1. Properties and conditions
6-6 Kites and Trapezoids
Properties and conditions

2. Kites
A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

3. Properties of Kites

4. Trapezoids
A 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.

5. Isosceles Trapezoids
If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

6. Properties of Isosceles Trapezoids

7. 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.

8. Trapezoid Midsegment Theorem

9. Lets apply!
Example 1
In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mBCD.
Kite  cons. sides 
∆BCD is isos.
2  sides isos. ∆
CBF  CDF
isos. ∆ base s 
mCBF = mCDF
Def. of   s
mBCD + mCBF + mCDF = 180°
Polygon  Sum Thm.

10. Lets apply! Example 1 Continued mBCD + mCBF + mCDF = 180°
mBCD + mCDF + mCDF = 180°
Substitute mCDF for mCBF.
mBCD + 52° + 52° = 180°
Substitute 52 for mCDF.
mBCD = 76°
Subtract 104 from both sides.

11. Lets apply!
Example 2
In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mFDA.
CDA  ABC
Kite  one pair opp. s 
mCDA = mABC
Def. of  s
mCDF + mFDA = mABC
 Add. Post.
52° + mFDA = 115°
Substitute.
mFDA = 63°
Solve.

12. Example 3: Applying Conditions for Isosceles Trapezoids
Lets apply!
Example 3: Applying Conditions for Isosceles Trapezoids
Find the value of a so that PQRS is isosceles.
S  P
Trap. with pair base s   isosc. trap.
mS = mP
Def. of  s
2a2 – 54 = a2 + 27
Substitute 2a2 – 54 for mS and a for mP.
a2 = 81
Subtract a2 from both sides and add 54 to both sides.
a = 9 or a = –9
Find the square root of both sides.

13. Example 4: Applying Conditions for Isosceles Trapezoids
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.
AD = BC
Def. of  segs.
12x – 11 = 9x – 2
Substitute 12x – 11 for AD and 9x – 2 for BC.
3x = 9
Subtract 9x from both sides and add 11 to both sides.
x = 3
Divide both sides by 3.

14. Example 5 Conditions of Midsegments
Lets apply!
Example 5 Conditions of Midsegments
Find EH.
Trap. Midsegment Thm. 1
16.5 = (25 + EH) 2
Substitute the given values.
Simplify.
33 = 25 + EH
Multiply both sides by 2.
13 = EH
Subtract 25 from both sides.