# HSPT Math : How to find the measure of an angle

## Example Questions

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### Example Question #1 : How To Find The Measure Of An Angle

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

Explanation:

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

### Example Question #2 : How To Find The Measure Of An Angle

Two of the interior angles of a triangle measure  and . What is the greatest measure of any of its exterior angles?

It cannot be determined from the information given.

Explanation:

The interior angles of a triangle must have measures whose sum is , so the measure of the third angle must be .

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:

### Example Question #1 : Lines

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

Explanation:

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:

### Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

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

Explanation:

Since the sum of the angles of a triangle is , 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:

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.

### Example Question #3 : How To Find The Measure Of An Angle

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

45
90
30
60
Explanation:

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

### Example Question #4 : How To Find The Measure Of An Angle

Call the three angles of a triangle

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

Explanation:

The measure of  is twenty degrees greater than the measure  of , so its measure is 20 added to that of  - that is, .

The measure of  is thirty degrees less than twice that of . Twice the measure of  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:

### Example Question #5 : How To Find The Measure Of An Angle

Call the three angles of a triangle

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

Explanation:

The measure of  is forty degrees less than the measure  of , so its measure is 40 subtracted from that of  - that is, .

The measure of  is ten degrees less than twice that of . Twice the measure of  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:

### Example Question #6 : How To Find The Measure Of An Angle

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

Explanation:

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

### Example Question #7 : How To Find The Measure Of An Angle

What is  of the measure of a right angle?

Explanation:

A right angle has a measure of .  One fifth of the angle is:

### Example Question #8 : How To Find The Measure Of An Angle

What angle is complementary to ?