﻿ Trapezoids and Kites Worksheet | Problems & Solutions

# Trapezoids and Kites Worksheet

Trapezoids and Kites Worksheet
• Page 1
1.
ABCD is an isosceles trapezoid. AP = CD. Select the correct statement / statements.
1. $m$$\angle$CBP = $m$$\angle$CPB
2. $m$$\angle$ADC = $m$$\angle$CPA
3. $m$$\angle$PCB = $m$$\angle$PCD

 a. 1 and 3 only b. All are correct c. 1 and 2 only d. 2 and 3 only

#### Solution:

AP = DC, AP || DC
[Given.]

APCD is a parallelogram.
[Step 1.]

[Step 2.]

[Given that ABCD is an isosceles trapezoid.]

ΔCPB is an isosceles triangle.
[CP = CB, steps 3 & 4.]

mCPB = mCBP
[Base angles.]

[Step 2.]

mPCB will be equal to mPCD only if ΔCPB is an equilateral triangle .

Only statements 1 and 2 are correct.

Correct answer : (3)
2.
Find the height of the given trapezoid if $a$ = 13.2 cm, $b$ = 19.8 cm, and $c$ = 5.5 cm.

 a. 5.5 cm b. 3.3 cm c. 4.4 cm d. 14.4 cm

#### Solution:

AE ^CD, BF ^CD, EF = 13.2 cm, DE = CF = 3.3 cm
[Figure.]

Considering the triangle AED, 3.32 + AE2 = 5.52
[Pythagorean theorem.]

AE = 4.4 cm
[Simplify.]

Correct answer : (3)
3.
Find the measures of $m$$\angle$A , $m$$\angle$B, $m$$\angle$D in the isosceles trapezoid.

 a. $m$$\angle$A = 45; $m$$\angle$B = 135; $m$$\angle$D = 90 b. $m$$\angle$A = 135; $m$$\angle$B = 135; $m$$\angle$D = 45 c. $m$$\angle$A =135; $m$$\angle$B=90; $m$$\angle$D = 135 d. $m$$\angle$A =135; $m$$\angle$B = 135; $m$$\angle$D = 90

#### Solution:

mC = mD = 45
[ABCD is an isosceles trapezoid.]

Because C and A are consecutive interior angles formed by parallel lines, they are supplementary. mA + mC = 180
[AB || CD, AC transversal.]

mA =180 - 45 = 135
[Solve for A.]

mB = mA = 135
[ABCD is an isosceles trapezoid.]

Correct answer : (2)
4.
Using the mid point theorem find the length of $\stackrel{‾}{\mathrm{RS}}$. In the picture, $\stackrel{‾}{\mathrm{PQ}}$ || $\stackrel{‾}{\mathrm{RS}}$ || $\stackrel{‾}{\mathrm{TU}}$. R and S are the mid points of $\stackrel{‾}{\mathrm{PT}}$ and $\stackrel{‾}{\mathrm{QU}}$. [Given $a$ = 20 units and $b$ = 30 units.]

 a. 40 units b. 25 units c. 50 units d. 15 units

#### Solution:

RS = 1 / 2 (PQ + TU)
[Mid segment theorem.]

RS = 1 / 2 (20 + 30)
[Substitute.]

RS = 25 units
[Simplify.]

Correct answer : (2)
5.
In the picture $\stackrel{‾}{\mathrm{MN}}$ || $\stackrel{‾}{\mathrm{PQ}}$ || $\stackrel{‾}{\mathrm{RS}}$. P and Q are the mid points of $\stackrel{‾}{\mathrm{MR}}$ and $\stackrel{‾}{\mathrm{NS}}$. If $a$ = 6.2 and $b$ = 12.2, find the length of $\stackrel{‾}{\mathrm{RS}}$.

 a. 24 cm b. 12.3 cm c. 18.2 cm d. 18.4 cm

#### Solution:

PQ = 1 / 2 (MN + RS)
[Midsegment theorem.]

MN / 2 + RS / 2 = PQ 6.2 / 2 + RS / 2 = 12.2 cm
[Substitute.]

RS = 18.2 cm
[Simplify.]

Correct answer : (3)
6.
Find the variable $x$ in the figure, if $\stackrel{‾}{\mathrm{AB}}$ || $\stackrel{‾}{\mathrm{CD}}$ || $\stackrel{‾}{\mathrm{EF}}$ where C and D are the mid points of $\stackrel{‾}{\mathrm{AE}}$ and $\stackrel{‾}{\mathrm{BF}}$.

 a. $x$ = 6 (or) 3 b. $x$ = 6.7 (or) 0.3 c. $x$ = 5 (or) 4 d. $x$ = 18 (or) 1

#### Solution:

1 / 2 (8 - x + x² + 10) = 4x
[According to mid segment theorem.]

x2- 9 x + 18 = 0
[Simplify.]

= (x - 6)(x - 3)
[Factor.]

x = 6 or 3
[Simplify.]

Correct answer : (1)
7.
Find the measure of the angles P and R, if $\stackrel{‾}{\mathrm{RQ}}$ || $\stackrel{‾}{\mathrm{PS}}$.

 a. $m$$\angle$P = 90, $m$$\angle$R = 138 b. $m$$\angle$P = 90, $m$$\angle$R = 128 c. $m$$\angle$P = 90, $m$$\angle$R = 160 d. $m$$\angle$P = 90, $m$$\angle$R = 118

#### Solution:

PS || RQ
[Given.]

mR + mS = 180
[Same side interior angles.]

Therefore mR = 180 - 52 = 128

R + Q + P + S = 360°
[Sum of angles in a quadrilateral is 360°.]

mP = 90

Correct answer : (2)
8.
The bases of a trapezoid are 16 cm and 5 cm. The height of the trapezoid is 18 cm. What is the area of the trapezoid?
 a. 85 cm² b. 10 cm² c. 199 cm2 d. 189 cm²

#### Solution:

Area of the trapezoid = sum of the bases2 × height

16+52 × 18
[From step1.]

= 189 cm²
[Simplify.]

Correct answer : (4)
9.
Two bases of a trapezoid are 9.3 cm and 5.3 cm. The area of that trapezoid is 45.99 square centimeter. Find the height of the trapezoid.
 a. 12.6 cm b. 3.15 cm c. 6.3 cm d. 7.30 cm

#### Solution:

Area = 45.99 sqcm
[Given.]

Average of bases = 14.6 / 2.
[Given.]

Area = average of bases × height

Height = AreaAverage of the bases

= 45.9914.62
[Substitute.]

= 6.3 cm
[Simplify.]

Correct answer : (3)
10.
Which of the following is correct?
 a. The diagonals of a kite bisect each other. b. The diagonals of a kite are perpendicular to each other. c. The diagonals of an isosceles trapeziod divide it into 4 congruent triangles. d. A trapezoid has two pair of parallel lines.

#### Solution:

The diagonals of a kite does not bisect each other.

A trapezoid does not have two pair of parallel lines.

The diagonals of a kite are perpendicular, which is true.

Diagonals of an isosceles trapezoid divide it into 4 triangles, but all of them are not congruent. Only triangles with one side as one of the congruent sides of the trapeziod will be be congruent.

So the correct answer is, " The diagonals of a kite are perpendicular".

Correct answer : (2)

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