Solved Examples and Worksheet for Proving Quadrilateral is a Parallelogram

Q1The length of AB is 3 in. and m∠A = 55o in a parallelogram ABCD. Which of the following measures cannot be found using the properties of parallelograms?

A. m∠C
B. length of BC
C. length of CD
D. m∠B

Solutions
Share 
Step: 1
If one angle of the parallelogram is known, we can find all other angles and if length of one side is known, we can find the measure of opposite side since opposite sides of parallelogram are equal in measures.
Step: 2
Length of side AB = 3 in., therefore, CD = 3 in.
  [Since AB and CD are opposite sides.]
Step: 3
We cannot find the length of the sides BC and AD.
Correct Answer is :   length of BC
Q2The vertices of a quadrilateral are A(11, 16), B(4, 17), C(7, 10), D(14, 9). Which of the following is correct?
A. ABCD is not a parallelogram
B. ABCD is a kite
C. ABCD is a parallelogram
D. ABCD is a trapezoid

Solutions
Share 
Step: 1
Mid point of diagonal AC¯ = (11 + 72, 16 + 102) = (9, 13)
  [Use mid point formula.]
Step: 2
Mid point of diagonal BD¯ = (4 + 142, 17 + 92) = (9, 13)
  [Use mid point formula.]
Step: 3
So, the diagonals of the quadrilateral bisect each other, and hence ABCD is a parallelogram.
Correct Answer is :   ABCD is a parallelogram
Q3Find the values of x, y for the parallelogram shown.

A. x = 3, y = 3
B. x = 3, y = 2
C. x = 3, y = 4
D. x = 3, y = 6

Solutions
Share 
Step: 1
In a parallelogram opposite sides are congruent.
Step: 2
3x- 4 = 5x- 10 and 2y = 6
  
Step: 3
x = 3 and y = 3
  [Solve for x.]
Correct Answer is :   x = 3, y = 3
Q4Find the values of P and Q in the parallelogram.

A. 3, 4
B. 3, - 4
C. 2, 5
D. - 4, 3

Solutions
Share 
Step: 1
In a parallelogram diagonals bisect each other.
Step: 2
2P = 6 and Q + 3 = 2Q - 1
  [From the figure.]
Step: 3
P = 3 and Q = 4
  [Solve for P,Q.]
Correct Answer is :   3, 4
Q5Find the measures of the angles of the parallelogram.


A. x = 100°, y = 80°, z = 100°
B. x = 80°, y = 80°, z = 80°
C. x = 100°, y = 100°, z = 100°
D. x = 80°, y = 100°, z = 80°

Solutions
Share 
Step: 1
Since opposite angles of a parallelogram are congruent, we have y = 80° and x = z
Step: 2
Since adjacent angles of a parallelogram are supplementary, x + y = 180°
Step: 3
x + 80° = 180°
  [Substitute y = 80°.]
Step: 4
x = 180° - 80° = 100°
  [Solve for x.]
Step: 5
So, z = x = 100°
Correct Answer is :   x = 100°, y = 80°, z = 100°
Q6Find the value of x in the parallelogram shown.


A. 60
B. 80
C. 70
D. 50

Solutions
Share 
Step: 1
Since in a parallelogram opposite angles are congruent, 2x - 60 = x + 20.
Step: 2
x = 80.
  [Solve for x.]
Correct Answer is :   80
Q7In a parallelogram ABCD, ∠ADC measures 83°. What are the measures of the other angles ?

A. ∠DCB = 97°, ∠CBA = 97° ∠BAD = 83°
B. ∠DCB = 97°, ∠CBA = 83° ∠BAD = 97°
C. ∠DCB = 97°, ∠CBA = 83° ∠BAD = 83°
D. ∠DCB = 83°, ∠CBA = 97° ∠BAD = 83°

Solutions
Share 
Step: 1
The opposite angles in a parallelogram are equal.
Step: 2
∠ADC = ∠CBA = 83°
  [∠ADC and ∠CBA are opposite angles.]
Step: 3
Let ∠BAD = ∠DCB = x°
  [∠BAD and ∠DCB are opposite angles.]
Step: 4
The sum of all the four angles in a parallelogram = 360°
Step: 5
∠ADC + ∠DCB + ∠CBA + ∠BAD = 360°
Step: 6
83° + x + 83° + x = 360°
  [Substitute ∠BAD = ∠DCB = x.]
Step: 7
166° + 2x = 360°
  [Combine like terms.]
Step: 8
2x = 360° - 166°
  [Subtract 166° on both sides.]
Step: 9
2x = 194°
  
Step: 10
x = 97°
  [Divide by 2 on both sides.]
Step: 11
Therefore, ∠DCB = 97°, ∠CBA = 83° and ∠BAD = 97°.
Correct Answer is :   ∠DCB = 97°, ∠CBA = 83° ∠BAD = 97°
Q8What is the value of x and y in the parallelogram ABCD ?

A. x = 43.6°, y = 45.4°
B. x = 40°, y = 40°
C. x = 40°, y = 46°
D. x = 37.6°, y = 43.6°

Solutions
Share 
Step: 1
∠A + ∠B = 180°
  [Supplementary angles.]
Step: 2
3x - 24 + 2y + 4 = 180
Step: 3
3x + 2y = 200 ---------- (1)
  [Simplify.]
Step: 4
∠B + ∠C = 180°
  [Supplementary angles.]
Step: 5
2y + 4 + 2x + 16 = 180
Step: 6
2x + 2y = 160 ---------- (2)
  [Simplify.]
Step: 7
x = 40
  [Subtract (2) from (1).]
Step: 8
3(40) + 2y = 200
  [Substitute the value of x in eq (1).]
Step: 9
2y = 200 - 120 ⇒ y = 40
  [Simplify.]
Step: 10
Hence the value of x = 40 and y = 40.
Correct Answer is :   x = 40°, y = 40°
Q9X and Y are the midpoints of AB and CD of the parallelogram ABCD. AY cuts DX at M and BY cuts CX at N. Then MXNY is ___________

A. a rhombus
B. a parallelogram
C. a square
D. a rectangle

Solutions
Share 
Q10Calculate the variables x and y.

A. x = 67, y = 47
B. x = 4, y = 4
C. x = 4, y = 6
D. x = 2, y = 2

Solutions
Share 
Step: 1
x + y = 2y - 2
  [Since, the diagonal bisects each other.]
Step: 2
x = y - 2
  [Simplify.]
Step: 3
3x = 2y ⇒ 3(y - 2) = 2y
  [Substitute the value of x,]
Step: 4
y = 6
  [Simplify.]
Step: 5
x + y = 2y - 2
Step: 6
x + 6 = 2 (6) - 2
  [Subtracting value of y.]
Step: 7
x = 4
  [Simplify.]
Correct Answer is :   x = 4, y = 6

HighSchool Math