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8.5 Trapezoids and Kites

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8.5 Trapezoids and Kites

- Use properties of trapezoids.
- Use properties of kites.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

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

Theorem 6.14

If a trapezoid is isosceles, then each pair of base angles is congruent.

A ≅ B, C ≅ D

Theorem 6.15

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

ABCD is an isosceles trapezoid

Theorem 6.16

A trapezoid is isosceles if and only if its diagonals are congruent.

ABCD is isosceles if and only if AC ≅ BD.

PQRS is an isosceles trapezoid. Find mP, mQ, mR.

50°

Ex.2: The stone above the arch in the diagram is an isosceles trapezoid. Find

Show that ABCD is a trapezoid.

Compare the slopes of opposite sides.

The slope of AB = 5 – 0 = 5 = - 1

0 – 5 -5

The slope of CD = 4 – 7 = -3 = - 1

7 – 4 3

The slopes of AB and CD are equal, so AB ║ CD.

The slope of BC = 7 – 5 = 2 = 1

4 – 0 4 2

The slope of AD = 4 – 0 = 4 = 2

7 – 5 2

The slopes of BC and AD are not equal, so BC is not parallel to AD.

So, because AB ║ CD and BC is not parallel to AD, ABCD is a trapezoid.

The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17 is similar to the Midsegment Theorem for triangles.

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)

LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

Ex.4: In the diagram, is the midsegment of trapezoid PQRS. Find MN.

Ex.5: In the diagram, is the midsegment of trapezoid DEFG. Find HK.

A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorem 6.18

If a quadrilateral is a kite, then its diagonals are perpendicular.

AC BD

Theorem 6.19

If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent.

A ≅ C, B ≅ D

Ex.7: Find in the kite shown below.

Ex.8: In a kite, the measures of the angles are 3xo, 75o, 90o, and 120o. Find the value of x. What are the measures of the angles that are congruent.