FINDING MISSING ANGLES OF PARALLELOGRAM

Consecutive Angles in Parallelogram

Opposite Angles in Parallelogram

Problem 1 :

From the parallelogram given below, find the value of x, y and z.

Solution :

In triangle EBC.

∠EBC + ∠BCE + ∠CEB = 180

x + 50 + 90 = 180

x + 140 = 180

x = 180 - 140

x = 40

Opposite angles are equal, x = z = 40

In triangle FDC.

∠FDC + ∠DCF + ∠CFD = 180

z + ∠DCF + 90 = 180

40 + 90 + ∠DCF = 180

130 + ∠DCF = 180

∠DCF = 180 - 130

∠DCF = 50

z + ∠DCF + y + 50 = 180

40 + 50 + y + 50 = 180

140 + y = 180

y = 180 - 140

y = 40

Problem 2 :

If one angle of a parallelogram is 24 degree less than twice the smallest angle then, find the largest angle of the parallelogram.

Solution :

Let the smallest angle be x

Larger angle measure = 2x - 24

Sum of consecutive angles = 180

x + 2x - 24 = 180

3x - 24 = 180

3x = 180 + 24

3x = 204

x = 204/3

x = 68

The largest angle measure = 2(68) - 24

= 136 - 24

= 112

So, the largest angle measure is 112.

Problem 3 :

In a parallelogram PQRS, if

∠P = (3x − 5)° and ∠Q = (2x + 15)°

then find the value of x.

Solution :

Since ∠P and ∠Q are consecutive integers, sum of the consecutive integers = 180.

∠P + ∠Q = (3x − 5) + (2x + 15)

3x + 2x - 5 + 15 = 180

5x + 10 = 180

5x = 170

x = 170/5

x = 34

Problem 4 :

If PQRS is a parallelogram, then ∠P − ∠R is

(a) 90° (b) 45° (c) 60° (d) 0°

Solution :

In any parallelogram, the opposite angles will be equal.

∠P  =  ∠R

∠P − ∠R = 0

Problem 5 :

Two adjacent angles of a parallelogram are 3x - 4 and 3x + 10. Find the angles of parallelogram.

Solution :

Adjacent angles are 3x - 4 and 3x + 10.

Sum of adjacent angles (Consecutive angles) = 180

3x - 4 + 3x + 10 = 180

6x + 6 = 180

6x = 180 - 6

6x = 174

x = 174/6

x = 29

So, the angles are 83 and 97.

Problem 6 :

One angle of a parallelogram is 60 degree. Find its opposite angle and the adjacent angle.

Solution :

Sum of adjacent angels = 180

One of the angle = 60

Adjacent angle = 180 - 60

= 120

In parallelogram, opposite angle will be equal. Opposite angle 60 degree is also 60 degree.

Problem 7 :

Find the indicated measure in parallelogram MNOP, explain your reasoning.

Solution :

a) PO

Opposite sides will be equal in parallelogram, then MN = PO = 24 units

b) QO

Diagonals will bisect each other, then OQ = QM = 14 units

c) NO

Opposite sides will be equal, then OP = NO = 26 units

d) PQ

Diagonals will bisect each other, then QN = PQ = 20.7 units

e) m∠PMN

Adjacent angles are supplementary, then

m∠PMN +  m∠MNO = 180

m∠PMN + 68 = 180

= 180 - 68

m∠PMN = 112

f)

m∠NOP = m∠NOM + m∠MOP 

m∠NOP = 180 - 68

= 112

m∠NOM + m∠MOP = 112

m∠NOM + 59 = 112

m∠NOM = 112 - 59

m∠NOM = 53

g) In parallelogram, the opposite angles will be equal. Then 

m∠OPM = 68

h) m∠NMO = 59 (Alternate interior angles are equal)

Find the value of each of the variable in the parallelogram.

Problem 8 :

Solution :

Adjacent angles are supplementary.

3v + 66 = 180

3v = 180 - 66

3v = 114

v = 114/3

v = 38

u + 3v = 180

u + 3(38) = 180

u + 114 = 180

u = 180 - 114

u = 66

Problem 9 :

Solution :

Opposite sides are equal in parallelogram.

5a - 9 = 3a + 5

5a - 3a = 5 + 9

2a = 14

a = 14/2

a = 7

Adjacent angles are supplementary.

Opposite angles are equal.

b + 84 = 3b

b - 3b = -84

-2b = -84

b = 41

Problem 10 :

Find the value(s) of the variable(s) in each parallelogram

Solution :

Since opposite angles are equal, 

6a + 10 = 130

6a = 130 - 10

6a = 120

a = 120/6

a = 20

So, the value of a is 20.

Problem 11 :

Two consecutive angles in a parallelogram have measures x + 5 and 4x - 10. Find the measure of the smaller angle.

Solution :

Sum of adjacent angles = 180

x + 5 + 4x - 10 = 180

5x - 5 = 180

5x = 180 + 5

5x = 185

x = 185/5

x = 37

Applying the value of x, we get

x + 5 ==> 37 - 5 ==> 32 degree

4x - 10 ==> 4(37) - 10 ==> 148 - 10

= 138

So, the smaller angle measure is 32 degree.