QUIZ ON PROPERTIES OF QUADRILATERALS

Part A

Answer the following with the response of True or False:

1) A trapezoid has one pair of parallel sides. ____________________

True

2) In an isosceles trapezoid, the non-parallel sides are congruent. ____________________

True

3) In a rhombus, all sides are congruent. ____________________

True

4) In a rectangle, diagonals are perpendicular. ____________________

False

In square and in rhombus, the diagonals are perpendicular. For Parallelogram, rectangle, and trapeziums, the diagonals are not perpendicular.

5) In an isosceles trapezoid, opposite angles are congruent. ____________________

False

The opposite angles are supplementary.

6) All quadrilaterals are rectangles. ____________________

False

Rectangle can be considered as quadrilateral but all quadrilateral not not rectangles.

Part B

Problem 1 :

Which of the following is true about every parallelogram ?

a) All four sides are congruent

b) The diagonals are perpendicular to each other.

c) Two pairs of opposite angles are congruent

d) The consecutive angles are congruent.

Solution :

In parallelogram, opposite sides are parallel and equal.

In triangle ADC and in ABC,

∠DCA = ∠BAC (DC || AB, alternate interior angles)

∠DAC = ∠BCA  (AD || BC, alternate interior angles)

AC = AC (Common)

So, triangles ADC and ABC are congruent. Using CPCT 

∠D = ∠B

Opposite angles are equal.

Problem 2 :

Which reason can be used to prove that a parallelogram is a rhombus ?

a) The diagonals are congruent

b) The opposite angles are congruent

c) The diagonals are perpendicular

d) The opposite sides are parallel.

Solution :

In rhombus, the diagonal will be perpendicular. But in parallelogram the diagonals will not be perpendicular.

Problem 3 :

For which quadrilateral are the diagonals congruent but do not bisect each other ?

a) Parallelogram

b) Isosceles trapezoid

c) Rectangle 

d) Rhombus

Solution :

In isosceles trapezoid, the diagonals are congruent but they will not bisect each other.

Problem 4 :

For which quadrilaterals are the diagonals congruent ?

a) Rhombus   b)  Parallelogram  c) Isosceles trapezoid

d) Rectangle      e) Square

Solution :

In  Isosceles trapezoid, Rectangle and Square the diagonals are congruent.

Problem 5 :

If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral could be a

a) Trapezoid   b) Square   c) Rectangle   d) Rhombus

Solution :

In trapezoid, the diagonals do not bisect each other.

Problem 6 :

One angle of a quadrilateral is 150°and other three angles are equal. What is the measure of each of these equal angles ?

(a) 75°     (b) 85°      (c) 95°         (d) 70°

Solution :

Let x be the equal angle measures.

Sum of interior angles of quadrilateral = 360

x + x + x + 150 = 360

3x + 150 = 360

3x = 360 - 150

3x = 210

x = 210/3

x = 70

So, the equal angle measure is 70 degree, option d is correct.

Problem 7 :

If the following quadrilaterals are parallelograms, find the values of x, y, and z.

Solution :

78 + z = 180

z = 180 - 78

z = 102

Since the given quadrilateral is a parallelogram, the opposite sides will be parallel.

x = 29 (alternate interior angles)

In a triangle, the sum of interior angles is 180

78 + x + y = 180

78 + 29 + y = 180

107 + y = 180

y = 180 - 107

= 73

Problem 8 :

If the following quadrilaterals are parallelograms, find the values of x, y, and z.

Solution :

Since it is parallelogram, co-interior angles is 180

44 + x + 105 = 180

149 + x = 180

x = 180 - 149

x = 31

y = 44 (alternate interior angles)

z = 105 (opposite angles)

Problem 9 :

Use rectangle ABCD and the given information to complete the following.

i) If 𝐴𝐶 = 4𝑥 − 60 and 𝐵𝐷 = 30 − 𝑥, find 𝐵𝐷.

ii) If ∠𝐵𝐴𝐶 = 4𝑥 + 5 and ∠𝐶𝐴𝐷 = 5𝑥 − 14, find ∠𝐶𝐴𝐷.

iii) If 𝐷𝐸 = 13, find 𝐶𝐸.

Solution :

i) If 𝐴𝐶 = 4𝑥 − 60 and 𝐵𝐷 = 30 − 𝑥

In a rectangle, diagonals will be equal.

AC = BD

4x - 60 = 30 - x

4x + x = 30 + 60

5x = 90

x = 90/5

x = 18

Applying the value of x, we get

𝐵𝐷 = 30 - 18

= 12

So, the length of BD is 12.

ii) If ∠𝐵𝐴𝐶 = 4𝑥 + 5 and ∠𝐶𝐴𝐷 = 5𝑥 − 14, find ∠𝐶𝐴𝐷.

∠𝐵𝐴𝐶 + ∠𝐶𝐴𝐷 = 90

4x + 5 + 5x - 14 = 90

9x - 9 = 90

9x = 90 + 9

9x = 99

x = 11

Applying the value of x, we get

∠𝐶𝐴𝐷 = 5(11) − 14

= 55 - 14

= 41

iii) If 𝐷𝐸 = 13, find 𝐶𝐸.

In a rectangle, diagonals will bisect each other.

DE = BE 

AE = EC

13 = CE

Problem 10 :

ABCD is an isosceles trapezoid with bases 𝐴𝐵 and 𝐶𝐷, and median 𝐸𝐹. Use the given information to solve each problem.

i) If 𝐷𝐶 = 30 and 𝐴𝐵 = 42, find 𝐸𝐹.

ii) If ∠𝐴 = 5𝑥 and ∠𝐷 = 4𝑥, find the value of 𝑥.

iii)  If 𝐸𝐹 = 𝑥 + 5 and 𝐴𝐵 +𝐶𝐷 = 4𝑥 + 6, find 𝐸𝐹.

Solution :

i) If 𝐷𝐶 = 30 and 𝐴𝐵 = 42

𝐸𝐹 = (1/2) (DC + AB)

= (1/2)(30 + 42)

= (1/2) 72

= 36

So, the length of EF is 36 units.

ii) If ∠𝐴 = 5𝑥 and ∠𝐷 = 4𝑥, find the value of 𝑥.

∠𝐴 + ∠D = 180

5x + 4x = 180

9x = 180

x = 180/9

x = 20

So, the value of x is 20.

iii)  If 𝐸𝐹 = 𝑥 + 5 and 𝐴𝐵 + 𝐶𝐷 = 4𝑥 + 6, find 𝐸𝐹.

𝐸𝐹 = (1/2) (DC + AB)

x + 5 = (1/2)(4x + 6)

2(x + 5) = 4x + 6

2x + 10 = 4x + 6

2x - 4x = 6 - 10

-2x = -4

x = 2

Applying the value of x, we get

EF = 2 + 5

= 7