HSPT Math - How to find the measure of an angle

What angle is complement to $\frac{\pi}{6}$ ?

$\frac{\pi}{3}$

$\frac{\pi}{9}$

$\frac{\pi}{4}$

$\frac{\pi}{6}$

$\frac{2\pi}{3}$

Explanation

The complement to an angle is ninety degrees subtract the angle since two angles must add up to 90. In this case, since we are given the angle in radians, we are subtracting from $\frac{\pi}{2}$ instead to find the complement. The conversion between radians and degrees is: $\pi \textup{ radians } = 180 \textup{ degrees}$

$\frac{\pi}{2}-\frac{\pi}{6}$

Reconvert the fractions to the least common denominator.

$\frac{\pi}{2}-\frac{\pi}{6} = \frac{3\pi}{6}-\frac{\pi}{6} = \frac{2\pi}{6}$

Reduce the fraction.

$\frac{2\pi}{6} = \frac{\pi}{3}$

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.

Varsity_question

AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is , what is the measure of angle 2?

$48^{\circ}$

$132^{\circ}$

$64^{\circ}$

$180^{\circ}$

$24^{\circ}$

Explanation

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.

Lines

Examine the above diagram. What is ?

Explanation

By angle addition,

$\left ( x - 13\right )+ 100 + (y + 25) = 180$

$x - 13\right+ 100 + y + 25 = 180$

$x + y - 13\right+ 100+ 25 = 180$

What is the sum of the interior angles of a triangle?

Explanation

The sum of the three interior angles of a triangle is degrees.

Spinner target 2

The above diagram shows a spinner. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A player spins the above spinner. What is the probability that the spinner will stop while pointing inside a red region?

$\frac{7}{24}$

$\frac{2}{7}$

$\frac{11}{24}$

$\frac{5}{13}$

Explanation

The size of the regions does not matter here. What matters is the angle measurement, or, equivalently, what part of a circle each sector is.

The two smaller red regions each comprise one fourth of one fourth of a circle, or

$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$ circle.

The two larger red regions each comprise one third of one fourth of a circle, or

$\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$ circle.

Therefore, the total angle measure comprises

$\frac{1}{16} + \frac{1}{16} + \frac{1}{12} + \frac{1}{12}$

$=\frac{3}{48} + \frac{3}{48} + \frac{4}{48} + \frac{4}{48}$

$= \frac{14}{48} = \frac{7}{24}$

of a circle.

This makes $\frac{7}{24}$ the correct probability.

$\angle 1$ and $\angle 2$ are supplementary; $\angle 1$ and $\angle 3$ are complementary.

$m \angle 2 = 100 ^{\circ }$ .

What is $m \angle 3$ ?

$10^{\circ }$

$80^{\circ }$

$100^{\circ }$

$50^{\circ }$

Explanation

Supplementary angles and complementary angles have measures totaling $180^{\circ }$ and $90^{\circ }$ , respectively.

$m \angle 2 = 100 ^{\circ }$ , so its supplement $\angle 1$ has measure

$m \angle 1 = 180^{\circ } - m \angle 2 = 180^{\circ } - 100 ^{\circ } = 80^{\circ }$

$\angle 3$ , the complement of $\angle 1$ , has measure

$m \angle 3 = 90^{\circ } - m \angle 1 = 90^{\circ } - 80 ^{\circ } = 10^{\circ }$

The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?

Explanation

Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.

Let be the vertex angle and be the base angle.

The equation to solve becomes , since the base angle occurs twice.

Now we can solve for the vertex angle.

The difference between the vertex angle and the base angle is .

Triangle_a

Figure NOT drawn to scale.

If $w = 125^{\circ }$ and $x = 81^{\circ }$ , evaluate .

$44^{\circ }$

$55^{\circ }$

$53^{\circ }$

$35^{\circ}$

More information is needed to solve the problem.

Explanation

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so

$y = w - x= 125 - 81 = 44^{\circ }$

In a right triangle ABC, the measure of angle C is greater than 60 degrees. Which of the following statements could describe the measures of angles A and B?

$\angle A=90^o\ \text{and}\ \angle B<30^o$

$\angle A=90^o\ \text{and}\ \angle B>30^o$

$\angle A>30^o\ \text{and}\ \angle B=90^o$

$\angle A=45^o\ \text{and}\ \angle B=90^o$

$\angle A =59^o\ \text{and}\ \angle B=60^o$

Explanation

Given that it is a right triangle, either angle A or B has to be 90 degrees. The other angle then must be less than 30 degrees, given that C is greater than 60 because there are 180 degrees in a triangle.

Example:

If angle C is 61 degrees and angle A is 90 degrees, then angle B must be 29 degrees in order for the angle measures to sum to 180 degrees.

How to find the measure of an angle

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation