Summary 2, theory

Angles

You should be able to state and prove following theorems.

Theorem 1 Vertical angles are equal.

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

Hint, Proof of Theorem 2 Given a triangle ΔABC.

Show that the sum of α and β is less than two right angles.

See the construction 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. You are only allowed to use theorems you have already proved.

Theorem 3 State the conjecture from Exercise 1 as a theorem. What are your premises? What is your proposition?

Hint, Proof of 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.

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

Theorem 5 State the conjecture from Exercise 2 as a theorem. What are your premises? What is your proposition?

Theorem 6 If two parallel lines are intersected by a third line, 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.

Hint, Proof of the Exterior Angle Theorem Use the picture below. The line l is parallel to AC.

Theorem 8 The sum of the interior angles of a triangle is two right angles.(Use the Exterior Angle Theorem).