Geometric Proofs

## Geometric Proofs

Geometric Proofs use a logical argument that always follows the same form.

Proofs start with a Given fact about the problem being solved. There then follows one or more steps, which consist of a Statement followed by the Reason that that statement is true.

The final statement is the fact being proven.

## Example 1

For the parallelogram given below, prove that angle CAB is congruent to angle BDC. You may assume that opposite sides in a parallelogram are equal.

Answer: Given: Opposite sides in a parallelogram are equal

Statement Reason
AB = CD Opposite sides in a parallelogram are equal
AC = BD Opposite sides in a parallelogram are equal
BC is shared Line BC shared by both triangles
Triangles are congruent Congruency by SSS
therefore ∠CAB = ∠BDC

## Example 2

Prove that, for the triangle below, a line drawn at a right angle from the midpoint of AB (point D) bisects the angle at C.

Answer: Given: The triangle is isosceles; D is the midpoint of AB

Statement Reason
DC is common Line shared by both smaller triangles
AC = CB Definition of an isosceles triangle
AD = DB D is given as the midpoint
Triangles are congruent Congruency by SSS
∠ACD = ∠DCB ∠ACB is bisected