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6-6 Kites and Trapezoids. Properties and conditions. Kites. A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Properties of Kites. Trapezoids. - PowerPoint PPT Presentation
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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