Topic 3 of 7
Triangle congruence and similarity
Step 12 · Geometry topics
Prove triangles congruent with SSS, SAS, ASA, AAS and HL, prove them similar with AA, SSS and SAS similarity, and use similar triangles to measure lengths you cannot reach.
Congruence shortcuts
Two triangles are congruent if all three sides and all three angles match. You do not have to check all six: any of these shortcuts is enough.
| Shortcut | What matches |
|---|---|
| SSS | three sides |
| SAS | two sides and the angle between them |
| ASA | two angles and the side between them |
| AAS | two angles and a side not between them |
| HL | the hypotenuse and a leg of two right triangles |
Example: Which shortcut?
In triangles ABC and DEF, AB = DE, ∠B = ∠E and BC = EF. Which shortcut shows they are congruent?
- Two sides match, and the angle between them matches.
AnswerSAS
Once triangles are proved congruent, every pair of matching parts is equal. Proofs often give this reason as CPCTC: corresponding parts of congruent triangles are congruent.
Similar triangles
Triangles are similar if they have the same shape: matching angles are equal and matching sides are in the same ratio. Three shortcuts prove similarity.
| Shortcut | What you need |
|---|---|
| AA | two pairs of equal angles |
| SSS similarity | all three side ratios equal |
| SAS similarity | two side ratios equal and the angle between them equal |
Example: Checking side ratios
Triangle ABC has sides 4, 6 and 8. Triangle DEF has sides 6, 9 and 12. Are they similar?
- 6 ÷ 4 = 1.5, 9 ÷ 6 = 1.5 and 12 ÷ 8 = 1.5.
AnswerYes, with scale factor 1.5
Example: A missing side
△ABC ~ △XYZ, with AB = 5, BC = 8 and XY = 15. Find YZ.
- The scale factor is 15 ÷ 5 = 3.
- YZ = 8 × 3 = 24.
Answer24
Using similarity to measure
Similar triangles let you measure things you cannot reach, using shadows, mirrors or a line parallel to one side of a triangle.
Example: Shadows
A 1.8 m person casts a 2.4 m shadow at the moment a flagpole casts a 16 m shadow. How tall is the flagpole?
- Height ÷ shadow is the same for both: 1.8 ÷ 2.4 = 0.75.
- 16 × 0.75 = 12.
Answer12 m
Example: A line parallel to a side
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. AD = 4, DB = 6 and AE = 5. Find EC.
- The parallel line cuts both sides in the same ratio: AD ÷ DB = AE ÷ EC.
- 4 ÷ 6 = 5 ÷ EC, so EC = 5 × 6 ÷ 4 = 7.5.
Answer7.5
Try it yourself
Work each one on paper first, then open the answer to check.
Two triangles match in two angles and the side between them. Which shortcut proves them congruent?
Show answer
ASA
Can SSA prove two triangles congruent?
Show answer
No: two different triangles can share two sides and an angle not between them.
Triangles have angles of 40°, 60°, 80° and 40°, 80°, 60°. Are they similar?
Show answer
Yes, by AA.
Similar triangles have scale factor 2.5. A side of the smaller one is 6. How long is the matching side of the larger?
Show answer
15
A 2 m stick casts a 3 m shadow while a tree casts a 21 m shadow. How tall is the tree?
Show answer
14 m
Similar triangles have sides in the ratio 1 : 4. What is the ratio of their areas?
Show answer
1 : 16