Properties of Parallelograms Objectives Prove and apply properties of parallelograms. Use properties of parallelograms to solve problems. Holt Geometry

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

A Diagonal of a polygon is a segment that joins two nonconsecutive vertices. Have students find more diagonals diagonals

A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .

 

Using Properties of Parallelograms PQRS is a parallelogram. Find the angle measure. m< R m< Q Q R 70° P S m< R = 70 ° To find m< Q 70 ° + m < Q = 180 ° m< Q = 110 °

Example 1: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find CF.  opp. sides  CF = 74 mm Subst. prop

Example 2: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find mEFC. mEFC + mFCD = 180°  cons. s supp. mEFC + 42 = 180 Subst. prop. mEFC = 138° - prop.

Using properties of parallelograms FGHJ is a parallelogram. Find the unknown length. JH JK 5 F 5 3 G 3 K J H

Example 3: Properties of Parallelograms In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find DF. DF = 2DG  diags. bisect each other. DF = 2(31) Subst. prop DF = 62 Simplify.

Examples Use the diagram of parallelogram JKLM. Complete the statement. LM K L KN <KJM N <LMJ LN MJ J M

Properties of Parallelograms Check It Out! Example 1a In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find KN.  opp. sides  LM = 28 in. Subst. prop

Properties of Parallelograms In KLMN, LM = 28 in., LN = 26 in., and mLKN = 74°. Find mNML. NML  LKN  opp. s  mNML = 74° Subst. prop.

Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ.  opp. s  8a–4 = 6a+10 Subst. prop 2a - 4 = 10 - prop 2a = 14 + prop a = 7   YZ = 8a – 4 = 8(7) – 4 = 52

Properties of Parallelograms WXYZ is a parallelogram. Find mZ . mZ + mW = 180°  cons. s supp. (9b+2) + (18b–11) = 180 Subst. prop 27b – 9 = 180 Simplify 27b = 189 + prop b = 7   mZ = (9b + 2)° = [9(7) + 2]° = 65°

6.2 Properties of Parallelograms EFGH is a parallelogram. Find JG.  diags. bisect each other. 3w = w + 8 Subst. prop 2w = 8 - prop   w = 4 JG = w + 8 = 4 + 8 = 12

1. ABCD is a parallelogram 1. Given Write a two-column proof. Given: ABCD is a parallelogram. Prove: ∆AEB  ∆CED Statements Reasons 1. ABCD is a parallelogram 1. Given 2.  opp. sides  3.  diags. bisect each other 4. SSS

Write a two-column proof. Given: RSTU is a parallelogram. Prove: ∆RSU  ∆TUS Statements Reasons 1. RSTU is a parallelogram. 4. ∆RSU  ∆TUS 3. R  T