6-6 Kites and TrapezoidsProperties and conditions

KitesA kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

Properties of Kites

TrapezoidsA 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 TrapezoidsIf the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Properties of Isosceles Trapezoids

Midsegments of Trapezoids

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

Trapezoid Midsegment Theorem

Lets apply!

Kite cons. sides

Example 1In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mBCD.

∆BCD is isos. 2 sides isos. ∆

isos. ∆ base s

Def. of s

Polygon Sum Thm.

CBF CDF

mCBF = mCDF

mBCD + mCBF + mCDF = 180°

Lets apply!

Substitute mCDF for mCBF.Substitute 52 for mCDF.

Subtract 104 from both sides.

mBCD + mCDF + mCDF = 180°

mBCD + 52° + 52° = 180°

mBCD = 76°

mBCD + mCBF + mCDF = 180°

Example 1 Continued

Lets apply!

Kite one pair opp. s

Def. of s

Add. Post.

Substitute.

Solve.

Example 2In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mFDA.

CDA ABC

mCDA = mABC

mCDF + mFDA = mABC

52° + mFDA = 115°

mFDA = 63°

Lets apply!Example 3: Applying Conditions for Isosceles Trapezoids

Find the value of a so that PQRS is isosceles.

a = 9 or a = –9

Trap. with pair base s isosc. trap.

Def. of s

Substitute 2a2 – 54 for mS and a2 + 27 for mP.

Subtract a2 from both sides and add 54 to both sides.

Find the square root of both sides.

S P

mS = mP

2a2 – 54 = a2 + 27

a2 = 81

Lets apply!Example 4: Applying Conditions for Isosceles Trapezoids

AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles.

Diags. isosc. trap.

Def. of segs.

Substitute 12x – 11 for AD and 9x – 2 for BC.Subtract 9x from both sides and add 11 to both sides.

Divide both sides by 3.

AD = BC

12x – 11 = 9x – 2

3x = 9

x = 3

Lets apply!Example 5 Conditions of Midsegments

Find EH.

Trap. Midsegment Thm.

Substitute the given values.

Simplify.

Multiply both sides by 2.33 = 25 + EH

Subtract 25 from both sides.13 = EH

116.5 = (25 + EH)2