Geometry Worksheets

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Triangle Sum Theorem

By Shyama Chandran

Learn about the triangle sum theorem and how it is calculated. Find the definition, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

he Triangle Sum Theorem is useful in calculating the angles of a triangle. In real life, for triangular shaped planes/ surfaces, knowing the triangle sum theorem helps calculate the angles for planes/ surfaces.

Where does this concept fit into the curriculum?

Middle School Trigonometry

What is the Triangle Sum Theorem?

The Triangle Sum Theorem states that the sum of the three interior angles in a triangle is always 180°.

Thus, we can express the Triangle Sum Theorem as

∠ABC + ∠BCA + ∠CAB = 180°

How to Apply the Triangle Sum Theorem

Consider a triangle ABC, such that ∠ABC = 34° and ∠BCA = 120°. Calculate ∠CAB

According to the Triangle Sum Theorem, the interior angles of a triangle add up to 180°

Therefore, ∠ABC + ∠BCA + ∠CAB = 180°

=> 34° + 120° + ∠CAB = 180°

=> 154° + ∠CAB = 180°

=> ∠CAB = 180° - 154°

=> ∠CAB = 26° (Ans)



Sample Math Problems

Question 1:

A triangle ABC has ∠ABC = 13° and ∠BCA = 36°. Calculate ∠CAB

Solution:

According to the Triangle Sum Theorem, the interior angles of a triangle add up to 180°

Therefore, ∠ABC + ∠BCA + ∠CAB = 180°

=> 13° + 36° + ∠CAB = 180°

=> 49° + ∠CAB = 180°

=> ∠CAB = 180° - 49°

=> ∠CAB = 131° (Ans)

Question 2:

An isosceles triangle has one triangle = 56°. What are the remaining angles if they are equal angles?

Solution:

According to the Triangle Sum Theorem, the interior angles of a triangle add up to 180°

Let us assume that the triangle is ABC,

where ∠ABC = 56° and ∠BCA = ∠CAB = x°

Therefore, ∠ABC + ∠BCA + ∠CAB = 180°

=> 56° + x + x = 180°

=> 56° + 2x = 180°

=> 2x = 180° - 56°

=> 2x = 124°

=> ∠BCA = ∠CAB = 62° (Ans)

Question 3:

A right-angled triangle has one angle equal to 43°. What is the value of the other angles?

Solution:

According to the Triangle Sum Theorem, the interior angles of a triangle add up to 180°

Let us assume that the triangle is ABC,

where ∠ABC = 90° (as the triangle is right angled) and ∠BCA = 43°

Therefore, ∠ABC + ∠BCA + ∠CAB = 180°

=> 90° + 43° + ∠CAB = 180°

=> 133° + ∠CAB = 180°

=> ∠CAB = 180° - 133°

=> ∠CAB = 47° (Ans)


Question 4:

Find the angles of a triangle where the second angle is twice the first angle and the third angle is one-third the first angle.

Solution:

Let us consider the first angle as x. Then the 2nd angle is 2x and the third is (1/3)x

According to the Triangle Sum Theorem, the interior angles of a triangle add up to 180°

Therefore, x + 2x+ (1/3)x = 180°

=> 3x + (1/3)x = 180°

=> (10/3)x = 180°

=> x = 180° X 3/10 = 18 X 3 = 54°

Thus 1st angle is 54°; 2nd angle is 108° and 3rd angle is 18° (Ans)


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Practice Math Problems

Question 1:

A triangle ABC has ∠ABC = 21° and ∠BCA = 57°. Calculate ∠CAB

Question 2:

An isosceles triangle has one triangle = 30°. What are the remaining angles if they are equal angles?

Question 3:

A right-angled triangle has one angle equal to 11°. What is the value of the other angles?

Question 4:

Find the angles of a triangle where the second angle exceeds the first angle by 15° and the third angle is 66° more than the second angle.

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