Grade 9 · Grade 9: Proof, Similarity and Trigonometry · Proof, Similarity and Right-Triangle Trigonometry
Congruence Proof and Similar Figures
Prove theorems about lines, angles and triangles, then scale lengths, areas and volumes.
Learning objectives
- Prove theorems about lines, angles, triangles and parallelograms.
- Apply SSS, SAS, ASA and RHS.
- Solve problems using similarity criteria.
- Use k, k² and k³ for length, area and volume scaling.
AERO Mathematics alignment
AERO.M9.GM.1
Prove theorems about lines, angles, triangles and parallelograms, and use congruence criteria such as SSS, SAS, ASA and RHS.
AERO.M9.GM.2
Use similarity criteria to establish relationships in geometric figures and solve problems involving lengths, areas and volumes of similar figures.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Two triangles have two equal sides and one equal angle that is not between them.
Think about it
Are they necessarily congruent? Try to draw a counterexample.
Hint: The angle position matters as much as the lengths.
EXPLAIN
Proof, criteria and scaling
| Criterion | Given | Congruent? |
|---|---|---|
| SSS | Three sides | Yes |
| SAS | Two sides and the included angle | Yes |
| ASA | Two angles and the included side | Yes |
| RHS | Right angle, hypotenuse and one side | Yes |
| SSA | Two sides and a non-included angle | Not always |
| AAA | Three angles | Similar only |
Writing a proof
A proof is a chain of justified statements. State what is given, apply known theorems such as vertical angles are equal or base angles of an isosceles triangle are equal, cite a congruence criterion, then conclude that corresponding parts are equal.
Scaling similar figures
If the scale factor is k, then lengths multiply by k, areas by k² and volumes by k³. Two similar solids with a length ratio 1:3 have a volume ratio 1:27, which is why small models are so much lighter than the real thing.
Common misconception
Equal angles alone never prove congruence. AAA guarantees the same shape, but the figures can be any size.
INVESTIGATION
Try it yourself
Construct two triangles with sides 5 cm and 7 cm and a non-included angle of 35°. Show that two different triangles are possible, then explain why SAS avoids the problem.
Watch
Introduction to Transformations
Khan Academy · 7:22
Grounds congruence criteria in transformations before you write proofs.
Before you watch: Is SSA enough to prove two triangles congruent?
- State the criterion you would use given two sides and the included angle.
- Two similar solids have scale factor 3. How do their volumes compare?
Interactive simulation · PhET
Area Builder
Create similar shapes with scale factors 2 and 3 and record perimeter and area.
While you explore
- Why does area scale by k²?
- Which measurements are unchanged by a rigid motion?
Key vocabulary
- Similar
- Same shape, possibly different size.
- Congruent
- Same shape and same size.
Practice questions
0/1 correct
Level 1 · Criterion A
Which condition guarantees two triangles are congruent?
MYP criterion tasks
Level 2 · Criterion D
Two similar cylinders have radii in the ratio 1:4. Compare their surface areas and volumes.
Reflect & track
Your learning log
Saved privately in this browser — no account needed.
How confident do you feel?