# Summary - Angles

## Definitions

- The angle of one revolution is 360°.
- Two angles sharing a common ray are called
**adjacent**. - Two adjacent angles lying along a line are called
**supplementary angles**. - If two supplementary angles are equal they are
**right angles**. - An angle that is smaller than one right angle is an
**acute angle**. - An angle that is larger than one right angle and smaller than two right angles is an
**obtuse angle**. - A line intersecting two other lines is called a
**transversal**. The angles are**corresponding angles**. - The angles are
**alternate angles**. - The angles are
**vertical angles**. - The angle is an
**exterior angle**to the triangle.

**Note:** Number 1 has been added to the list even though degrees are not mentioned in the Elements by Euclid.

## GeoGebra tasks

Make a line `a` through the points `A` and `B`,
and a line `b` through the points `C` and `D`. Enter
the point of intersection `E` and the angle `α`. Place
a point `F` on the line `b`.

#### Task 1

Make an angle `β` at the point `F` equal to `α`
, and such that `β` becomes an **alternate** angle
when a new line is drawn. What can you say about the line `a` and the
new line?

#### Task 2

Make an angle `β` at the point `F` equal to `α`
, and such that `β` becomes a **corresponding** angle
when a new line is drawn. What can you say about the line `a` and the
new line?

## Theorems

**Theorem 1** Vertical angles are equal.

**Theorem 2** In any triangle, the sum of two interior angles
is less than two right angles.

**Theorem 3** If two lines are intersected by a transversal, and if alternate angles are equal, then the
two lines are parallel.

**Theorem 4** If two parallel lines are intersected by a transversal, then alternate angles are equal.

**Theorem 5** If two lines are intersected by a transversal, and if corresponding angles are equal, then the
two lines are parallel.

**Theorem 6** If two parallel lines are intersected by a transversal, then corresponding angles are equal.

**Theorem 7 - The Exterior Angle Theorem** An exterior angle of
a triangle is equal to the sum of the two remote interior angles.

**Theorem 8** The sum of the interior angles of a triangle is two right angled.

**Theorem 9 The converse of the isosceles triangle theorem** If two angles in a triangle are equal, then the triangle is isosceles.

## Exercises

The theorems you should know by before doing this, are: the congruence cases SAS, SSS, ASA, and the theorem about angles in an isosceles triangle.

#### Exercise 1

Prove Theorem 1

#### Exercise 2

In the demonstration below, `D` is the midpoint of the segment `AC` and also the midpoint
of the segment `BE`. As long as the vertices of the triangle have the
counterclockwise order `A`, `B`, `C`; the sum of
`α` and `γ` is less than two right angles. Show
that `γ=β`. Then prove Theorem 2. You are only allowed to use theorems that have already
been proved.

#### Exercise 3

Prove Theorem 3. Try to do a proof by contradiction, i.e. assume that your proposition is not true; then show that this assumption leads to a contradiction. Then use Theorem 3 to prove Theorem 4, a proof by contradiction works in this case as well.

#### Exercise 4

Use some of the theorems proved so far to prove Theorem 5 and 6.

#### Exercise 5

Prove Theorem 7 - The Exterior Angle Theorem. Use the picture below. The line `l` is parallel to `AC`.

#### Exercise 6

Prove Theorem 8.

#### Exercise 7

Prove Theorem 9! Hint: draw an angle bisectris at one of the vertices of the triangle.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License