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.

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

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 |

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 |

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