6.5 Trapezoids

Objectives: Use properties of trapezoids.

Using properties of trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD  D and  C are one pair of base angles. The other pair is  A and  B. The nonparallel sides are the legs of the trapezoid.

Using properties of trapezoids If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

Trapezoid Theorems Theorem If a trapezoid is isosceles, then each pair of base angles is congruent.  A ≅  B,  C ≅  D

Trapezoid Theorems If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. ABCD is an isosceles trapezoid

Trapezoid Theorems A trapezoid is isosceles if and only if its diagonals are congruent. ABCD is isosceles if and only if AC ≅ BD.

Ex. 1: Using properties of Isosceles Trapezoids PQRS is an isosceles trapezoid. Find m  P, m  Q, m  R. PQRS is an isosceles trapezoid, so m  R = m  S = 50 °. m  P = 180 °- 50° = 130°, and m  Q = m  P = 130° 50 ° You could also add 50 and 50, get 100 and subtract it from 360°. This would leave you 260/2 or 130°.

Midsegment of a trapezoid The midsegment of a trapezoid is the segment that connects the midpoints of its legs.

Theorem 6.17: Midsegment of a trapezoid The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases. MN ║AD, MN║BC MN = ½ (AD + BC)

Example:

Practice GF = ½ (QR + PS) 12 = ½ ( 8+ x) Multiply both sides by 2 24 = 8 + x X = 16

Practice RS = ½ (FG + EH) x = ½ ( ) x = 10.6 / 2 x = 5.2

Practice ED = ½ (QR + PS) 4x - 10 = ½ (15+ 29) 4x - 10 = 22 4x = 32 X = 8

Practice KJ= ½ (RS + QT) 2x + 16 = ½ ( ) 2x + 16 = 28 2x = 12 X = 6

Practice ∟N = ∟ K and ∟M = ∟ L ∟N + ∟ M = 180 and ∟K + ∟L = 180 ∟N + ∟L = ∟L = 180 ∟L = ∟L =112