How to find an angle in an acute / obtuse triangle

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

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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 largest angle in an obtuse scalene triangle is degrees. The second largest angle in the triangle is $\frac{1}{3}$ the measurement of the largest angle. What is the measurement of the smallest angle in the obtuse scalene triangle?

$20^\circ$

$40^\circ$

$30^\circ$

$25^\circ$

$15^\circ$

Explanation

Since this is a scalene triangle, all of the interior angles will have different measures. However, it's fundemental to note that in any triangle the sum of the measurements of the three interior angles must equal degrees.

The largest angle is equal to degrees and second interior angle must equal:

$120\div3=40$

Therefore, the final angle must equal:

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 .

A triangle has sides of lengths 19.5, 46.8, and 50.7. Is the triangle acute, right, or obtuse?

Right

Acute

Obtuse

Explanation

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

$19.5 ^{2}+ 46.8^{2} = 380.25+2,190.24 =2,570.49$

Calculate the square of the length of the longest side:

$50.7^{2} =2,570.49$

The two quantities are equal, so by the Converse of the Pythagorean Theorem, the triangle is right.

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:

$x+3x+5x=180$

$9x=180$

$x=20$

If the smallest angle is 20 degrees, then given that the middle angle is in ratio of 1:3, the middle angle would be 3 times as large, or 60 degrees.

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