Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
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pg. 432 (14; 16; 24; 28-29, 41; 42; 48 why4,
ch. 45)
pg. 432 (18-22 even, 30-34 even, 35, 43, 47
why4, ch. 51)
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
How do you use properties of kites to solve problems?
How do you use properties of trapezoids to solve problems?
Essential Questions
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Kite � cons. sides ≅
Example 1A: Using Properties of Kites
In kite ABCD, m∠∠∠∠DAB = 54°, and m∠∠∠∠CDF = 52°. Find m∠∠∠∠BCD.
∆BCD is isos. 2 ≅ sides �isos. ∆
isos. ∆ �base ∠∠∠∠s ≅
Def. of ≅ ∠∠∠∠ s
Polygon ∠∠∠∠ Sum Thm.
∠CBF ≅ ∠CDF
m∠CBF = m∠CDF
m∠BCD + m∠CBF + m∠CDF = 180°
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Example 1A Continued
Substitute m∠CDF for m∠CBF.
Substitute 52 for m∠CDF.
Subtract 104 from both sides.
m∠BCD + m∠CDF + m∠CDF = 180°
m∠BCD + 52° + 52° = 180°
m∠BCD = 76°
m∠BCD + m∠CBF + m∠CDF = 180°
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Kite � one pair opp. ∠∠∠∠s ≅
Example 1B: Using Properties of Kites
Def. of ≅ ∠∠∠∠s
Polygon ∠∠∠∠ Sum Thm.
In kite ABCD, m∠∠∠∠DAB = 54°, and m∠∠∠∠CDF = 52°. Find m∠∠∠∠ABC.
∠ADC ≅ ∠ABC
m∠ADC = m∠ABC
m∠ABC + m∠BCD + m∠ADC + m∠DAB = 360°
m∠ABC + m∠BCD + m∠ABC + m∠DAB = 360°
Substitute m∠ABC for m∠ADC.
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Example 1B Continued
Substitute.
Simplify.
m∠ABC + m∠BCD + m∠ABC + m∠DAB = 360°
m∠ABC + 76° + m∠ABC + 54° = 360°
2m∠ABC = 230°
m∠ABC = 115° Solve.
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Kite � one pair opp. ∠∠∠∠s ≅
Example 1C: Using Properties of Kites
Def. of ≅ ∠∠∠∠s
∠∠∠∠ Add. Post.
Substitute.
Solve.
In kite ABCD, m∠∠∠∠DAB = 54°, and m∠∠∠∠CDF = 52°. Find m∠∠∠∠FDA.
∠CDA ≅ ∠ABC
m∠CDA = m∠ABC
m∠CDF + m∠FDA = m∠ABC
52° + m∠FDA = 115°
m∠FDA = 63°
Holt McDougal Geometry
6-6 Properties of Kites and 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.
Holt McDougal Geometry
6-6 Properties of Kites and 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.
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Isos.� trap. ∠∠∠∠s base ≅
Example 2A: Using Properties of Isosceles
Trapezoids
Find m∠∠∠∠A.
Same-Side Int. ∠∠∠∠s Thm.
Substitute 100 for m∠∠∠∠C.
Subtract 100 from both sides.
Def. of ≅ ∠∠∠∠s
Substitute 80 for m∠∠∠∠B
m∠C + m∠B = 180°
100 + m∠B = 180
m∠B = 80°
∠A ≅ ∠B
m∠A = m∠B
m∠A = 80°
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Example 2B: Using Properties of Isosceles
Trapezoids
KB = 21.9 and MF = 32.7.
Find FB.
Isos. � trap. ∠∠∠∠s base ≅
Def. of ≅ segs.
Substitute 32.7 for FM.
Seg. Add. Post.
Substitute 21.9 for KB and 32.7 for KJ.
Subtract 21.9 from both sides.
KJ = FM
KJ = 32.7
KB + BJ = KJ
21.9 + FB = 32.7
FB = 10.8
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Example 3A: 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 m∠∠∠∠S and a2 + 27 for m∠∠∠∠P.
Subtract a2 from both sides and add 54 to both sides.
Find the square root of both sides.
∠S ≅ ∠P
m∠S = m∠P
2a2 – 54 = a2 + 27
a2 = 81
Holt McDougal Geometry
6-6 Properties of Kites and Trapezoids
Example 3B: 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
Holt McDougal Geometry
6-6 Properties of Kites and 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.