# What is the measure of angle <ABC?

### 9 Answers

- PuzzlingLv 71 month agoFavorite Answer
The flaw with this question is we aren't explicitly told that this is a regular pentagon (all sides and angles congruent). Without that statement, we shouldn't make assumptions.

That being said, it seems you are supposed to assume that ABCDE is a *regular* pentagon.

There are a few ways to figure out the measure of an interior angle of a regular polygon.

I actually like to find the measure of an *exterior* angle first. Look at the picture below. The exterior angles add up to a full 360° turn around the shape.

Equilateral triangle --> 360°/3 = 120° exterior angle

Square --> 360°/4 = 90° exterior angle

Regular pentagon --> 360°/5 = 72° exterior angle

Once you have the exterior angle, the interior angle is always supplementary (they add to 180°).

Equilateral triangle --> 180° - 120° = 60° interior angle

Square --> 180° - 90° = 90° interior angle

Regular pentagon --> 180° - 72° = 108° interior angle

Answer:

108° interior angle (in an assumed regular pentagon ABCDE)

Another method is use the formula for the sum of the interior angles and divide by the number of sides.

S = 180°(n - 2)

S/n = 180°(n - 2)/n

For example, with a pentagon:

S/n = 180°(5 - 2)/5

= 180°*3/5

= 540°/5

= 108°

There are other methods too.

- Engr. RonaldLv 71 month ago
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° ..

So ∠ABC = 108° ...........Answer//

- sparrowLv 71 month ago
I imagine it like this. There is a circle in the center of the pentagon. Each angle coming out of the center is 360/5=72

Draw a spoke from the center to each point.The triangles that are formed will have the two equal angles, plus the angle formed by the center. They must all add up to 180.

180 - 72 = 108

108/2 = 54

The outer angles are all equal to 54

Since all the outer angles are equal, remove the drawn spokes, and the depicted angle should equal 2 x 54 = 108

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- PinkgreenLv 71 month ago
If the figure is a regular pentagon, then

According to the formula:

the measure of an interior angle of a regular n-gon

=[(n-2)180*]/n

=>

<ABC=[(5-2)180*]/5=108*, here n=5.

- lenpol7Lv 71 month ago
Remember

Triangle sum to 180

Quadrilaterals sum to 360

Pentagons sum to 540

Hexagons sum to 720

For each extra side add 180 .

Assuming the pentagon to equilateral , then 540 / 5 = 108

Hence each angle is 108 degrees . It follows angle B is 108 degrees.

- KrishnamurthyLv 71 month ago
There are two types of pentagon; regular and irregular.

A regular pentagon has five sides of equal length.

An irregular pentagon has five sides of varying length.

A regular pentagon has five sides of equal length, as well as five equal angles.

A standard "STOP" sign in the US is a regular pentagon,

as is the Pentagon building in the District of Columbia.

The measure of angle ∠ABC = 108°

- geezerLv 71 month ago
The shape ABCDE is a pentagon

and, looking at it, all the sides are the same length

and all the internal angles are the same

which (in a pentagon) is 108 degrees.

How do you work it out if you don't just 'know' the answer ?

Well ..

We know that all the internal angles of a triangle are 180 degrees

and all the internal angles of a square are 360 degrees

So just keep adding 180 degrees for every extra side.

Three sides ... 180 degrees

Four sides ..... 360 degrees

Five sides ...... 540 degrees

and if they are all the same that's ... 540 / 5 .... 108 degrees

- llafferLv 71 month ago
I always forget the equation for the number of degrees based on the number of sides of a shape so will need to derive it knowing that 3 sides are 180° and 4 sides are 360°.

Finding the equation of the line that goes through the points (3, 180) and (4, 360) we get:

y = mx + b

180 = m(3) + b and 360 = m(4) + b

180 - 3m = b and 360 - 4m = b

180 - 3m = 360 - 4m

m = 180

180 = 180(3) + b

180 = 540 + b

-360 = b

y = 180x - 360

y = 180(x - 2)

Now that we have this we can find the total number of degrees for a 5-sided shape:

180(5 - 2)

180(3)

540°

Divide this by 5 since all angles are the same to get the measurement of a single angle:

540 / 5 = 108°