Geometric Proofs

Geometric Proofs

GCSE(H),

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.

See Congruent Triangles for reasons that can be used in these proofs.

Examples

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: Parallelogram ABCD with a diagonal BC;

Opposite sides in a parallelogram are equal

StatementReason
AB = CDOpposite sides in a parallelogram are equal
AC = BDOpposite sides in a parallelogram are equal
BC is sharedLine BC shared by both triangles
Triangles are congruentCongruency by SSS; therefore ∠CAB = ∠BDC

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

StatementReason
DC is commonLine shared by both smaller triangles
AC = CBDefinition of an isosceles triangle
AD = DBD is given as the midpoint
Triangles are congruentCongruency by SSS
∠ACD = ∠DCB∠ACB is bisected