#### Solved Examples and Worksheet for Angle Sum Theorem of a Triangle

Q1What are the angles of a triangle which are in the ratio 4 : 5 : 6?
A. 58o, 81o, 77o
B. 38o, 55o, 72o
C. 48o, 60o, 72o
D. 96o, 60o, 97o

Step: 1
Let 4x, 5x and 6x be the angles.
Step: 2
4x + 5x + 6x = 180o
[Triangle angle sum theorem.]
Step: 3
15x = 180o
x = 180°15
x = 12o
[Divide each side by 15.]
Step: 4
So, 4x = 4 × 12o = 48o
5x = 5 × 12o = 60o
6x = 6 × 12o = 72o
[Substitute x = 12o.]
Step: 5
The angles of the triangle are 48o, 60o and 72o .
Correct Answer is :   48o, 60o, 72o
Q2If mA + mB = 90, then what all can be concluded from this?
(i) One arm of A and one arm of B are perpendicular
(ii). A and B are complementary.
(iii). mA = 90 - mB
(iv) mB = 90 - mA
(v). A and B cannot be the angles of the same triangle.

A. (i), (ii) and (v)
B. (ii), (iii), (iv) and (v)
C. All statements
D. (ii), (iii) and (iv)

Step: 1
The two angles can be with two different arms. Statement (i) need not be correct.
Step: 2
The two angles can be the angles of the same triangle since sum of the angles of a triangle is 180 degrees.
Step: 3
(ii), (iii) and (iv) are correct.
Correct Answer is :   (ii), (iii) and (iv)
Q3Find the value of 2y + 3z.

A. 200
B. 300
C. 350
D. 250

Step: 1
y + 120 = 180
[Linear pair.]
Step: 2
y = 180 - 120 = 60
[Simplify.]
Step: 3
The sum of the measures of the angles of a triangle is 180.
[Triangle angle sum theorem.]
Step: 4
60 + y + z = 180
[From step 3.]
Step: 5
60 + 60 + z = 180
[Substitute 60 for y.]
Step: 6
z = 180 - 120 = 60
[Simplify.]
Step: 7
So, 2y + 3z = 120 + 180 = 300
[Substitute 60 for y and z.]
Q4Find the values of x and y.

A. 72 and 36
B. 36 and 72
C. 36 and 36
D. 72 and 72

Step: 1
The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
[Exterior angle theorem.]
Step: 2
y = 2x
[From step 1.]
Step: 3
The sum of the measure of the angles of a triangle is 180.
[Triangle angle-sum theorem.]
Step: 4
x + y + y = 180
[From step 3.]
Step: 5
x + 2x + 2x = 180
[Substitute y = 2x.]
Step: 6
5x = 180 x = 36
[Simplify.]
Step: 7
y = 2x 2(36) = 72
[From step 2.]
Step: 8
Hence the value of x is 36 and y is 72.
Correct Answer is :   36 and 72
Q5If the angles of a triangle are in the ratio of 2 : 4 : 6, find the largest angle.

A. 90°
B. 120°
C. 60°
D. 100°

Step: 1
Since the measures of the angles of the triangle are in the ratio 2: 4: 6, the measures of the angles of triangle can be taken as 2x, 4x and 6x.
Step: 2
The sum of the measure of angles of a triangle is 180.
[Triangle angle sum theorem.]
Step: 3
2x + 4x + 6x =180
[ From steps 1 and 2 ]
Step: 4
12x= 180
[Simplify]
Step: 5
x = 15
[solve for x]
Step: 6
The measures of the angles of triangle are,
Step: 7
2x = 2(15) = 30,
Step: 8
4x=4(15) = 60 and
Step: 9
6x = 6(15) = 90
Step: 10
Therefore, the largest angle of a triangle = 90°.
Q6The measures of angles of a triangle are perfect squares. Find the difference between the largest and the smallest angles.
A. 84°
B. 6°
C. 36°
D. 48°

Step: 1
The sum of all the three angles in a triangle is 180°.
[Angle sum theorem of a triangle.]
Step: 2
The measures of angles of a triangle are perfect squares.
[Given.]
Step: 3
If the angles of a triangle are a°, b°, and c°, then a + b + c = 180°, where a, b, and c are perfect squares.
Step: 4
Let the angles be a = 16°, b = 64°, and c = 100°.
Step: 5
⇒ 16 + 64 + 100 = 180
[16 = 42, 64 = 82, and 100 = 102.]
Step: 6
Here, the largest angle is 100 and the smallest angle is 16.
Step: 7
So, the difference between the largest and the smallest angles = 100 - 16 = 84°.
Q7If 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.]
Q8What are the measures of the angles of a triangle, if they are in the ratio 5 : 6 : 7 ?

A. 100°, 60°, 95°
B. 40°, 55°, 70°
C. 60°, 81°, 75°
D. 50°, 60°, 70°

Step: 1
Let 5x, 6x and 7x be the angles.
Step: 2
Sum of the measures of the angles in a triangle = 180°
Step: 3
5x + 6x + 7x = 180°
18x = 180°
x = 180°18
x = 10°
[Divide each side by 18.]
Step: 4
5x = 5 × 10° = 50°
6x = 6 × 10° = 60°
7x = 7 × 10° = 70°
[Substitute x = 10° .]
Step: 5
The measures of the angles are 50°, 60° and 70° .
Correct Answer is :   50°, 60°, 70°
Q9Find the value of x.

A. 32
B. 42
C. 29
D. 34

Step: 1
The sum of the measures of the angles of a triangle is 180°.
[Triangle angle sum theorem.]
Step: 2
x° + (2x + 4)°+ (3x + 2)° = 180°.
Step: 3
6x = 174.
[Simplify.]
Step: 4
x = 29.
[Solve for x.]
Q10The measures of angles of ΔPQR are 2x, 3x and 4x. Find the value of x.
A. 324°
B. 400°
C. 256°
D. 200°

Step: 1
The measures of angles of ΔPQR are 2x, 3x and 4x.
[Given.]
Step: 2
The sum of the measure of angles of a triangle is 180°.
[Triangle angle sum theorem.]
Step: 3
2x + 3x + 4x = 180°
Step: 4
9x = 180°
[Simplify.]
Step: 5
x = 1809 = 20°
[Simplify.]
Step: 6
x = (20)2 = 400°
[Squaring on both sides.]
Step: 7
Therefore, x = 400°.
Q11The ratio of the angle measures in a triangle is 2 : 3 : 5. The triangle is ______.
A. an obtuse angled triangle
B. a right triangle
C. an equilateral triangle
D. an isosceles triangle

Step: 1
Since the measures of the angles of the triangle are in the ratio 2 : 3 : 5, the measures of the angles of it can be taken as 2x, 3x and 5x.
Step: 2
The sum of the measure of angles of a triangle is 180°.
[Triangle angle sum theorem.]
Step: 3
2 x + 3 x + 5 x = 180 °
Step: 4
10x = 180°
[Simplify.]
Step: 5
x = 18010 = 18°
[Simplify.]
Step: 6
2 x = 2 × 18° = 36°,
3 x = 3 × 18× = 54° and 5 x = 5 × 18° = 90°
Step: 7
So, the measures of the angles of the triangle are 36° , 54° and 90° and hence the triangle is right triangle.
Correct Answer is :   a right triangle
Q12Find the value of x in the figure shown.

A. 48°
B. 42°
C. 46°
D. 36°

Step: 1
The sum of all the angle measures of a triangle is 180°.
Step: 2
90° + 44° + x = 180°
[Equate the sum of angles of the triangle to 180°.]
Step: 3
134° + x = 180°
Step: 4
x = 180° - 134°
[Simplify.]
Step: 5
Therefore, x = 46°.
Q13In triangle ABC, BAC = 64° and ABC = 76°. What is the measure of BCA?

A. 46°
B. 30°
C. 50°
D. 40°

Step: 1
The sum of all the angle measures of a triangle is 180° .
Step: 2
So, ∠BAC + ∠ABC + ∠BCA = 180°.
Step: 3
BCA + 64° + 76° = 180°
[Substitute the given values of ∠BAC and ∠ABC.]
Step: 4
BCA + 140° = 180°
Step: 5
BCA = 180° - 140°
[Simplify.]
Step: 6
Therefore, the measure of BCA is 40°.
Q14In triangle ABC, ∠BAC = 64° and ∠ABC = 76°. What is the measure of ∠BCA?

A. 50°
B. 40°
C. 30°
D. 46°

Step: 1
The sum of all the angle measures of a triangle is 180°.
Step: 2
∠BCA + 64° + 76° = 180°
Step: 3
∠BCA + 140° = 180°
Step: 4
∠BCA = 180° - 140°
[Simplify.]
Step: 5
Therefore, the measure of ∠BCA is 40°.