Congruence & Similarity
Congruence and similarity are fundamental geometry concepts that describe how two figures relate in shape and size. Congruent figures are identical copies; similar figures are scaled versions of each other.
This guide covers key definitions, all five congruence criteria (SSS, SAS, ASA, AAS, HL), three similarity criteria (AA, SSS, SAS), proportional reasoning, worked examples, common mistakes, and a practice quiz.
1Introduction
Imagine you have two identical cookies from the same batch. They are the same shape and the same size. In geometry, we call such figures congruent. Now imagine a small cookie and a giant cookie, both made with the same cutter but scaled differently. They are the same shape but different sizes. In geometry, we call such figures similar.
Understanding the distinction between congruent and similar figures is key to solving a wide range of geometric problems, from proofs and constructions to real-world applications in architecture, engineering, and design.
A blueprint of a house and the actual house are similar figures — same shape, different size. Two houses built from the same blueprint are congruent figures — same shape, same size.
Real-World Uses
Architecture & Engineering
Blueprints use similarity to scale buildings. Identical structural beams rely on congruence for safety.
Map Reading
Maps are similar to the actual terrain — distances scale proportionally using the map's scale factor.
Shadow Problems
Using similar triangles formed by objects and their shadows to calculate unknown heights.
Art & Photography
Enlarging or reducing images while maintaining proportions is a direct application of similarity.
2Key Definitions
Congruent Figures (≅)
Two figures with exactly the same shape and size. All corresponding angles are equal and all corresponding sides are equal in length.
Similar Figures (~)
Two figures with the same shape but potentially different sizes. Corresponding angles are equal and corresponding sides are proportional.
Scale Factor (k)
The ratio of corresponding side lengths in similar figures. If k > 1 it is an enlargement; if 0 < k < 1 it is a reduction.
Corresponding Angles
Angles in the same relative position in two different figures. Equal in both congruent and similar figures.
Corresponding Sides
Sides in the same relative position. Equal in congruent figures, proportional in similar figures.
△ABC ≅ △DEF — all corresponding sides and angles are equal
△PQR ~ △XYZ with scale factor k = 2 — same shape, proportional sides
3Congruence Criteria
To prove two triangles are congruent, you do not need to check all six parts (3 sides + 3 angles). These shortcut criteria guarantee congruence:
SSS (Side-Side-Side)
If three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent.
△ABC ≅ △DEF if AB = DE, BC = EF, CA = FD
SAS (Side-Angle-Side)
If two sides and the included angle (the angle between those sides) are equal, the triangles are congruent.
△ABC ≅ △DEF if AB = DE, ∠B = ∠E, BC = EF
ASA (Angle-Side-Angle)
If two angles and the included side (the side between those angles) are equal, the triangles are congruent.
△ABC ≅ △DEF if ∠A = ∠D, AB = DE, ∠B = ∠E
AAS (Angle-Angle-Side)
If two angles and a non-included side are equal, the triangles are congruent. (Since two angles determine the third, this is essentially a variation of ASA.)
△ABC ≅ △DEF if ∠A = ∠D, ∠B = ∠E, BC = EF
HL (Hypotenuse-Leg) — Right Triangles Only
If the hypotenuse and one leg of a right triangle are equal to those of another right triangle, the triangles are congruent.
△ABC ≅ △DEF if ∠B = ∠E = 90°, AC = DF, AB = DE
Not Valid for Congruence
SSA (Side-Side-Angle with non-included angle) is ambiguous — two different triangles can share the same SSA measurements. AAA (three equal angles) only proves similarity, not congruence.
4Similarity Criteria
For two triangles to be similar, all corresponding angles must be equal and all corresponding sides must be proportional. These shortcut criteria guarantee similarity:
AA Similarity
∠A = ∠D, ∠B = ∠E
If two angles are equal, the triangles are similar. (The third angle is automatically determined.)
SSS Similarity
AB/DE = BC/EF = CA/FD
If all three sides are proportional, the triangles are similar.
SAS Similarity
AB/DE = BC/EF, ∠B = ∠E
If two sides are proportional and the included angle is equal, the triangles are similar.
AA is the most commonly used similarity criterion. Since the angles in a triangle always sum to 180°, knowing two angles automatically determines the third. This makes AA especially powerful for problems involving parallel lines and shared angles.
5Proportional Reasoning
The concept of proportionality is at the heart of similarity. If two figures are similar with scale factor k, the ratio of any pair of corresponding sides equals k.
Side₁/Side₁′ = Side₂/Side₂′ = k
The ratio of any pair of corresponding sides in similar figures is constant — the scale factor.
How Scale Factor Affects Measurements
Perimeter
k
Scales linearly with the scale factor
Area
k²
Scales by the square of the factor
Volume (3D)
k³
Scales by the cube of the factor
Steps to Find Missing Sides
- Prove similarity: Use AA, SSS, or SAS to establish that the figures are similar.
- Identify corresponding sides: Match sides by the angles they are opposite to, or by the naming convention.
- Set up a proportion: Write an equation using the ratios of corresponding sides.
- Solve: Use cross-multiplication to find the unknown value.
6Worked Examples
Example 1: SAS Congruence
Given △PQR and △STU with PQ = ST = 5 cm, ∠Q = ∠T = 40°, QR = TU = 7 cm. Are they congruent?
PQ = ST = 5 cm (Side)
∠Q = ∠T = 40° (Included Angle)
QR = TU = 7 cm (Side)
△PQR ≅ △STU by SAS
Example 2: AAS Congruence
△ABC: ∠A = 30°, ∠B = 70°, BC = 8 cm. △DEF: ∠D = 30°, ∠E = 70°, EF = 8 cm.
∠A = ∠D = 30° (Angle)
∠B = ∠E = 70° (Angle)
BC = EF = 8 cm (Non-included Side)
△ABC ≅ △DEF by AAS
Example 3: AA Similarity — Shadow Problem
A 2 m tall person casts a 3 m shadow. A flagpole casts an 18 m shadow. How tall is the flagpole?
Both form right angles with ground (∠ = 90°)
Same sun angle → AA Similarity
height/shadow = height/shadow
2/3 = h/18
3h = 36
h = 12 meters
Example 4: SSS Similarity — Scale Factor
Triangle A: 6, 8, 10 cm. Triangle B: 9, 12, 15 cm. Are they similar?
9/6 = 1.5
12/8 = 1.5
15/10 = 1.5
All ratios equal → SSS Similarity
Scale factor k = 1.5
Example 5: Overlapping Triangles — Parallel Lines
In △ABC, D is on AB and E is on AC such that DE ∥ BC. AD = 3, DB = 2, DE = 5. Find BC.
DE ∥ BC → ∠ADE = ∠ABC, ∠AED = ∠ACB
△ADE ~ △ABC by AA
AB = AD + DB = 3 + 2 = 5
AD/AB = DE/BC
3/5 = 5/BC
3 × BC = 25
BC = 25/3 ≈ 8.33 cm
7Common Mistakes
Confusing congruence and similarity
Congruent means identical (same shape AND size). Similar means same shape but potentially different sizes. Don't use the wrong term or symbol (≅ vs ~).
Incorrectly identifying corresponding parts
Always match angles to angles and sides to sides based on their relative positions. The side opposite the largest angle corresponds to the side opposite the largest angle in the other triangle.
Using SSA or AAA for congruence
SSA is NOT valid for congruence (ambiguous case). AAA is NOT valid for congruence either — it only proves similarity. A small and large equilateral triangle both have 60°-60°-60° angles but are not congruent.
Setting up proportions incorrectly
Ensure your ratios are consistent. Either always put the sides of one figure in the numerator, or match corresponding sides within each ratio. Mixing the order will give wrong answers.
Forgetting to state the criterion
In proofs, always state why figures are congruent or similar (e.g., "by SAS" or "by AA similarity"). This is required for full marks on exams.
8Tips & Memory Aids
"Congruent = Carbon Copy, Similar = Same Shape Scaled"
Congruent figures are exact duplicates. Similar figures keep the same shape but change size.
"For congruence, you need a Side in the sandwich (SAS, ASA) or all three Sides (SSS)"
The key criteria always involve sides being equal, not just angles. That is why AAA alone cannot prove congruence.
"Perimeter: k. Area: k². Volume: k³."
The exponent matches the dimension: length (1D), area (2D), volume (3D). So scale factor is raised to the 1st, 2nd, or 3rd power.
"Look for parallel lines — they create equal corresponding angles for AA similarity"
Whenever you see parallel lines cutting through triangles, corresponding and alternate interior angles are your ticket to proving similarity.
"Always write the full congruence/similarity statement with vertices in matching order"
Writing △ABC ≅ △DEF tells the reader that A corresponds to D, B to E, and C to F. This prevents errors in identifying corresponding parts.
Quick Revision Summary
- Congruent figures have the exact same shape and size (≅).
- Similar figures have the same shape but different sizes (~). Corresponding sides are proportional.
- Congruence criteria: SSS, SAS, ASA, AAS, and HL (right triangles only).
- Similarity criteria: AA, SSS (proportional), and SAS (proportional + included angle).
- Scale factor (k) is the ratio of corresponding side lengths. Perimeter scales by k, area by k², volume by k³.
- SSA and AAA are NOT valid congruence criteria.
- To find missing sides: prove similarity → identify corresponding sides → set up proportion → solve.
- Always state the criterion you use and write vertices in matching order.
Frequently Asked Questions
- What is the fundamental difference between congruence and similarity?
- Congruence means identical in both shape and size. Similarity means identical in shape only, but potentially different in size (scaled versions of each other). All congruent figures are similar (with a scale factor of 1), but not all similar figures are congruent.
- Why isn't SSA a valid congruence criterion?
- SSA (Side-Side-Angle) is ambiguous because it's possible to construct two different triangles with the same two side lengths and the same non-included angle. Imagine swinging a side of a specific length from a vertex — it might intersect the third side at two different points, creating two different triangles.
- Can AAA prove congruence?
- No, AAA (Angle-Angle-Angle) only proves similarity, not congruence. For example, an equilateral triangle with side length 5 cm and another with side length 10 cm both have angles of 60°, 60°, 60°. They are similar but clearly not congruent.
- How do I know which sides are corresponding?
- Look at the angles: the side opposite a particular angle in one triangle corresponds to the side opposite the equal angle in the other. Follow the naming convention: if △ABC ∼ △DEF, then AB corresponds to DE, BC to EF, and AC to DF. Angle A corresponds to D, B to E, and C to F.
- What if the triangles are overlapping?
- When triangles overlap, it helps to redraw them separately, or carefully trace the two distinct triangles to identify their corresponding parts and shared angles/sides. Look for parallel lines, as they often create corresponding or alternate interior angles, which are key for AA similarity.
Practice Quiz
Test your understanding of congruence and similarity — select the correct answer for each question.
1.What does congruent mean?
2.What does similar mean?
3.SSS congruence means:
4.AA similarity means:
5.If the scale factor is 2:3 and the smaller perimeter is 10, what is the larger perimeter?
6.Which is NOT a congruence theorem?
7.If triangles are similar with a scale factor of 1:4, what is the ratio of their areas?
8.HL congruence applies to:
9.Corresponding angles in similar triangles are:
10.The scale factor from a small figure to a large figure is 3/5. If the small area is 9, what is the large area?
Final Study Advice
- 1.Always draw and label the figures clearly before attempting any proof or calculation.
- 2.When triangles overlap or are rotated, redraw them separately to clearly see corresponding parts.
- 3.Practice writing full congruence/similarity statements with vertices in the correct order.
- 4.In word problems, identify the similar triangles first, then set up proportions to solve for unknowns.
- 5.Remember that all congruent figures are similar (with k = 1), but not all similar figures are congruent.