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

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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

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• 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