Similarity and Congruence Proofs

Grade 10 · Math · Study Guide

Understand the criteria for triangle similarity and congruence and use them in formal geometric proofs.

Two figures are congruent if they have the same shape and size — one can be mapped onto the other by rigid motions (reflections, rotations, translations). Two figures are similar if they have the same shape but possibly different sizes — one can be mapped onto the other by rigid motions combined with a dilation.

Triangle congruence can be established using: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), AAS (two angles and a non-included side equal), or RHS/HL (right angle, hypotenuse, and one side for right triangles). Note that SSA is NOT a valid congruence criterion.

Triangle similarity is established using: AA (two pairs of equal angles — the third follows automatically), SAS~ (two sides in proportion with the included angle equal), or SSS~ (all three pairs of sides in proportion).

In similar triangles with ratio k:1, corresponding sides are in ratio k, areas are in ratio k², and volumes (for 3D extensions) in ratio k³.

Formal proofs use a two-column or paragraph format. Each statement must be justified by a definition, postulate, or theorem. Common justifications include 'corresponding parts of congruent triangles are congruent' (CPCTC) and properties such as reflexivity (a segment equals itself) and transitivity.

Key terms

CPCTC
Corresponding Parts of Congruent Triangles are Congruent — used after establishing triangle congruence to conclude individual sides or angles are equal.
SSS Congruence
If all three sides of one triangle are equal to the corresponding sides of another, the triangles are congruent.
AA Similarity
If two angles of one triangle equal two angles of another, the triangles are similar.
Scale Factor
The ratio of corresponding lengths in two similar figures.
Dilation
A transformation that enlarges or reduces a figure by a scale factor from a fixed centre point.
Reflexive Property
Any geometric figure is congruent to itself (e.g., a shared side or angle in two triangles).

Flashcards

SSS Congruence
Three sides of one triangle equal three sides of another → congruent.
SAS Congruence
Two sides and the included angle equal → congruent.
ASA Congruence
Two angles and the included side equal → congruent.
AAS Congruence
Two angles and a non-included side equal → congruent.
RHS/HL Congruence
Right angle + hypotenuse + one side equal → congruent right triangles.
AA Similarity
Two pairs of equal angles → similar triangles.
SAS Similarity
Two sides in proportion with the included angle equal → similar.
SSS Similarity
All three pairs of sides in proportion → similar triangles.
CPCTC
After proving triangles congruent, corresponding parts (sides and angles) are congruent.
Area Ratio
If scale factor is k:1, area ratio is k²:1.

Questions

  1. Triangles ABC and DEF have AB = DE, angle A = angle D, and AC = DF. Which congruence criterion applies?

    1. SAS
    2. SSS
    3. ASA
    4. AAS
    Show answer

    Answer: SAS

    AB = DE (side), angle A = angle D (included angle), AC = DF (side) → SAS congruence.

  2. SSA is a valid triangle congruence criterion.

    Show answer

    Answer: False

    SSA can give two possible triangles (the ambiguous case) and is therefore not a valid congruence criterion.

  3. Two similar triangles have a scale factor of 3:2. What is the ratio of their areas?

    1. 9:4
    2. 3:2
    3. 6:4
    4. 27:8
    Show answer

    Answer: 9:4

    Area ratio = (scale factor)² = 3² : 2² = 9:4.

  4. If two angles of one triangle equal two angles of another triangle, the triangles are ______.

    Show answer

    Answer: similar

    The AA criterion states that two pairs of equal angles are sufficient to prove triangle similarity.

  5. In a proof, you have shown triangle PQR ≅ triangle STU. What justification allows you to conclude that PQ = ST?

    1. CPCTC
    2. SSS
    3. Reflexive Property
    4. AA Similarity
    Show answer

    Answer: CPCTC

    CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is the standard justification for concluding that individual parts are equal after proving congruence.

Get a personalized course on Similarity and Congruence Proofs built around your child.