#### Solved Examples and Worksheet for Finding Missing Angles in Polygons

Q1Find the measure of A, if C = 61° and B = 60°.

A. 47°
B. 59°
C. 71°
D. 44°

Step: 1
The sum of three angles in a triangle is 180°.
Step: 2
A + B + C = 180°.
Step: 3
A + 60° + 61° = 180°
[Substitute B = 60° and C = 61°.]
Step: 4
A + 121° = 180°
[Combine like terms.]
Step: 5
A = 180° - 121°
[Subtract 121° from both sides.]
Step: 6
A = 59°
Q2Find A and B for the triangle ABC, if C = 34°.

A. 93°
B. 107°
C. 83°
D. 73°

Step: 1
From the figure, AC = BC.
Step: 2
So, ΔABC is isosceles and A = B.
Step: 3
Let A = B = x
Step: 4
In a triangle, the sum of all the angles is equal to 180°.
Step: 5
A + B + C =180°
Step: 6
x + x + 34° = 180°
[Substitute A = B = x and C = 34°.]
Step: 7
2x = 180° - 34°
[Subtract 34° from each side.]
Step: 8
2x = 146°
Step: 9
x = 146°2
[Divide each side by 2.]
Step: 10
x = 73°
Q3ΔABC is an isosceles triangle and A = 30°. Which of the following cannot be the measures of other two angles?

A. 40°, 120°
B. 75°, 75°
C. 30°, 120°
D. None of the above

Step: 1
In an isosceles triangle, two angles are equal.
Step: 2
A triangle with angle measures 30°, 30°, 120° and 30°, 75°, 75° forms an isosceles triangle.
Step: 3
Among the choices, a triangle with angle measures 30°, 40°, and 120° cannot form an isosceles triangle.
Correct Answer is :   40°, 120°
Q4Three angles of a quadrilateral are 123o, 57o, and 97o. What is the measure of the fourth angle?
A. 69.25o
B. 83o
C. 277o
D. 123o

Step: 1
Sum of the four angle measures in a quadrilateral is 360o.
Step: 2
Let x be the measure of the fourth angle.
Step: 3
x + 123o + 57o + 97o = 360o
Step: 4
x + 277o = 360o
Step: 5
x = 360o - 277o
[Subtract 277 from both sides.]
Step: 6
x = 83o
Step: 7
So, the measure of the fourth angle is 83o.
Q5mA = 50° in the parallelogram ABCD. Find the measures of B, C and D.

A. 50°, 50°, 50°
B. 130°, 50°, 50°
C. 130°, 50°, 130°
D. None of the above

Step: 1
The adjacent angles in a parallelogram are supplementary.
Step: 2
In parallelogram ABCD, A and B are adjacent angles.
Step: 3
mA + mB = 180°

Step: 4
50° + mB = 180°
[Substitute mA = 50°.]
Step: 5
B = 180° - 50°
[Subtract 50° on both sides.]
Step: 6
mB = 130°

Step: 7
In a parallelogram, opposite angles are congruent.
Step: 8
A and C are opposite angles.
Step: 9
mC = mA = 50°
Step: 10
B and D are opposite angles.
Step: 11
mD = mB = 130°
Step: 12
mB = 130°, mC = 50° and mD = 130°
Correct Answer is :   130°, 50°, 130°
Q6The sum of two angles in a triangle is 120°. Find the third angle.

A. 40°
B. 65°
C. 60°
D. 240°

Step: 1
The sum of three angles in a triangle is 180°.
Step: 2
Let x be the third angle.
Step: 3
x + Sum of the other two angles = 180°
Step: 4
x + 120° = 180°
[Substitute sum of two angles = 120°.]
Step: 5
x = 180° - 120°
[Subtract 120° from each side.]
Step: 6
x = 60°

Step: 7
The third angle is 60°.
Q7If two angles of a triangle measure 50 and 40, then find the measure of the third angle.

A. 85
B. 90
C. 95
D. 80

Step: 1
The sum of the measures of all the three angles of a triangle is 180.
Step: 2
The measure of the third angle = 180 - (50 + 40).
Step: 3
= 90
[Simplify]
Step: 4
So, the measure of the third angle is 90.
Q8Find the measure of x in the quadrilateral.

A. 121°
B. 110°
C. 109°
D. 100°

Step: 1
The given diagram is a quadrilateral with four angles.
Step: 2
In a quadrilateral, the sum of the four angles = 360°
Step: 3
So, ∠50 + ∠120 + ∠ 80 + ∠x = 360°
Step: 4
∠250 + ∠x = 360°
Step: 5
Therefore, ∠x = 360° - 250° = 110°
Q9Find the measure of the unknown angle.

A. 75°
B. 100°
C. 85°
D. 101°

Step: 1
The given diagram is a quadrilateral with four angles.
Step: 2
In a quadrilateral, the sum of the four angles = 360°.
Step: 3
So, 79° + 100° + 106° + y° = 360°.
Step: 4
y° + 285° = 360°
Step: 5
Therefore, y° = 360° - 285° = 75°.
[Simplify.]
Q10In quadrilateral PQRS, mP = 87°, mQ = 78°, and mR = 90°. Find the mS.

A. 100°
B. 105°
C. 45°
D. 205°

Step: 1
In a quadrilateral, the sum of the four angles = 360°.
Step: 2
mP + mQ + mR + mS = 360°
Step: 3
87° + 78° +90° + mS = 360°
[Substitute the given values.]
Step: 4
255° + mS = 360°
[Simplify.]
Step: 5
mS = 360° - 255° = 105°
[Subtract.]
Step: 6
So, mS is 105°.
Q11Find the measure of the unknown angle.

A. 125°
B. 128°
C. 50°
D. 232°

Step: 1
Let the unknown angle be x.
Step: 2
90° + 90° + 52° + x° = 360°
[The given figure is a quadrilateral.]
Step: 3
232° + x° = 360°
[Simplify.]
Step: 4
x° = 360° - 232° = 128°
[Subtract.]
Step: 5
So, the measure of the unknown angle is 128°.
Q12Three angles of a quadrilateral are 82°, 65°, 93°. What is the measure of the fourth angle?

A. 120°
B. 110°
C. 150°
D. 80°

Step: 1
Sum of the four angle measures in a quadrilateral is 360°.
Step: 2
Let x be the measure of the fourth angle.
Step: 3
x + 82° + 65° + 93° = 360°
Step: 4
x + 240° = 360°
Step: 5
x = 360° - 240°
[Subtract 240 from both sides.]
Step: 6
x = 120°
Step: 7
So, the measure of the fourth angle is 120°.
Q13Three angles of a quadrilateral are 120°, 55°, and 75°. What is the measure of the fourth angle?
A. 95°
B. 100°
C. 110°
D. 105°

Step: 1
Sum of the four angle measures in a quadrilateral is 360°.
Step: 2
Let x be the measure of the fourth angle.
Step: 3
x + 120° + 55° + 75° = 360°.
Step: 4
x = 360° - 250°
[Subtract 250 from both sides.]
Step: 5
x = 110°
Step: 6
So, the measure of the fourth angle is 110°.
Q14If x = 62°, then find the missing angle shown in the figure.

A. 28°
B. 45°
C. 36°
D. cannot be determined

Step: 1
The sum of all the angle measures of a triangle is 180°.
Step: 2
90° + 62° + y° = 180°
[Equate the sum of angles of the triangle to 180°.]
Step: 3
152° + y° = 180°
Step: 4
y° = 28°
[Subtract 152° from each side.]
Q15Find the measure of the third angle.

A. 120°
B. 100°
C. 90°
D. 110°

Step: 1
The sum of all the angle measures of a triangle is 180°.
Step: 2
In the triangle, the sum of all the 3 angles measures is 30° + 40° + X.
Step: 3
30° + 40° + X = 180°
[Equate the sum of all the three angles measures of the triangle to 180°.]
Step: 4
70° + X = 180°
Step: 5
X = 110°
[Subtract 70° on both sides.]
Step: 6
So, the measure of the third angle of the triangle is 110°.