Trapezoids and Kites Essential Questions How do I

Essential Questions How do I use properties of trapezoids? How do I use properties

Vocabulary Trapezoid – a quadrilateral with exactly one pair of parallel sides. base leg

Base Angles Trapezoid Theorem If a trapezoid is isosceles, then each pair of base

Diagonals of a Trapezoid Theorem A trapezoid is isosceles if and only if its

Example 1 PQRS is an isosceles trapezoid. Find m P, m Q and m

Definition Midsegment of a trapezoid – the segment that connects the midpoints of the

Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base

If given both base length… A 12 F D For trapezoid ABCD, F and

If given the mid segment length and a base length… A F D 7

Definition Kite – a quadrilateral that has two pairs of consecutive congruent sides, but

Theorem: Perpendicular Diagonals of a Kite If a quadrilateral is a kite, then its

Theorem: Opposite Angles of a Kite If a quadrilateral is a kite, then exactly

Example 2 Find the side lengths of the kite.

Example 2 Continued We can use the Pythagorean Theorem to find the side lengths.

Example 3 Find m G and m J. Since GHJK is a kite G

Try This! RSTU is a kite. Find m R, m S and m T.

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Trapezoids and Kites

Essential Questions How do I use properties of trapezoids? How do I use properties of kites?

Vocabulary Trapezoid – a quadrilateral with exactly one pair of parallel sides. base leg base A trapezoid has two pairs of base angles. In this example the base angles are A & B and C & D

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

Diagonals of a Trapezoid Theorem A trapezoid is isosceles if and only if its diagonals are congruent.

Example 1 PQRS is an isosceles trapezoid. Find m P, m Q and m R = 50 since base angles are congruent m P = 130 and m Q = 130 (consecutive angles of parallel lines cut by a transversal are )

Definition Midsegment of a trapezoid – the segment that connects the midpoints of the legs.

Midsegment Theorem for Trapezoids 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 given both base length… A 12 F D For trapezoid ABCD, F and G Are midpoints of the legs. B If AB = 12 and DC = 24, Find FG. G 24 C

If given the mid segment length and a base length… A F D 7 21 B For trapezoid ABCD, F and G Are midpoints of the legs. If AB = 7 and FG = 21, Find DC. G C Multiply both sides by 2 to get rid of fraction: 42 = 12 +DC 30 = DC

Definition Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorem: Perpendicular Diagonals of a Kite If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem: Opposite Angles of a Kite If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent A C, B D

Example 2 Find the side lengths of the kite.

Example 2 Continued We can use the Pythagorean Theorem to find the side lengths. 122 + 202 = (WX)2 122 + 122 = (XY)2 144 + 400 = (WX)2 144 + 144 = (XY)2 544 = (WX)2 288 = (XY)2

Example 3 Find m G and m J. Since GHJK is a kite G J So 2(m G) + 132 + 60 = 360 2(m G) =168 m G = 84 and m J = 84

Try This! RSTU is a kite. Find m R, m S and m T. 125 x +30 + 125 + x = 360 2 x + 280 = 360 2 x = 80 x = 40 So m R = 70 , m T = 40 and m S = 125

TEST ON FRIDAY! QUADRILATERALS! OH MY!