Geometric Proofs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Proofs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Geometric proofs establish that a geometric claim is true by chaining justified statements from definitions, theorems, and givens.

It is a legal argument where each line needs a valid reason.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A geometric proof chains statements, each justified by a given, definition, or theorem, to reach a conclusion.

Common stuck point: The procedure for geometric proofs is the easy part; the trap is skipping justifications. Asking "Am I required to justify a claim with reasons for each step, rather than calculate a number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I required to justify a claim with reasons for each step, rather than calculate a number?

Worked Examples

Example 1

medium

Prove that the base angles of an isosceles triangle are equal.

Answer

The base angles

\angle ABC = \angle ACB

are equal by SSS congruence.

First step

Given: Triangle

ABC

with

AB = AC

. Draw the median from

A

to the midpoint

M

BC

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Example 2

hard

Prove by contradiction: if two lines are cut by a transversal so that alternate interior angles are equal, the lines are parallel.

Example 3

medium

Given

\overline{AC}\cong\overline{BD}

and

\overline{AB}\cong\overline{CD}

, prove

\triangle ABD\cong\triangle DCA

where

A,B,C,D

are arranged so

\overline{AD}

is shared.

Example 4

medium

Prove: If two angles are supplementary to the same angle, then they are congruent.

Example 5

medium

Prove that if

\angle 1

and

\angle 2

are a linear pair, they are supplementary.

Example 6

medium

Prove: The diagonals of a parallelogram bisect each other.

Example 7

medium

Prove: The sum of the interior angles of a triangle is

180°

Example 8

hard

Prove: The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Example 9

hard

Prove by contradiction: A triangle cannot have two right angles.

Example 10

hard

Prove: If

\overline{AB}\parallel\overline{CD}

and

\overline{AB}\cong\overline{CD}

, then

ABDC

is a parallelogram (where

ABDC

means

A\to B\to D\to C\to A

Example 11

hard

Prove: In an isosceles triangle, the median to the base is also the perpendicular bisector of the base.

Example 12

challenge

Prove that in any triangle, the three perpendicular bisectors of the sides meet at a single point (the circumcenter).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Prove that vertical angles are equal. Given: Lines

AB

and

CD

intersect at

O

; prove

\angle AOC = \angle BOD

Example 2

medium

Given that

\triangle ABC \cong \triangle DEF

, prove that the perimeters are equal.

Example 3

easy

Given:

M

is the midpoint of

\overline{AB}

. Justify

AM=MB

Example 4

easy

Given:

\overline{BD}

bisects

\angle ABC

. Justify

\angle ABD\cong\angle DBC

Example 5

medium

Given

\angle A\cong\angle D

\overline{AB}\cong\overline{DE}

\overline{AC}\cong\overline{DF}

. Which congruence shortcut proves

\triangle ABC\cong\triangle DEF

Example 6

medium

State the converse of: 'If a triangle is equilateral, then it is equiangular.' Is the converse true?

Example 7

medium

\triangle ABC

\angle B=\angle C

and

\overline{BD}\cong\overline{CD}

where

D

lies on

\overline{BC}

. State the reason these triangles

\triangle ABD

and

\triangle ACD

might be congruent.

Example 8

medium

Write the contrapositive of: 'If a quadrilateral is a square, then it is a rhombus.'

Example 9

hard

In a paragraph proof, what is the role of the 'Given' information?

Example 10

hard

Why is 'AAA' (three pairs of congruent angles) NOT a valid triangle congruence shortcut?

Example 11

hard

Given parallel lines cut by a transversal: name three pairs of angles you may treat as congruent without further proof.

Example 12

medium

Given

\angle 1\cong\angle 2

and

\angle 1

and

\angle 3

are vertical angles, prove

\angle 2\cong\angle 3

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Related Concepts

Proof (Intuition)Congruence Criteria Triangle Angle Sum

Background Knowledge

These ideas may be useful before you work through the harder examples.

proof intuitioncongruence criteriatriangle angle sum