Lesson 8-6

Trapezoids

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LMNO is a rhombus.

1. Find x

2. Find y

QRST is a square.

3. Find n if mTQR = 8n + 8.

4. Find w if QR = 5w + 4 and RS = 2(4w – 7).

5. Find QU if QS = 16t – 14 and QU = 6t + 11.

6. What property applies to a square, but not to a rhombus?Standardized Test Practice:

A C

B D

Opposite sides are congruent

Opposite angles are congruent All angles are right angles

Diagonals bisect each other

Q R

ST

U

L

M

N

O P

(8y – 6)°(3x + 12)°

(5x – 2)°

7

12

10.25

6

65

D

Objectives

• Recognize and apply the properties of trapezoids

• Solve problems involving medians of trapezoids

Vocabulary

• Trapezoid – a quadrilateral with only one pair of parallel sides

• Isosceles Trapezoid – a trapezoid with both legs (non parallel sides) congruent

• Median – a segment that joins the midpoints of the legs of a trapezoid

Polygon Hierarchy

Polygons

Squares

RhombiRectangles

Parallelograms Kites Trapezoids

IsoscelesTrapezoids

Quadrilaterals

TrapezoidsTrapezoid CharacteristicsBases ParallelLegs are not ParallelLeg angles are supplementary (mA + mC = 180, mB + mD = 180)Median is parallel to basesMedian = ½ (base + base) ½(AB + CD)

Isosceles Trapezoid CharacteristicsLegs are congruent (AC BD)Base angle pairs congruent (A B, C D)Diagonals are congruent (AD BC)

A B

C D

base

base

legmidpoint median

legmidpoint

A B

C D

M

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.

The sides of a picture frame appear to be two adjacent trapezoids. Determine if they are isosceles trapezoids.

Answer: yes

DEFG is an isosceles trapezoid with median Find DG if and

Theorem 8.20

Multiply each side by 2.

Substitution

Subtract 20 from each side.

Answer:

DEFG is an isosceles trapezoid with median Find , and if and

Since EF // DG, 1 and 3 are supplementary Because this is an isosceles trapezoid, 1 2 and 3 4

Substitution

Combine like terms.

Divide each side by 9.

Answer: If x = 20, then m1 = 65° and 3 = 115°. Because 1 2 and 3 4, 2 = 65° and 4 = 115°

WXYZ is an isosceles trapezoid with median

Answer:

a.

b.

Answer: Because

Quadrilateral Characteristics Summary

Convex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

Summary & Homework

• Summary:– In an isosceles trapezoid, both pairs of base

angles are congruent and the diagonals are congruent.

– The median of a trapezoid is parallel to the bases and its measure is one-half the sum of the measures of the bases

• Homework: – pg 442-444; 10, 13-16, 22-25