Trapezoids Section 6.5

Objectives Use properties of trapezoids.

Key Vocabulary Trapezoid Bases of a trapezoid Legs of a trapezoid Base angles of a trapezoid Isosceles trapezoid Midsegment of a trapezoid

Theorems 6.12 & 6.13 Isosceles Trapezoid Theorems

Trapezoids What makes a quadrilateral a trapezoid?

Definition: Trapezoid trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.

Isosceles Trapezoid If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Isosceles Trapezoid Theorem 6.12 If a trapezoid is isosceles, then each pair of base angles is congruent.

Isosceles Trapezoid Theorem 6.13 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. If ∠ D ≅∠ C, then trapezoid ABCD is an isosceles.

Isosceles Trapezoids If the legs of a trapezoid are ≅, then it is called an isosceles trapezoid. Theorem 6.12: Both pairs of base  s of an isosceles trapezoid are ≅. (  A ≅  D and  B ≅  C) Theorem 6.13: If one pair of base  s are ≅, then it is an isosceles trapezoid. (If ∠ B ≅∠ C, then trapezoid ABCD is isosceles)

The top of this work station appears to be two adjacent trapezoids. Determine if they are isosceles trapezoids. Each pair of base angles is congruent, so the legs are the same length. Answer: Both trapezoids are isosceles. Example 1:

The sides of a picture frame appear to be two adjacent trapezoids. Determine if they are isosceles trapezoids. Answer: yes Your Turn:

Example 2 A. BASKET Each side of the basket shown is an isosceles trapezoid. If m  JML = 130, KN = 6.7 feet, and LN = 3.6 feet, find m  MJK.

Example 2 Since JKLM is a trapezoid, JK ║ LM. m  JML + m  MJK=180Consecutive Interior Angles Theorem m  JML + 130=180Substitution m  JML=50Subtract 130 from each side. Answer: m  JML = 50

Your Turn: A.124 B.62 C.56 D.112 A. Each side of the basket shown is an isosceles trapezoid. If m  FGH = 124, FI = 9.8 feet, and IG = 4.3 feet, find m  EFG.

Example 3 PQRS is an isosceles trapezoid. Find the missing angle measures. SOLUTION PQRS is an isosceles trapezoid and  R and  S are a pair of base angles. So, m  R = m  S = 50°. 1. Because  S and  P are same-side interior angles formed by parallel lines, they are supplementary. So, m  P = 180° – 50° = 130°. 2. Because  Q and  P are a pair of base angles of an isosceles trapezoid, m  Q = m  P = 130°. 3.

Your Turn: ABCD is an isosceles trapezoid. Find the missing angle measures. ANSWER m  A = 80° ; m  B = 80° ; m  C = 100° ANSWER m  A = 110° ; m  B = 110° ; m  D = 70° ANSWER m  B = 75° ; m  C = 105° ; m  D = 105°

Trapezoid Midsegment midsegment A midsegment of a trapezoid is a segment that connects the midpoints of the legs of a trapezoids.

Medians of Trapezoids  The segment that joins the midpoints of the legs of a trapezoid is called the median (MN). It is also referred to as the midsegment. BC || AD median

Trapezoid Midsegment Properties The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. If

Review: Trapezoids Trapezoid Characteristics Bases Parallel Legs are not Parallel Leg angles are supplementary (m  A + m  C = 180, m  B + m  D = 180) Midsegment is parallel to bases Midsegment = ½ (base + base) ½(AB + CD) Isosceles Trapezoid Characteristics Legs are congruent (AC  BD) Base angle pairs congruent (  A   B,  C   D) AB CD base leg midpoint midsegment leg midpoint A B CD M

Example 4 In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

Example 4 Trapezoid Midsegment Theorem Substitution Multiply each side by 2. Subtract 20 from each side. Answer: x = 40

Your Turn: A.XY = 32 B.XY = 25 C.XY = 21.5 D.XY = 11 WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25.

Example 5 Find the length of the midsegment DG of trapezoid CEFH. SOLUTION Use the formula for the midsegment of a trapezoid. DG = 1 2 (EF + CH) Formula for midsegment of a trapezoid = 1 2 (8 + 20) Substitute 8 for EF and 20 for CH. = 1 2 (28) Add. = 14 Multiply. ANSWER The length of the midsegment DG is 14.

Your Turn: ANSWER 11 Find the length of the midsegment MN of the trapezoid ANSWER 8 21

DEFG is an isosceles trapezoid with median Find m ∠ 1, m ∠ 2, m ∠ 3, and m ∠ 4. if and Because this is an isosceles trapezoid, Example 6:

Consecutive Interior Angles Theorem Substitution Combine like terms. Divide each side by 9. Answer: Because Example 6:

WXYZ is an isosceles trapezoid with median Answer: Because Your Turn:

REVIEW TRAPEZOIDS

Polygon Hierarchy Polygons Squares RhombiRectangles ParallelogramsTrapezoids Isosceles Trapezoids Quadrilaterals

ONE PAIR OF PARALLEL SIDES

leg base Leg angles are supplementary Leg angle 1 Leg angle 2

Base (b 2 ) Base (b 1 ) Midsegment is ½ the sum of the lengths of the bases Midsegment =½ (b 1 + b 2 )

leg base Isosceles: Base angles are congruent Base angle 2 Base angle 1

Joke Time Why does a room full of married people look so empty? There's not a Single person in it... What do you find in a clean nose? Fingerprints!

Assignment  Pg #1 – 25 odd, 35, 37