How to find the measure of one interior angle

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Find the measure of angle 𝐴𝐶𝐵.

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An Interior Angle is an angle inside a shape
Quadrilaterals (Squares, etc)
Because there are 2 triangles in a square ...
The General Rule
Example: What about a Regular Decagon (10 sides) ?
How do you find one interior angle of a polygon?

Let’s have a closer look at the figure we’ve been given. There is a triangle formed by connecting the points 𝐴, 𝐵, and 𝐶. The measure of one of the interior angles in this triangle, angle 𝐴𝐵𝐶, has been marked as 35 degrees. And the measure of the angle we’ve been asked to calculate, angle 𝐴𝐶𝐵, has been marked as 𝑥. We’re also given the measure of an angle outside the triangle, which is 65 degrees. The final thing to note is that line 𝐵𝐴 and line 𝐶𝐷, as we’ve now labeled it, are parallel, as indicated by the arrows along their lengths.

Now, we need to consider how we can use all this information to determine the measure of angle 𝐴𝐶𝐵. Because lines 𝐵𝐴 and 𝐶𝐷 are parallel, angle 𝐴𝐶𝐷 and angle 𝐶𝐴𝐵 are alternate angles. Alternate angles are of equal measure, so this tells us that the measure of angle 𝐶𝐴𝐵 is also 65 degrees.

We now know the measures of two of the interior angles in triangle 𝐴𝐵𝐶, and we wish to calculate the measure of the third. We can recall that the sum of the measures of the interior angles in any triangle is 180 degrees. We can therefore form an equation using the measures of the interior angles of triangle 𝐴𝐵𝐶. 𝑥 plus 35 degrees plus 65 degrees equals 180 degrees. This equation simplifies to 𝑥 plus 100 degrees equals 180 degrees. 𝑥 is therefore equal to 180 degrees minus 100 degrees, which is 80 degrees.

Remember 𝑥 represents the measure of angle 𝐴𝐶𝐵. So using the sum of the measures of the interior angles in a triangle, we found that the measure of angle 𝐴𝐶𝐵 is 80 degrees.

Christine L.

asked • 01/24/13

round to the nearest tenth if necessary.

2 Answers By Expert Tutors

Kevin S. answered • 01/24/13

The sum of the angles of any polygon is (n-2)*180, where n = number of sides. Therefore, for a regular polygon we would divide the result by n to find the measure of one interior angle: [(n-2)*180]/n I'll leave the calculations to the student.

Mary C. answered • 02/25/13

Math, Science, & Special Needs Tutoring Experience!

The sum of all the interior angles of a regular polygon can be found with the equation (n-2) × 180°. "n" in this equation refers to the number of sides in the polygon. (We can recall this equation easily if we remember that regular polygons can be divided up into triangles, and that the sum of angle triangles interior angles must equal 180°)

So, the sum of all interior angles in our regular 11-gon polygon = (11-2) × 180° = 1620°

Regular polygons are regular because they are equiangular (all angles of polygon are congruent) and equilateral (all sides of polygon are congruent). So if we know that the 11-gon polygon has 11 equal interior angles, we can divide the total sum of the interior angles by "n" to determine the measure of one interior angle.

[(n-2) × 180°] ÷ n = measure of one interior angle

[(11-2) × 180°] ÷ 11 = 1620° ÷ 11 = 147.27°

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An Interior Angle is an angle inside a shape

Another example:

Triangles

The Interior Angles of a Triangle add up to 180°

Let's try a triangle:

90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°:

80° + 70° + 30° = 180°

It still works!
One angle went up by 10°,
and the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Let's try a square:

90° + 90° + 90° + 90° = 360°

A Square adds up to 360°

Now tilt a line by 10°:

80° + 100° + 90° + 90° = 360°

It still adds up to 360°

The Interior Angles of a Quadrilateral add up to 360°

Because there are 2 triangles in a square ...

The interior angles in a triangle add up to 180° ...

... and for the square they add up to 360° ...

... because the square can be made from two triangles!

Pentagon

A pentagon has 5 sides, and can be made from three triangles, so you know what ...

... its interior angles add up to 3 × 180° = 540°

And when it is regular (all angles the same), then each angle is 540° / 5 = 108°

(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)

The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

So the general rule is:

Sum of Interior Angles = (n−2) × 180°

Each Angle (of a Regular Polygon) = (n−2) × 180° / n

Perhaps an example will help:

Example: What about a Regular Decagon (10 sides) ?

Sum of Interior Angles = (n−2) × 180°

= (10−2) × 180°

= 8 × 180°

= 1440°

And for a Regular Decagon:

Each interior angle = 1440°/10 = 144°

Note: Interior Angles are sometimes called "Internal Angles"

How do you find one interior angle of a polygon?

In order to find the value of the interior angle of a regular polygon, the equation is (n−2)180∘n where n is the number of sides of the regular polygon.

find measure interior angle