Transcript Use Properties of Trapeziods and Kites

Warm-Up
• Pg 537-539
• 4,10,20,23,43,49
• Answers to evens:
•
•
•
•
•
6.Always
12. Always
36. About 6
42. About 8.3
48. 2
Use Properties of Trapezoids and
Kites
8.5
Trapezoid
• A quadrilateral with
exactly one pair of
parallel sides, called
bases.
Diagonals
• If a trapezoid is isosceles,
then each pair of base
angles is congruent.
• If a trapezoid has a pair of
congruent base angles, then
it is an isosceles trapezoid.
• A trapezoid is isosceles if
and only if its diagonals are
congruent.
EXAMPLE 1
Use a coordinate plane
Show that ORST is a trapezoid.
SOLUTION
Compare the slopes of
opposite sides.
Slope of RS =
Slope of OT =
4–3
2–0=
2–0
4–0=
1
2
1
2
=
2
4
The slopes of RS and OT are the same, so RS
OT .
EXAMPLE 2
Use properties of isosceles trapezoids
Arch
The stone above the arch in the
diagram is an isosceles trapezoid.
Find m K, m M, and m J.
SOLUTION
STEP 1
Find m K. JKLM is an
isosceles trapezoid, so K
and L are congruent base
angles, and m K = m L= 85o.
Use properties of isosceles trapezoids
EXAMPLE 2
STEP 2
Find m M. Because L and M are consecutive
interior angles formed by LM intersecting two parallel
lines, they are supplementary.
So, m M = 180o – 85o = 95o.
STEP 3
Find m J. Because
J and
M are a pair of base
angles, they are congruent, and m J = m M = 95o.
ANSWER
So, m
J = 95o, m
K = 85o, and m
M = 95o.
Midsegment
• The Midsegment is
parallel to both bases
and half the length of
the sum of the bases,
Use the midsegment of a trapezoid
EXAMPLE 3
In the diagram, MN is the midsegment of trapezoid
PQRS. Find MN.
SOLUTION
Use Theorem 8.17 to find MN.
MN =
1
2
(PQ + SR)
1
= 2 (12 + 28)
= 20
ANSWER
Apply Theorem 8.17.
Substitute 12 for PQ and
28 for XU.
Simplify.
The length MN is 20 inches.
GUIDED PRACTICE
for Examples 2 and 3
In Exercises 3 and 4, use the diagram of trapezoid EFGH.
3. If EG = FH, is trapezoid EFGH isosceles?
Explain.
ANSWER
yes, Theorem 8.16
GUIDED PRACTICE
for Examples 2 and 3
4. If m
HEF = 70o and m FGH = 110o, is
trapezoid EFGH isosceles? Explain.
SAMPLE ANSWER
Yes;
m EFG = 70° by Consecutive Interior Angles
Theorem making EFGH an isosceles trapezoid
by Theorem 8.15.
for Examples 2 and 3
GUIDED PRACTICE
5. In trapezoid JKLM, J and M are right angles,
and JK = 9 cm. The length of the midsegment NP
of trapezoid JKLM is 12 cm. Sketch trapezoid
JKLM and its midsegment. Find ML. Explain your
reasoning.
ANSWER
J
N
M
9 cm K
12 cm P
L
1
15 cm; Solve 2 ( 9 + x ) = 12 for x to find ML.
Kites
• A quadrilateral that has two
pairs of consecutive
congruent sides, but
opposite sides are not
congruent.
• Diagonals are
perpendicular.
• Exactly one pair of opposite
angles are congruent.
Apply Theorem 8.19
EXAMPLE 4
Find m
D in the kite shown at the right.
SOLUTION
By Theorem 8.19, DEFG has exactly
one pair of congruent opposite angles.
Because E
G,
D and F must
be congruent. So, m D = m
F. Write
and solve an equation to find m D.
Apply Theorem 8.19
EXAMPLE 4
m
D+m
F +124o + 80o = 360o
Corollary to Theorem 8.1
m
D+m
D +124o + 80o = 360o
Substitute m
2(m
D) +204o = 360o
m
D = 78o
D for m
Combine like terms.
Solve for m
D.
F.
GUIDED PRACTICE
6.
for Example 4
In a kite, the measures of the angles are 3xo, 75o,
90o, and 120o. Find the value of x. What are the
measures of the angles that are congruent?
ANSWER
25; 75o
Classwork
• Pg 546-547
• 4,8,12,14,18,22,26
Homework
• Pg 546-547
• 3-27 odd