Ever wondered why a tiny map can show a whole country perfectly?
Two triangles are called similar when they have the same shape but not necessarily the same size. That means their angles match up and their sides grow or shrink by the same factor.
What is Triangle Similarity?
A similar triangle is a triangle that has exactly the same angles as another triangle, and the lengths of its sides are proportional – think of it like a photo that’s been resized without stretching.
We use the term corresponding to describe parts that line up with each other: corresponding angles are the angles that sit in the same position, and corresponding sides are the sides opposite those angles.
Key Theorems for Similar Triangles
AA (Angle‑Angle) Criterion
If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. The third angle automatically matches because the sum of angles in any triangle is 180°.
SSS (Side‑Side‑Side) Criterion
When the three sides of one triangle are in the same proportion as the three sides of another triangle, the triangles are similar. In other words, if you can write a single number k (called the scale factor) so that each side of triangle A equals k times the matching side of triangle B, they’re similar.
SAS (Side‑Angle‑Side) Criterion
If two sides of one triangle are proportional to two sides of another triangle and the included angles (the angles between those sides) are equal, the triangles are similar.
Imagine water flowing through two pipes of different diameters. If the pressure (angle) stays the same and the pipe widths (sides) change by the same factor, the flow pattern stays alike – that’s the idea behind the SAS rule.
How to Prove Triangles are Similar – A Step‑by‑Step Guide
- Identify which theorem (AA, SSS, or SAS) fits the given information.
- Write down the known angles or side ratios.
- Show that the required condition of the chosen theorem holds.
- Conclude that the triangles are similar and state the scale factor if needed.
Worked Example 1: Using the AA Criterion
Problem: In triangle ABC, ∠A = 40°, ∠B = 70°. In triangle DEF, ∠D = 40°, ∠E = 70°. Prove that the triangles are similar and find the ratio of AB to DE if AB = 8 cm and DE = 5 cm.
Solution:
- We have two pairs of equal angles: ∠A = ∠D and ∠B = ∠E.
- By the AA criterion, ΔABC ∼ ΔDEF.
- Corresponding sides opposite the equal angles are AB ↔ DE, BC ↔ EF, and AC ↔ DF.
- Scale factor k = AB ÷ DE = 8 ÷ 5 = 1.6.
- Therefore every side of ΔABC is 1.6 times the matching side of ΔDEF.
Worked Example 2: Using the SSS Criterion
Problem: Triangle PQR has sides 9 cm, 12 cm, 15 cm. Triangle XYZ has sides 6 cm, 8 cm, 10 cm. Show that the triangles are similar and state the scale factor.
Solution:
- Compute the ratios: 9/6 = 1.5, 12/8 = 1.5, 15/10 = 1.5.
- All three side ratios are equal, so the sides are proportional.
- By the SSS criterion, ΔPQR ∼ ΔXYZ.
- The common ratio 1.5 is the scale factor from ΔXYZ to ΔPQR.
Quick Comparison of the Three Similarity Tests
| Test | What You Need | How You Use It |
|---|---|---|
| AA | Two equal angles | Match the remaining angle automatically |
| SSS | Three side ratios equal | Show a single scale factor works for all sides |
| SAS | Two side ratios equal + included angle equal | Confirm the angle sits between the proportional sides |
📝 Likely Exam Questions
- Q1. In ΔABC, ∠A = 30° and ∠B = 60°. In ΔDEF, ∠D = 30° and ∠E = 60°. Prove the triangles are similar and find the ratio of AC to DF if AC = 9 cm and DF = 6 cm.
Answer: AA criterion → similar. Scale factor = 9/6 = 3/2. - Q2. The sides of ΔGHI are 5 cm, 12 cm, 13 cm. ΔJKL has sides 10 cm, 24 cm, 26 cm. State whether they are similar and give the scale factor.
Answer: Ratios 5/10 = 0.5, 12/24 = 0.5, 13/26 = 0.5 → SSS → similar, scale factor 0.5 (ΔJKL is twice as big). - Q3. Two triangles have a pair of equal angles and the sides including those angles are in the ratio 4:7. Are the triangles similar? Explain.
Answer: Yes, by SAS because the included angle is equal and the adjacent sides are proportional (4/7). - Q4. In a right‑angled triangle, a line drawn from the right‑angle vertex to the hypotenuse divides the triangle into two smaller triangles. Show that the two smaller triangles are similar to each other and to the original triangle.
Answer: Use AA: each small triangle shares the right angle and one acute angle with the original, so all three are similar. - Q5. If ΔMNO ∼ ΔPQR and the area of ΔMNO is 36 cm² while the area of ΔPQR is 81 cm², find the ratio of corresponding sides.
Answer: Area ratio = (scale factor)² → √(36/81) = 2/3. So side ratio = 2:3.