Angles in a polygon

Angles between lines

If a line is split into 2 and you know one angle you can always find the other one.

Example: If we know one angle is 45° what is angle “a” ?

Angle a is 180° − 45° = 135°

This method can be used for several angles on one side of a straight line.

Example: What is angle “b” ?

Angle b is 180° less the sum of the other angles.

Sum of known angles = 45° + 39° + 24°
Sum of known angles = 108°

Angle b = 180° − 108°
Angle b = 72°

Vertically opposite angles are equal

Vertically Opposite Angles are the angles opposite each other when two lines cross

“Vertical” in this case means they share the same Vertex (or corner point), not the usual meaning ofup-down.
In this example, a° and b° are vertically opposite angles. The interesting thing here is that vertically opposite angles are equal:

a⁰ = b⁰

Angles in a triangle

In a triangle, the three interior angles always add to 180°:

A + B + C = 180°

Example: Find the Missing Angle “C”

Start with:A + B + C = 180°

Fill in what we know:38° + 85° + C = 180°

Rearrange: C = 180° – 38° – 85°

Calculate: C = 57°

This is a proof that the angles in a triangle equal 180°:

The top line (that touches the top of the triangle) is
running parallel to the base of the triangle.

So, angles A are the same, angles B are the same

And you can easily see that A + C + B does a complete rotation from one side of the straight line to the other, or 180°

Angles in a quadrilateral

Any quadrilateral can be divided into two triangles by forming its diagonal

Quadrilateral just means “four sides”
(quad means four, lateral means side).

Any four-sided shape is a Quadrilateral.

But the sides have to be straight, and it has to be 2-dimensional.

Angles in a polygon

Polygons

A polygon is any plane figure with straight sides. Thus a triangle is a three-sided polygon and a quadrilateral is a four-sided polygon Polygons are named after the number of sides they have.

Sum of the interior angles of a polygon

In each polygon, one vertex is joined to all the other vertices. This divides the polygons into triangles. The number of triangles depends on the number of sides of the polygon.

In each case, the number of triangles is two less than the number of sides. For a polygon with n sides there will be n – 2 triangles. The sum of the angles of a triangle is 180⁰.

Thus, in degrees:

the sum of the angles of an n-sided polygon

= (n – 2) x 180⁰

and, in right angles:

the sum of the angles of an n – sided polygon

= (n – 2) x 2 right angles

= 2n – 4 right angles

Notice that these formulae are true for the polygons we already know.

In a triangle, n = 3.

Sum of angles = (3 – 2) x 180⁰

= 1 x 180⁰ = 180⁰

In a quadrilateral, n = 4.

Sum of angles = 2 x 4 – 4 right angles

= 4 right angles (= 360⁰)

Example

Example 1:Find the number of degrees in the sum of the interior angles of an octagon.

An octagon has 8 sides.  So n = 8.  Using the formula from above, 180(n – 2) = 180 (8 – 2) = 180(6) = 1080 degrees.

Example

How many sides does a polygon have if the sum of its interior angles is 720°?

Since, the number of degrees is given, set the formula above equal to 720°, and solve for n.

180(n – 2) = 720
n – 2 = 4
n = 6

Assessment

• How many sides does a polygon have if the sum of its interior angles is 540