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)
