How to find the measure of an angle - HSPT Math

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Question

The measure of 3 angles in a triangle are in a 1:2:3 ratio. What is the measure of the middle angle?

Answer

The angles in a triangle sum to 180 degrees. This makes the middle angle 60 degrees.

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Question

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

Answer

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.

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Question

Two angles are supplementary and have a ratio of 1:4. What is the size of the smaller angle?

Answer

Since the angles are supplementary, their sum is 180 degrees. Because they are in a ratio of 1:4, the following expression could be written:

$x+4x=180$

$5x=180$

$x=36^{\circ}$

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Question

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

Answer

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

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Question

Two of the interior angles of a triangle measure $50^{\circ }$ and $45^{\circ}$ . What is the greatest measure of any of its exterior angles?

Answer

The interior angles of a triangle must have measures whose sum is $180^{\circ }$ , so the measure of the third angle must be $180 - (50+45) = 85^{\circ }$ .

By the Triangle Exterior-Angle Theorem, an exterior angle of a triangle measures the sum of its remote interior angles; therefore, to get the greatest measure of any exterior angle, we add the two greatest interior angle measures: $50+85= 135^{\circ }$

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Question

Call the three angles of a triangle $\angle 1, \angle 2, \angle 3$ .

The measure of $\angle2$ is twenty degrees greater than that of $\angle 1$ ; the measure of $\angle 3$ is thirty degrees less than twice that of $\angle 1$ . If is the measure of $\angle 1$ , then which of the following equations would we need to solve in order to calculate the measures of the angles?

Answer

The measure of $\angle2$ is twenty degrees greater than the measure of $\angle 1$ , so its measure is 20 added to that of $\angle 1$ - that is, .

The measure of $\angle 3$ is thirty degrees less than twice that of $\angle 1$ . Twice the measure of $\angle 1$ is , and thirty degrees less than this is 30 subtracted from - that is, .

The sum of the measures of the three angles of a triangle is 180, so, to solve for - thereby allowing us to calulate all three angle measures - we add these three expressions and set the sum equal to 180. This yields the equation:

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Question

Call the three angles of a triangle $\angle 1, \angle 2, \angle 3$ .

The measure of $\angle2$ is forty degrees less than that of $\angle 1$ ; the measure of $\angle 3$ is ten degrees less than twice that of $\angle 1$ . If is the measure of $\angle 1$ , then which of the following equations would we need to solve in order to calculate the measures of the angles?

Answer

The measure of $\angle2$ is forty degrees less than the measure of $\angle 1$ , so its measure is 40 subtracted from that of $\angle 1$ - that is, .

The measure of $\angle 3$ is ten degrees less than twice that of $\angle 1$ . Twice the measure of $\angle 1$ is , and ten degrees less than this is 10 subtracted from - that is, .

The sum of the measures of the three angles of a triangle is 180, so, to solve for - thereby allowing us to calulate all three angle measures - we add these three expressions and set the sum equal to 180. This yields the equation:

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Question

Two interior angles of a triangle adds up to degrees. What is the measure of the other angle?

Answer

The sum of the three angles of a triangle add up to 180 degrees. Subtract 64 degrees to determine the third angle.

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Question

What is $\frac{1}{5}$ of the measure of a right angle?

Answer

A right angle has a measure of $90$^{\circ}$$ . One fifth of the angle is:

$\angle A=\frac{90}{5}= 18$^{\circ}$$

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Question

What angle is complementary to $65$^{\circ}$$ ?

Answer

To find the other angle, subtract the given angle from $90$^{\circ}$$ since complementary angles add up to $90$^{\circ}$$ .

The complementary is:

$\angle 90-\angle65 = \angle25$

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Question

What is the supplementary angle to $100$^{\circ}$$ ?

Answer

Supplementary angles add up to $180$^{\circ}$$ . In order to find the correct angle, take the known angle and subtract that from $180$^{\circ}$$ .

$180$^{\circ}$ -100$^{\circ}$ = 80$^{\circ}$$

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Question

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

Answer

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.

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

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Question

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?

Answer

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.

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Question

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 are the odds against the spinner landing while pointing inside one of the blue regions?

Answer

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.

Two of the blue sectors are each one third of one quarter-circle, and thus are

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

of one circle.

The other two blue sectors are each one fourth of one quarter-circle, and thus are

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

of one 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. As odds, this translates to

, or odds against the spinner landing in blue.

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Question

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 are the odds against the spinner landing while pointing inside the purple region?

Answer

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 purple region is one third of one quarter of a circle, or, equvalently,

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

of a circle, so its central angle is $\frac{1}{12}$ of the total measures of the angles of the sectors. This makes $\frac{1}{12}$ the probability of the spinner stopping inside the purple region; this translates to

or odds against this occurrence.

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Question

In parallelogram , $m \angle A = (t+ 56 )^{\circ }$ . Give the measure of $\angle B$ in terms of .

Answer

$\angle A$ and $\angle B$ are a pair of adjacent angles of the parallelogram, and as such, they are supplementary - that is, their degree measures total 180. Therefore,

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

$m \angle B = 180 ^{\circ } - m \angle A$

$= 180 ^{\circ } - (t+ 56 )^{\circ }$

$= (180 - 56 - t )^{\circ }$

$= (124-t )^{\circ }$

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Question

The measures of the angles of $\bigtriangleup ABC$ are as follows:

$m \angle A = (3t-14)^{\circ }$

$m \angle B = (2t+16)^{\circ }$

$m \angle C =( 5t)^{\circ }$

Is this triangle acute, obtuse, right, or nonexistent?

Answer

The sum of the measures of the angles of a triangle is 180 degrees, so solve for in the equation:

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

$10t \div 10 = 178 \div 10$

$m \angle A = (3t-14)^{\circ } = (3 \cdot 17.8-14)^{\circ } = (53.4-14)^{\circ } = 39.4^{\circ }$

$m \angle B = (2t+16)^{\circ } = (2 \cdot 17.8 + 16)^{\circ }= (35.6+ 16) ^{\circ }= 51.6^{\circ }$

$m \angle C =( 5t)^{\circ } = ( 5 \cdot 17.8)^{\circ } = 89 ^{\circ }$

All three angles measure less than 90 degrees and are therefore acute angles; that makes $\bigtriangleup ABC$ an acute triangle.

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Question

If you have a right triangle, what is the measure of the two of the angles if they are equal?

Answer

The total degrees of the angles in a triangle are .

Since it is a right triangle, one of the three angles must be

That leaves you with for the other two angles .

If they are equal, you just divide the remaining degrees by to get .

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Question

If you have a right triangle with an angle measuring 45 degrees, what is the third angle measurement?

Answer

A right triangle has one 90 degree angle and all three angles must equal 180 degrees.

To find the answer, just subtract the two angles you have from the total to get

$180^\circ-\textup{Angle 1}-\textup{Angle 2}=\textup{Angle 3}$

The angle we have are,

$\ \textup{Angle 1}=90^\circ \ \textup{Angle 2}=45^\circ$ .

Substituting these into the formula results in the solution.

$180^\circ-90^\circ-45^\circ=45^\circ$ .

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Question

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°)

B C

Answer

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°

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