TRAPEZOID
and
KITE
P
TRAPEZOID
Q
S
R
• a quadrilateral with one pair of parallel sides
which are called the bases
•non-parallel sides are called the legs
•the angles on both ends of each base are called
base angles
MEDIAN/MIDSEGMENT/MIDLINE
The line segment joining the midpoints of the legs of
the trapezoid is called the median or midsegment.
Median of a trapezoid is parallel to its bases.
Median of a trapezoid half the sum of lengths of the
bases.
ISOSCELES TRAPEZOID
- a trapezoid with
congruent legs
PROPERTIES:
1. each pair of base angles are congruent
2. diagonals are congruent
CONDITIONS FOR A TRAPEZOID TO
BE ISOSCELES
1. If a pair of base angles of a trapezoid are
congruent, then the trapezoid is isosceles.
2. If the diagonals of a trapezoid are
congruent, then the trapezoid is isosceles.
ABCD is an isosceles trapezoid,
m∠𝐴 = (3𝑥 + 40)° and m∠𝐷 = (𝑥 + 60)°.
Find m∠𝐵.
B
A
C
D
C
D
ABCD is an isosceles trapezoid with
median 𝑋𝑌 and diagonals 𝐴𝐶 and 𝐵𝐷.
X
Y
A
B
1. If AC= 5 (x-10) + 10 and BD= 6 (x-10), what is AC and BD?
AC= BD
5 (x-10) + 10 = 6 (x-10),
5x – 50 +10 = 6x-60
5x – 40 = 6x – 60
5x- 40 + 40 = 6x – 60 + 40
5x = 6x – 20
5x- 6x = 6x-6x - 20
-x = - 20
x=20
AC = 5 (x-10) + 10
AC = 5 (20-10) +10
AC = 5 (10) + 10
AC = 50+ 10
AC = 60
BD = AC
BD = 60
C
D
ABCD is an isosceles trapezoid with
median 𝑋𝑌 and diagonals 𝐴𝐶 and 𝐵𝐷.
X
Y
A
B
2. If AD= 4 (x-5) and BC= 2x+5, what is AD and BC?
AD = BC
4 (x-5) = 2x+5
4x – 20 = 2x + 5
4x – 20 + 20 = 2x + 5 + 20
4x = 2x + 25
4x – 2x = 2x – 2x + 25
2x = 25
25
x= 2
BC = 2x+5
25
BC = 2 ( ) + 5
2
BC = 25 + 5
BC = 30
AD = BC
AD = 30
D
ABCD is an isosceles trapezoid with
median 𝑋𝑌 and diagonals 𝐴𝐶 and 𝐵𝐷.
X
A
3. If 𝑚∠𝐷𝐴𝐵 = 2𝑥 − 30 and 𝑚∠𝐶𝐵𝐴 = 3 𝑥 − 20 − 10, what is
𝑚∠𝐷𝐴𝐵 and 𝑚∠𝐶𝐵𝐴?
𝑚∠𝐷𝐴𝐵 = 𝑚∠𝐶𝐵𝐴
2𝑥 − 30 = 3 𝑥 − 20 − 10
2x – 30 = 3x – 60 – 10
2x – 30 = 3x – 70
2x – 30 + 30 = 3x – 70 + 30
2x = 3x – 40
2x – 3x = 3x – 3x – 40
-x = - 40
x = 40
𝑚∠𝐷𝐴𝐵
𝑚∠𝐷𝐴𝐵
𝑚∠𝐷𝐴𝐵
𝑚∠𝐷𝐴𝐵
= 2𝑥 − 30
= 2(40) − 30
= 80 − 30
= 50°
𝑚∠𝐶𝐵𝐴 = 𝑚∠𝐷𝐴𝐵
𝑚∠𝐶𝐵𝐴 = 50°
C
Y
B
WXYZ is an isosceles trapezoid with 𝐴𝐵 as the median.
m∠𝑍 = 60° , 𝑀𝑌 ⊥ 𝑊𝑍 , AB=9, YZ=5 and XY=8.
Find WY.
X
A
W
Y
B
Z
5 3
Given:
SUPE is an isosceles trapezoid with bases 𝑆𝑈 and 𝑃𝐸 and
diagonals 𝑆𝑃 and 𝑈𝐸.
Prove: ∠1 ≅ ∠2.
STATEMENTS
REASONS
S
1
2
1. SUPE is an isosceles
trapezoid with bases 𝑆𝑈 and
𝑃𝐸 and diagonals 𝑆𝑃 and 𝑈𝐸.
U
E
3. ∠𝐸𝑆𝑈 ≅ ∠𝑃𝑈𝑆
2. Legs of isosceles trapezoid are
congruent.
3. Base angles of an isosceles
trapezoid are congruent.
4. 𝑆𝑈 ≅ 𝑆𝑈
4. Reflexivity
2. 𝑆𝐸 ≅ 𝑈𝑃
R
P
1. Given
5.∆𝑃𝑈𝑆 ≅ ∆𝐸𝑆𝑈
6. ∠1 ≅ ∠2
5. SAS Postulate
6. CPCTC.
KITE
2 distinct pairs of consecutive congruent sides.
• One diagonal is the ⊥ bisector of the other.
• Non-vertex angles are congruent.
• One diagonal bisects both vertex angles.
Vertex Angles
Non-vertex Angles
Complete the Statement
1.
2.
3.
4.
5.
6.
7.
8.
𝐸𝐹 ≅
𝐹𝐺 ≅
𝐹𝐼 ≅
∠𝐹 ≅
F
∠𝐹𝐸𝐺 ≅
∠𝐻𝐺𝐸 ≅
𝐸𝐺 ⊥
E
I
𝐸𝐺 bisects _____, _____, _____
H
G
In kite WXYZ, mWXY = 104°, and
mVYZ = 49°. Find each measure.
1. mVZY
2. mVXW
X
Y
V
3. mXWZ
Z
W
Find the perimeter of kite ABCD.
3
4
8