Math - How to find an angle in an acute / obtuse triangle

Exterior_angle

If the measure of $\angle A=52^{\circ}$ and the measure of $\angle B= 43^{\circ}$ then what is the meausre of $\angle\theta$ ?

$95^{\circ}$

$85^{\circ}$

$82^{\circ}$

$98^{\circ}$

Not enough information to solve

Explanation

The key to solving this problem lies in the geometric fact that a triangle possesses a total of $180^{\circ}$ between its interior angles. Therefore, one can calculate the measure of $\angle C$ and then find the measure of its supplementary angle, $\angle\theta$ .

$180^{\circ}=\angle A + \angle B + \angle C$

$\angle C= 180^{\circ}- 52^{\circ} - 43^{\circ}$

$\angle C= 85^{\circ}$

$\angle C$ and $\angle\theta$ are supplementary, meaning they form a line with a measure of $180^{\circ}$ .

$\angle\theta =180^{\circ} - 85^{\circ}$

$\rightarrow 95^{\circ}$

One could also solve this problem with the knowledge that the sum of the exterior angle of a triangle is equal to the sum of the two interior angles opposite of it.

The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?

Explanation

Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

Exterior_angle

If the measure of $\angle A=56^{\circ}$ and the measure of $\angle B=47^{\circ}$ then what is the meausre of $\angle\theta$ ?

$103^{\circ}$

$77^{\circ}$

$87^{\circ}$

Not enough information to solve

$107^{\circ}$

Explanation

$180^{\circ}=\angle A + \angle B + \angle C$

$\angle C= 180^{\circ}- 56^{\circ} - 47^{\circ}$

$\angle C=77^{\circ}$

$\angle C$ and $\angle\theta$ are supplementary, meaning they form a line with a measure of $180^{\circ}$ .

$\angle\theta =180^{\circ} - 77^{\circ}$

$\rightarrow 103^{\circ}$

One could also solve this problem with the knowledge that the sum of the exterior angle of a triangle is equal to the sum of the two interior angles opposite of it.

Which of the following can NOT be the angles of a triangle?

45, 45, 90

1, 2, 177

30.5, 40.1, 109.4

45, 90, 100

30, 60, 90

Explanation

In a triangle, there can only be one obtuse angle. Additionally, all the angle measures must add up to 180.

In a given triangle, the angles are in a ratio of 1:3:5. What size is the middle angle?

$60^{\circ}$

$20^{\circ}$

$90^{\circ}$

$45^{\circ}$

$75^{\circ}$

Explanation

Since the sum of the angles of a triangle is $180^{\circ}$ , and given that the angles are in a ratio of 1:3:5, let the measure of the smallest angle be , then the following expression could be written: