8.5 Trapezoids and Kites. Objectives:. Use properties of trapezoids. Use properties of kites. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Using properties of trapezoids. If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
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8.5 Trapezoids and 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.
If a trapezoid is isosceles, then each pair of base angles is congruent.
A ≅ B, C ≅ D
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
ABCD is an isosceles trapezoid
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.
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 = 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.
If a quadrilateral is a kite, then its diagonals are perpendicular.
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.