6.2 Parallelograms

Objectives Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals of parallelograms.

Key Vocabulary Parallelogram

Theorems 6.2 – 6.5 Properties of Parallelograms

Introduction Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names. Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.

Parallelograms What makes a polygon a parallelogram?

Parallelogram 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.”

PROPERTIES OF PARALLELOGRAMS

Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. ► PQ ≅ RS and SP ≅ QR P Q R S

Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent.  P ≅  R and  Q ≅  S P Q R S

Properties of Parallelograms Theorem 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 Q R S

Summary Parallelogram Theorems Theorem 6.2 – Opposite sides of are ≅. Theorem 6.3 – Opposite  s in are ≅. Theorem 6.4 – Consecutive  s in are supplementary.

FGHJ is a parallelogram. Find JH and FJ. Substitute 5 for FG. = 5= 5 SOLUTION JH = FG Opposite sides of a are congruent. Substitute 3 for GH. = 3= 3 FJ = GH Opposite sides of a are congruent. ANSWER In FGHJ, JH = 5 and FJ = 3. Example 1

ANSWER AB = 9; AD = 8 ABCD is a parallelogram. Find AB and AD. 1. Your Turn:

PQRS is a parallelogram. Find the missing angle measures. SOLUTION By Theorem 6.3, the opposite angles of a parallelogram are congruent, so m  R = m  P = 70° By Theorem 6.4, the consecutive angles of a parallelogram are supplementary. Consecutive angles of a are supplementary. m  Q + m  P = 180° Substitute 70 ° for m  P. m  Q + 70° = 180° Subtract 70 ° from each side. m  Q = 110° Example 2

By Theorem 6.3, the opposite angles of a parallelogram are congruent, so m  S = m  Q = 110°. 3. ANSWER The measure of  R is 70°, the measure of  Q is 110°, and the measure of  S is 110°. Example 2

ABCD is a parallelogram. Find the missing angle measures ANSWER m  B = 120°; m  C = 60°; m  D = 120° ANSWER m  A = 75°; m  B = 105°; m  C = 75° Your Turn:

Example 3A A. CONSTRUCTION In suppose m  B = 32, CD = 80 inches, BC = 15 inches. Find AD.

Example 3A AD=BCOpposite sides of a are . =15Substitution Answer: AD = 15 inches

Example 3B B. CONSTRUCTION In suppose m  B = 32, CD = 80 inches, BC = 15 inches. Find m  C.

Example 3B Answer: m  C = 148 m  C + m  B=180 Cons.  s in a are supplementary. m  C + 32=180 Substitution m  C=148 Subtract 32 from each side.

Example 3C C. CONSTRUCTION In suppose m  B = 32, CD = 80 inches, BC = 15 inches. Find m  D.

Example 3C Answer: m  D = 32 m  D=m  BOpp.  s of a are . =32 Substitution

Your Turn A.10 B.20 C.30 D.50 A. ABCD is a parallelogram. Find AB.

Your Turn A.36 B.54 C.144 D.154 B. ABCD is a parallelogram. Find m  C.

Your Turn A.36 B.54 C.144 D.154 C. ABCD is a parallelogram. Find m  D.

Diagonals of Parallelograms The diagonals of a parallelogram have special properties as well. Next theorems using diagonals of parallelograms.

Theorem 6.5 If a quadrilateral is a parallelogram, then its diagonals bisect each other. Point M is the midpoint of both diagonals.

More on Diagonals If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles. AB C D If ABCD is a parallelogram, then ∆ ABD ≅∆ CDB.

More on Diagonals A parallelogram with two diagonals divides the figure into pairs of congruent triangles. If ABCD is a parallelogram, then ∆ AZD ≅∆ BZC and ∆ AZB ≅∆ DZC. AB C D Z

Substitute 3 for XV. = 3= 3 TUVW is a parallelogram. Find TX. SOLUTION TX = XV Diagonals of a bisect each other. Example 4

Example 5A A. If WXYZ is a parallelogram, find the value of r. Opposite sides of a parallelogram are . Definition of congruence Substitution Divide each side by 4. Answer: r = 4.5

Example 5B B. If WXYZ is a parallelogram, find the value of s. 8s=7s + 3Diagonals of a bisect each other. Answer: s = 3 s=3Subtract 7s from each side.

Example 5C C. If WXYZ is a parallelogram, find the value of t. ΔWXY  ΔYZWDiagonal separates a parallelogram into 2  triangles.  YWX  WYZCPCTC m  YWX=m  WYZDefinition of congruence

Example 2C 2t=18Substitution t=9Divide each side by 2. Answer: t = 9

Your Turn: A.2 B.3 C.5 D.7 A. If ABCD is a parallelogram, find the value of x.

Your Turn: A.4 B.8 C.10 D.11 B. If ABCD is a parallelogram, find the value of p.

Your Turn: A.4 B.5 C.6 D.7 C. If ABCD is a parallelogram, find the value of k.

Assignment Pg #1 – 49 odd