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 $\angle\theta$ ?

$95^{\circ}$

CORRECT

$85^{\circ}$

$82^{\circ}$

$98^{\circ}$

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?

CORRECT

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 $\angle\theta$ ?

$103^{\circ}$

CORRECT

$77^{\circ}$

$87^{\circ}$

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

CORRECT

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?

$60^{\circ}$

CORRECT

$20^{\circ}$

$90^{\circ}$

$45^{\circ}$

$75^{\circ}$

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.

In $\Delta ABC$ , , , and . To the nearest tenth, what is $m \angle A$ ?

A triangle with these sidelengths cannot exist.

CORRECT

$168.5^{\circ }$

$158.5^{\circ }$

$148.5^{\circ }$

$153.5^{\circ }$

Explanation

The sum of the two smallest sides is less than the greatest side:

By the Triangle Inequality, this triangle cannot exist.

In the triangle below, AB=BC (figure is not to scale) . If angle A is 41°, what is the measure of angle B?

A (Angle A = 41°)

Act_math_108_02

B C

CORRECT

Explanation

If angle A is 41°, then angle C must also be 41°, since AB=BC. So, the sum of these 2 angles is:

41° + 41° = 82°

Since the sum of the angles in a triangle is 180°, you can find out the measure of the remaining angle by subtracting 82 from 180:

180° - 82° = 98°

Rt_triangle_letters If angle $a=38^\circ$ and angle $c=82^\circ$ , what is the value of ?

$60^\circ$

CORRECT

$120^\circ$

$42^\circ$

$52^\circ$

$60^\circ$

Explanation

For this problem, remember that the sum of the degrees in a triangle is $180^\circ$ .

This means that $a+b+c=180^\circ$ .

Plug in our given values to solve:

$38^\circ+b+82^\circ=180^\circ$

$120^\circ+b=180^\circ$

$c=60^\circ$

The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?

$73$

CORRECT

$34$

$47$

$62$

$55$

Explanation

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

Let $x$ = the vertex angle and $2x+5$ = the base angle

So the equation to solve becomes $x+(2x+5)+(2x+5)=180$

Thus the vertex angle is 34 and the base angles are 73.

You are given a triangle with angles degrees and degrees. What is the measure of the third angle?

degrees

CORRECT

degrees

Explanation

Recall that the sum of the angles of a triangle is degrees. Since we are given two angles, we can then find the third. Call our missing angle .

We combine the like terms on the left.

Subtract from both sides.

Thus, we have that our missing angle is degrees.

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