5-minute checkIn a brief paragraph explain all the properties
you know about parallelograms.Explain the properties of the following:RhombusRectangleSquare
Objectives:Students will analyze and apply properties of
trapezoids and kites. Why? So you can measure part of a building,
as in Example 2.Mastery is 80% or better on 5-minute checks
and practice problems.
Using properties of trapezoidsA trapezoid is a
quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. For instance in trapezoid ABCD D and C are one pair of base angles. The other pair is A and B. The nonparallel sides are the legs of the trapezoid.
base
base
legleg
A B
D C
Using properties of trapezoidsIf the legs of a
trapezoid are congruent, then the trapezoid is an isosceles trapezoid.
Trapezoid TheoremsTheorem 8.14If a trapezoid is
isosceles, then each pair of base angles is congruent.
A ≅ B, C ≅ D
A B
D C
Trapezoid TheoremsTheorem 8.15If a trapezoid has a
pair of congruent base angles, then it is an isosceles trapezoid.
ABCD is an isosceles trapezoid
A B
D C
Trapezoid TheoremsTheorem 8.16A trapezoid is
isosceles if and only if its diagonals are congruent.
ABCD is isosceles if and only if AC BD.≅
A B
D C
Skill DevelopEx. 1: Using properties of Isosceles Trapezoids
PQRS is an isosceles trapezoid. Find mP, mQ, mR.
PQRS is an isosceles trapezoid, so mR = mS = 50°. Because S and P are consecutive interior angles formed by parallel lines, they are supplementary. So mP = 180°- 50° = 130°, and mQ = mP = 130°
m PS = 2.16 cm
m RQ = 2.16 cm
S R
P Q
50°
You could also add 50 and 50, get 100 and subtract it from 360°. This would leave you 260/2 or 130°.
Skill DevelopEx. 2: Using properties of trapezoids
Show that ABCD is a trapezoid. Compare the slopes of opposite sides.
The slope of AB = 5 – 0 = 5 = - 1 0 – 5 -5
The slope of CD = 4 – 7 = -3 = - 1 7 – 4 3
The slopes of AB and CD are equal, so AB ║ CD. The slope of BC = 7 – 5 = 2 = 1
4 – 0 4 2 The slope of AD = 4 – 0 = 4 = 2
7 – 5 2 The slopes of BC and AD are not equal,
so BC is not parallel to AD. So, because AB ║ CD and BC is not parallel
to AD, ABCD is a trapezoid.
8
6
4
2
5 10 15A(5, 0)
D(7, 4)
C(4, 7)
B(0, 5)
Midsegment of a trapezoidThe midsegment of a
trapezoid is the segment that connects the midpoints of its legs. Theorem 8.17 is similar to the Midsegment Theorem for triangles.
midsegment
B C
DA
Theorem 8.17: Midsegment of a trapezoidThe midsegment of a
trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases.
MN║AD, MN║BCMN = ½ (AD + BC)
NM
A D
CB
Ex. 3: Finding Midsegment lengths of trapezoids
A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?
Ex. 3: Finding Midsegment lengths of trapezoids
Use the midsegment theorem for trapezoids.
HG = ½(EF + CD)=½ (8 + 20) = 14”
C
H
E
D
G
F
Using properties of kitesA kite is a
quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Kite theoremsTheorem 8.18If a quadrilateral is a
kite, then its diagonals are perpendicular.
AC BD
B
C
A
D
Kite theoremsTheorem 8.19If a quadrilateral is a
kite, then exactly one pair of opposite angles is congruent.
A ≅ C, B ≅ D
B
C
A
D
Think…Ink…ShareEx. 4: Using the diagonals of a kiteWXYZ is a kite so
the diagonals are perpendicular. You can use the Pythagorean Theorem to find the side lengths.
WX = √202 + 122 ≈ 23.32XY = √122 + 122 ≈ 16.97Because WXYZ is a kite,
WZ = WX ≈ 23.32, and ZY = XY ≈ 16.97
12
1220
12
U
X
Z
W Y
Ex. 5: Angles of a kite Find mG and mJ in the diagram at the right.SOLUTION:
GHJK is a kite, so G ≅ J and mG = mJ.2(mG) + 132° + 60° = 360°Sum of measures of int. s of a quad. is 360° 2(mG) = 168°Simplify mG = 84° Divide each side by 2.
So, mJ = mG = 84°
J
G
H K132° 60°
What was the objective for today?Students will analyze and apply properties of
trapezoids and kites. Why? So you can measure part of a building,
as in Example 2.Mastery is 80% or better on 5-minute checks
and practice problems.
HomeworkPage 546-547# 1-34 allQuiz tomorrow 8.4-8.5Test Coming Soon