use properties of parallelograms in real-life situations. •

Concept: Exterior Angle-Sum Thm. The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. For the pentagon

Concept: Exterior Angle-Sum Thm. What is the measure of ONE exterior angle of a regular nonagon?

Concept: Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Ex:

Concept: Opposite Sides Property The opposite sides of a parallelogram are congruent. PQ≅RS and SP≅QR Q R P S

Concept: Opposite Angles Property The opposite angles of a parallelogram are congruent. Q R LP ≅ LR and LQ ≅ LS P S

Concept: Concsecutive Angles Property The consecutive angles of a parallelogram are supplementary, (add up to 180º) Q R mLP +mLQ = 180°, mLQ +mLR = 180°, mLR + mLS = 180°, mLS + mLP = 180° P S

Concept: Using the Properties Problem #1 PQRS is parallelogram. Find the measures of the given angles: mLR mLQ 70º 110º

Concept: Diagonals Property If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM ≅ SM and PM ≅ RM Q R P S

Concept: Using the Properties SOLUTION: JH = FG Opposite sides of a are ≅. JH = 5 Since FG = JH then JH = 5 Find the JH and JK

Concept: Using the Properties cont… SOLUTION: JK = GK Diagonals of a bisect each other GK = 3 Since JK = GK then JK = 3 Find the JH and JK

Concept: Using the Properties cont… Solve for x and y 2y + 3 x HK = y + 5, KF = x, KG = x + 2, KE = 2y + 3 y + 5 x + 2 1. Label the diagram y + 5 = x 2y + 3 = x + 2 2. Set up equations

Concept: Using the Properties cont… Solve for x and y 2y + 3 x HK = y + 5, KF = x, KG = x + 2, KE = 2y + 3 y + 5 x + 2 3. Since there are 2 different variables in each equation, we will need to substitute. (Replace x.) y + 5 = x 2y + 3 = x + 2 2y + 3 = y + 5 + 2

Concept: Using the Properties cont… Solve for x and y HK = y + 5, KF = x, KG = x + 2, KE = 2y + 3 2y + 3 x 2y + 3 = y + 5 + 2 y + 5 x + 2 2y + 3 = y + 7 –y –y y + 3 = 7 4. Solve for y –3 = –3 y = 4 Substitute the value you found for y so you can find x. y + 5 = x 4 + 5 = x 9 = x