What Is a Geometric Proof?

A geometric proof is a logical argument that establishes a geometric fact as an inevitable consequence of given information (hypotheses) and known theorems (geometric facts). Every step must be justified.

Structure of a Proof

Given: State what information you are told
To prove: State clearly what you need to show
Proof: Each step is a statement, and each statement has a reason
Conclusion: State what has been proved, citing the result

Common Reasons Used in Proofs

Vertically opposite angles are equal
Alternate interior angles (parallel lines)
Co-interior angles are supplementary
Corresponding angles are equal
Angles in a triangle sum to 180°
Base angles of isosceles triangle are equal

Congruence Conditions (SSS, SAS, AAS, RHS)

SSS: Three sides equal
SAS: Two sides and included angle equal
AAS: Two angles and one side equal
RHS: Right angle, hypotenuse, one side equal

Congruence Proofs

Two triangles are congruent if they have exactly the same shape and size. All corresponding sides and angles are equal. We write ▵ABC ≅ ▵DEF.

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Worked Example: Congruence proof using SAS

Given: In the diagram, AB = DC and BC = AD (opposite sides of quadrilateral ABCD are equal). Prove ▵ABC ≅ ▵CDA.

Statement	Reason
AB = DC	Given
BC = AD	Given
AC = CA (common side)	Common side (reflexive property)
▵ABC ≅ ▵CDA	SSS (three sides equal)

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Worked Example: Proving lines parallel

Given: ∠ABD = ∠CDB (alternate interior angles). Prove that AB ∥ CD.

Step 1: We are told ∠ABD = ∠CDB.

Step 2: These are alternate interior angles formed by transversal BD cutting AB and CD.

Conclusion: Since alternate interior angles are equal, AB ∥ CD. (Converse of alternate angles theorem)

Similarity Proofs

Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are in proportion.

Similarity Conditions for Triangles

AA: Two pairs of equal angles (most common)
SAS: Two sides in proportion and included angle equal
SSS: All three pairs of sides in the same ratio

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Worked Example: Similarity proof using AA

Given: In ▵PQR, S is a point on PQ and T is a point on PR such that ST ∥ QR. Prove that ▵PST ∼ ▵PQR.

Statement	Reason
∠P = ∠P (common angle)	Common angle
∠PST = ∠PQR	Corresponding angles, ST ∥ QR
▵PST ∼ ▵PQR	AA similarity

Key Vocabulary

Congruent

Two figures are congruent if they are identical in shape and size. Notation: ≅

Similar

Two figures are similar if corresponding angles are equal and sides are in proportion. Notation: ∼

Theorem

A mathematical statement that has been logically proved from axioms and previously established theorems.

Scale Factor

In similar figures, the ratio of corresponding side lengths. If scale factor k, then areas are in ratio k².

Knowledge Check

Test your understanding of geometric reasoning and proofs.

Question 1

Which congruence condition applies when two triangles have three pairs of equal sides?

SAS AAS RHS SSS

Question 2

What is the minimum number of angle equalities needed to prove two triangles are similar (using AA)?

2 1 3 All three, plus one side

Question 3

In a proof, after writing a statement, you must always provide a:

Diagram Reason Example Calculation

Question 4

Two similar triangles have a scale factor of 3. If the smaller triangle has an area of 10 cm², what is the area of the larger triangle?

30 cm² 900 cm² 90 cm² 60 cm²

Question 5

Which congruence condition is most appropriate when two right-angled triangles share the same hypotenuse and one pair of equal legs?

SSS SAS AAS RHS

Geometric Proofs