Summarizing Angles

Angles in a triangle

For any triangle,For any triangle,

ab

c

a + b + c = 180°a + b + c = 180°

Angles in a triangle

We can prove that the sum of the angles in a triangle is 180° by drawing a line parallel to one of the sides through the opposite vertex.

These angles are equal because they are alternate angles.

a

a

b

b

Call this angle c.

c

a + b + c = 180° because they lie on a straight line.

The angles a, b and c in the triangle also add up to 180°.

Sum of the interior angles in a pentagon

What is the sum of the interior angles in a pentagon?

We can work this out by using lines from one vertex to divide the pentagon into three triangles .

a + b + c = 180° and d + e + f = 180°

So, (a + b + c) + (d + e + f ) + (g + h + i) = 540°

The sum of the interior angles in a pentagon is 540°.The sum of the interior angles in a pentagon is 540°.

c

a

b

and g + h + i = 180°

d

f

eg

ih

Sum of the interior angles in a polygon

We’ve seen that a quadrilateral can be divided into two triangles …

… and a pentagon can be divided into three triangles.

How many triangles can a hexagon be divided into?A hexagon can be divided into

four triangles.

To find the interior angles of a convex polygon:(n-2) x 180

Angles made with parallel lines

Types of angle

Acute angle0º < a < 90º

Acute angle0º < a < 90º a

Right anglea = 90º

Right anglea = 90º

a

Obtuse angle90º < a < 180º

Obtuse angle90º < a < 180º

a

Reflex angle 180º < a < 360º

Reflex angle 180º < a < 360º

a

Angles at a point add up to 360

ab

cd

a

b

Angles on a line add up to 180

c

Angles on a straight line and at a point

a + b + c + d = 360

because there are 360 in a full turn.

a + b + c = 180°

because there are 180° in a half turn.

Complementary and supplementary angles

a

b

a + b = 90°

Two complementary angles add up to 90°.

Two complementary angles add up to 90°.

Two supplementary angles add up to 180°.

Two supplementary angles add up to 180°.

a b

a + b = 180°

Intersecting lines

Vertically opposite angles

When two lines intersect, two pairs of vertically opposite angles are formed.

a

b

c

d

a = c and b = d

Vertically opposite angles are equal.Vertically opposite angles are equal.

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

ef

g

hThis line is called a

traversal.

Alternate angles are

equal

Alternate angles are

equal

a

b

a = ba = b

Corresponding, alternate and interior angles

Look for an F-shape

Look for a Z-shape

Corresponding angles are

equal

Corresponding angles are

equal

a

b

a = ba = b

Look for a C- or U-shape

Interior angles add up to 180°

Interior angles add up to 180°

a

b

a + b = 180°a + b = 180°

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 equalCorresponding 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 equalCorresponding angles are equal

gg

cc

Corresponding angles

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

c = g because

a

bd

e

fh

Corresponding angles are equalCorresponding angles are equal

ff

Corresponding angles

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

b = f because

a

b

c

d

e

g

h

b

Corresponding angles are equalCorresponding angles are equal

ff

dd

Alternate angles

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

d = f because

Alternate angles are equalAlternate angles are equal

a

b

ce

g

h

ccee

Alternate angles

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

c = e because

a

b

g

h

d

f

Alternate angles are equalAlternate angles are equal