The degree measures of the angles A and B are related as 36: 72. Find the degree measure of the angle C

The degree measures of the angles A and B are related as 36: 72. Find the degree measure of the angle C, if it is known that the degree measures of all angles of the triangle are expressed in integers

By condition, ∠А: ∠В = 36: 72. Introducing the proportionality coefficient, we obtain that:

∠А = 36x °;

∠В = 72x °.

We do not know the third corner of the triangle: ∠C = y °.

By the theorem on the sum of the interior angles of a triangle, the following equation will be obtained:

36x ° + 72x ° + y ° = 180 °;

From where we get:

108x ° = 180 ° – y °;

By condition, each angle is expressed as an integer number of degrees, which means 108x ° can be equal to:

108 °, 126 °, etc.

From the equation, it is clear that 108x ° can only be equal to 108 °.

108x ° = 108 °;

x = 1 °;

∠А = 36 °;

∠В = 72 °.

A ∠C = y ° = 180 ° – 36 ° – 72 ° = 72 °.

This triangle is isosceles, AB = AC.

Answer: ∠С = 72 °