If two angles of one triangle equal two of another, the triangles are similar.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why two is enough
The three angles sum to 180°, so matching two forces the third to match as well.
What similarity gives
Corresponding sides are in one ratio, so a missing length can be found by proportion.
The side-splitter theorem
A line parallel to one side of a triangle cuts the other two sides proportionally.
Two angles suffice
If two angles of one triangle equal two of another, the triangles are similar. The third angle follows automatically since all three must total 180°, so checking a third would add nothing.
Special to triangles
Equal angles do not force similarity for other polygons — a square and a long rectangle have identical angles and different shapes. Triangles are rigid in a way other polygons are not, which is why AA works only here.
Finding the equal angles
Shared angles, vertical angles, and angles made equal by parallel lines are the usual sources. In a figure with overlapping triangles, identifying the shared angle is often the whole of the insight.
Similarity established, ratios follow
Once AA gives similarity, corresponding sides are proportional and a missing length comes from a proportion. Prove first with angles, compute afterwards with ratios — attempting the reverse order is where proofs go astray.
Step 2: Try It Yourself
Tap and try it out.
| Small triangle | Large triangle |
|---|---|
| 3 | 9 |
| 6 | 18 |
Every row is the same ratio. 3 to 9 is the same comparison as 6 to 18.
Step 3: Watch an Example
One step at a time.
Watch Sam Find a Height by Shadow
A 2 m post casts a 3 m shadow. A tree casts a 24 m shadow at the same moment.
- Step 1
The sun is at the same angle for both, and both stand vertically.
Step 4: Your Turn
Practice makes it stick.
The Shadow
Problem 1 of 2
A 3 m post casts a 4 m shadow. A tree casts a 20 m shadow. How tall is the tree?
m
The Proportion
Problem 2 of 2
Similar triangles. 4 matches 10, and another side is 6. Its match?
Use the Ratio
1 of 4
Similar. 5 matches 15, another side is 4. Its match?
2 of 4
Similar. 6 matches 9, another side is 8. Its match?
3 of 4
Two triangles share angles of 40° and 65°. Are they similar?
4 of 4
Similar triangles, one angle is 55°. The matching angle?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Similar. 8 matches 20, another side is 6. Its match?
Question 2 of 2
Why do two matching angles prove similarity?
What You Learned
- Two matching angles prove triangles similar.
- Corresponding sides are then in one ratio.
- A line parallel to a side splits the other two proportionally.