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7.5 NOTES Trapezoids and Kites
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LESSON 7.5 - Trapezoids and Kites
• A TRAPEZOID is a quadrilateral with exactly one pair of parallel sides.
• The parallel sides are called the BASES.
• The nonparallel sides are called the LEGS.
• If the legs of the trapezoid are congruent, then the trapezoid is ISOSCELES.
• A trapezoid has two pairs of BASE ANGLES.
In the diagram, C and D are one pair of base angles, and A and B are the other pair.
• Each pair of non‑base angles ( A & D and B & C) are supplementary by the Consecutive Interior Angles Thm.
A
D C
B
•
TRAPEZOID BASE ANGLES THEOREM
If ABCD is isosceles, then A ≅ B and C ≅ D
If a trapezoid is isosceles, then each pair of base angles are congruent.
A
D C
B
TRAPEZOID BASE ANGLES CONVERSE
If A ≅ B then ABCD is isosceles.
If a trapezoid has a pair of congruent angles, then it is isosceles. A
D C
B
ISOSCELES TRAPEZOID DIAGONALS THEOREM
ABCD is isosceles iff AC ≅ BD
A trapezoid is isosceles if and only if its diagonals are congruent. A
D C
B
Given that TRAP is an isosceles trapezoid, find
the measure of each missing angle.
m∠A = m∠P
m∠A = 110°
m∠T + m∠P = 180°
m∠T + 110° = 180°
m∠T = 70°
m∠R = m∠T
m∠R = 70°
7.5 NOTES Trapezoids and Kites
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Given that ZOID is an isosceles trapezoid, find
the measure of each angle.
m∠Z = m∠D
8x - 8 = 6x + 18
2x - 8 = 18
2x = 26
x = 13
Given that ZOID is a trapezoid, find the values
of each variable.
m∠Z + m∠D = 180°
y° + 54° = 180°
y = 126°
m∠O + m∠I = 180°
97° + x° = 180°
x = 83°
Given that TRAP is a trapezoid, find the values
of each variable.
m∠P + m∠A = 180°
x° + 90° = 180°
x = 90°
m∠T + m∠R = 180°
y° + 81° = 180°
y = 99°
Prove that HIJK is a trapezoid.
8 - 64 - 0
24
12
mHK = = =
3 - 09 - 3
36
12
mIJ = = =
Use the Slope Formula to show that one pair of sides are parallel.
• The MIDSEGMENT of a trapezoid is the segment that connects the midpoints of its legs.
MIDSEGMENT OF A TRAPEZOID THEOREM
MS AB DC
MS = (AB + DC)
The midsegment of a trapezoid is parallel to both bases and its length is one half the sum of the lengths of the bases.
Find the measure of the midsegment MS.
MS = (15 + 9)
MS = (24)
MS = 12
12
12
7.5 NOTES Trapezoids and Kites
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Find the measure of the midsegment MS.
MS = (7 + 4)
MS = (11)
MS = 5.5
12
12
• A KITE is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
PERPENDICULAR DIAGONALS OF A KITE THEOREM
If KITE is a kite, then KT IE
If a quadrilateral is a kite, then its diagonals are perpendicular.
If KITE is a kite, then I ≅ E and K ≅ T
OPPOSITE ANGLES OF A KITE THEOREM
If a quadrilateral is a kite, then one pair of opposite angles is congruent.
Find the measures of the missing angles.
m S + m T + m U + m V = 360°
102° + 89° + 102° + m V = 360°
293° + m V = 360°
m V = 67°
m S = m U
Find the measures of the missing angles.
m J + m L + m K + m M = 360°
m K = m M
36° + 74° + m K + m M = 360°
36° + 74° + m K + m K = 360°
110° + 2(m K) = 360°
2(m K) = 250°
m K = 125°
m M = 125°
Find the lengths of each side of the kite.
PYTHAGOREAN THEOREM:
a2 + b2 = c2
(a and b are legs of a right triangle, and c is the hypotenuse)
QR2 + MR2 = QM2
R
202 + 82 = QM2
400 + 64 = QM2
464 = QM2
QM = 21.5QP = 21.5
RM2 + RN2 = MN2
82 + 92 = MN2
64 + 81 = MN2
145 = MN2
MN = 12.04
NP = 12.04
7.5 NOTES Trapezoids and Kites
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HOMEWORK:
7.5 Worksheet ‑ Trapezoids and Kites