ANGLESKHM

These figures are not polygons These figures are polygons

Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints.

Polygons

Interior Angle of a Polygon

For example, ∆ABC has interior angles:

ABC, BAC, BCA

A

B

C

The interior angles of a polygon are the angles inside the polygon, formed by two adjacent sides.

Exterior Angle of a Polygon

For example, ∆ABC has exterior angle:

ACD. It forms a linear pair with ACB.

An exterior angle of a polygon is an angle that forms a linear pair with an interior angle. It is an angle outside the polygon formed by one side and one extended side of the polygon.

A

BC

D

Exterior AngleInterior Angles

What is the sum of the measures of the interior angles of a convex n-gon?

PolygonNumber of

Sides

Sum of Measures of

Interior Angles

Triangle 3 180°

Quadrilateral 4 360°

Pentagon 5 540°

Hexagon 6 720°

n-gon n (n - 2)180°

What is the sum of the measures of the exterior angles of a 3-gon? (a triangle)

Sum of Measures of Exterior Angles = 360

What is the measure of each exterior angle of a regular 3-gon?

?

?

?

What is the measure of each exterior angle of a regular 4-gon?

?

??

?

Angles made with parallel lines

When a straight line crosses two parallel lines eight angles are formed.

Which angles are equal to each other?

a

b

c

d

e

f

g

h

dd

hh

a

b

ce

f

g

Corresponding angles

There are four pairs of corresponding angles, or F-angles.

a

b

ce

f

g

d = h because

Corresponding angles are equal

ee

aa

b

c

d

f

g

h

Corresponding angles

There are four pairs of corresponding angles, or F-angles.

b

c

d

f

g

h

a = e because

Corresponding angles are equal

gg

cc

Corresponding angles

There are four pairs of corresponding angles, or F-angles.

c = g because

Corresponding angles are equal

a

bd

e

fh

ff

Corresponding angles

There are four pairs of corresponding angles, or F-angles.

b = f because

Corresponding angles are equal

a

b

c

d

e

g

h

b

ff

dd

Alternate angles

There are two pairs of alternate angles, or Z-angles.

d = f because

Alternate angles are equal

a

b

ce

g

h

ccee

Alternate angles

There are two pairs of alternate angles, or Z-angles.

c = e because

Alternate angles are equal

a

b

g

h

d

f

Angles in a triangle

For any triangle,

a b

c

a + b + c = 180°

The angles in a triangle add up to 180°.

Calculating angles in a triangle

Calculate the size of the missing angles in each of the following triangles.

233°

82°31°

116°

326°

43°49°

28°

ab

c

d

33°64°

88°

25°

Interior and exterior angles in a triangle

ab

c

Any exterior angle in a triangle is equal to the sum of the two opposite interior angles.

a = b + c

We can prove this by constructing a line parallel to this side.

These alternate angles are equal.

These corresponding angles are equal.

b

c

Interior and exterior angles in a triangle

Calculating angles

Calculate the size of the lettered angles in each of the following triangles.

82°31°64° 34°

ab

33°116°

152°d25°

127°

131°

c

272°

43°

Calculating angles

Calculate the size of the lettered angles in this diagram.

56°a

73°b86° 69°

104°

Base angles in the isosceles triangle = (180º – 104º) ÷ 2

= 76º ÷ 2

= 38º

38º 38º

Angle a = 180º – 56º – 38º = 86º

Angle b = 180º – 73º – 38º = 69º