use properties of trapezoids and kites
DESCRIPTION
Use Properties of Trapezoids and Kites. Goal: Use properties of trapezoids and kites. Vocabulary. Trapezoid: A trapezoid is a quadrilateral with exactly one pair of parallel sides. Bases of a trapezoid: The parallel sides of a trapezoid are the bases. - PowerPoint PPT PresentationTRANSCRIPT
Use Properties of Trapezoids and Kites
Goal: Use properties of trapezoids and kites.
Vocabulary
Trapezoid: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Bases of a trapezoid: The parallel sides of a trapezoid are the bases.
Base angles of a trapezoid: A trapezoid has two pairs of base angles. Each pair shares a base as a side.
Vocabulary (Cont.)Legs of a trapezoid: The nonparallel sides of
a trapezoid are the legs.
Isoceles trapezoid: An isosceles trapezoid is a trapezoid in which the legs are congruent.
Midsegment of a trapezoid: The midsegment of a trapezoid is the segment that connects the midpoints of its legs.Kite: A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Example 1: Use a coordinate plane
Show that CDEF is a trapezoid.
Solution
Compare the slopes of oppositesides.
4-3 1Slope of DE
4-1 3
2- 0 2 1Slope of CF
6-0 6 3
The slopes of and are the same, so DE CF DE .CF
Example 1 (Cont.)
2-4 -2Slope of EF -1
6-4 2
3-0 3Slope of CD 3
1-0 1
The slopes of and are not EF CD Ethe same, so F is
to .not CD
Because quadrilateral CDEF has exactly one pair ofparallel sides, it is a trapezoid.
Theorem 6.14
If a trapezoid is isoceles, then each pair of base angles is congruent.
If trapezoid ABCD is isosceles, then
and .A D B C
Theorem 6.15
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
If (or if ), then trapezoid ABCD is isosceles.A D B C
Theorem 6.16
A trapezoid is isosceles iff its diagonals are congruent.
Trapezoid ABCD is isosceles iff .AC BD
Example 2: Use properties of isosceles trapezoids
Kitchen A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid.
. and ,, Find MmLmNm
Solution
Step 1 Find . KLMN is an , so
and are congruent base angles, and
isosceles t
50
rapezoid
.
m N
N K
m N m K
Example 2 (Cont.)Step 2 Find . Because and are consecutive
interior angles formed by KL are intersecting two
parallel lines, they are . So,
supplementa
180 50 130 .
ry
m L K L
m L
.130 and ,130,50 So,
.130
and congruent, are they angles, base of
pair a are and Because . Find 3 Step
MmLmNm
LmMm
LMMm
Checkpoint 1
1. In Example 1, suppose the coordinates of point E are (7,5). What type of quadrilateral is CDEF? Explain.
Parallelogram; opposite pairs of sides areparallel.
Checkpoint 2
shown. trapezoidin the and ,, Find 2. DmAmCm
45,45,135 DmAmCm
Theorem 6.17: Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
).(2
1 and
,DC toparallel is MN,AB toparallel is MN
thenABCD, trapezoidof midsegment theis MN If
CDABMN
Example 3: Use the midsegment of a trapezoid
MN. Find PQRS. trapezoidof
midsegment theis MN diagram, In the
Solution
Use Theorem 6.17 to find MN.
Simplify. 12.5
SR.for 9 and PQfor 16 Substitute )916(2
1
6.17. TheoremApply )(2
1
SRPQMN
The length MN is 12.5 inches.
Checkpoint 3
3. Find MN in the trapezoid at the right.
MN = 21 ft
Theorem 6.18
If a quadrilateral is a kite, then its diagonals are perpendicular.
.ACthen
kite, a is ralquadrilate If
BD
ABCD
Theorem 6.19
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
D. not is B and C A then
,BABC and kite a is ABCD ralquadrilate If
Example 4:Apply Theorem 6.19right. at theshown kite in the Find Tm
Solution
. find to
equationan solve and Write.
. bemust T and R S, not is Q
Because angles. opposite congruent ofpair
oneexactly has QRST 6.19, TheoremBy
Tm
TmRm
Example 4 (Cont.)
Corollary to Theorem 8.1
Substitut
70 88 360
70 88 360
2 15
e m T for m R.
( ) Combine li8 360 ke terms.
101 Solve for m T.
m T m R
m T m T
m T
m T
Checkpoint 4
below. at theshown kite in theG m Find
100Gm