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Example of finding angles in a triangle
ABC is a triangle. Extended CB to D to make the angle ABD = 3 times of the angle A. Find the angle A, angle C and the angle ABC.
interior angles of a triangle
Let the angle A be x and the angle C be y
since AB = AC (Given), so the angle C = angle ABC = y
Principles used: If two sides of a triangle are equal, then the triangle is an isosceles triangle. In an isosceles triangle, its two base angles are equal.
Given: the angle ABD = 3 times the angle A
since extended CB to D, so the angle ABD is an exterior angle of the triangle ABC.
angle ABD = angle A + angle C (exterior angle of a triangle theorem)
so 3x = x + y (exterior theorem)
In triangle ABC, angle A + angle C + angle ABC = 180o
x + y + y = 180o (the sum of the interior angles in a triangle is 180o
we get two equations:
3x = x + y ...equation1
x + y + y = 180o ...equation2
from equation1: 2x = y
from equation2: x + 2y = 180o
substitute y = 2x into the equation x + 2y = 180o
x + 2(2x) = 180o
x + 4x = 180o
5x = 180o
x = 180o/5 = 36o
y = 2x = 2(36o) = 72o
so the angle A is 36o, angle C is 72o and the angle ABC is also 72o.