Two figures are similar if a sequence of rigid motions and a dilation carries one onto the other.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What that gives you
Matching angles are equal, and matching sides are all in the same ratio.
Two angles are enough for triangles
If two triangles share two equal angles, the third must match too, so they are similar.
Finding a missing side
Set up the ratio from a known pair, then apply it to the pair you want.
Similar means reachable by rigid motions and a dilation
Two figures are similar if some sequence of translations, reflections, rotations and dilations carries one onto the other. Congruence is the special case where the scale factor is 1.
What similarity guarantees
Corresponding angles are equal and corresponding sides are in a constant ratio. These two conditions together are equivalent to similarity, and either one alone is not enough for general polygons.
Working with the ratio
If two similar triangles have sides in ratio 3 : 5, then every pair of corresponding sides has that ratio. Finding a missing length means setting up a proportion — the ratio work from Grade 6 doing real geometric work.
Areas scale by the square
Similar figures with side ratio 3 : 5 have areas in ratio 9 : 25. This is the same squaring seen in Grade 7 scale drawings. Forgetting it is one of the most common errors on similarity problems involving area.
Step 2: Try It Yourself
Tap and try it out.
| Small figure | Large figure |
|---|---|
| 3 | 6 |
| 6 | 12 |
| 9 | 18 |
Every row is the same ratio. 3 to 6 is the same comparison as 9 to 18.
Step 3: Watch an Example
One step at a time.
Watch Leo Find a Missing Side
Two similar triangles. The small has sides 4 and 6; the large has 10 matching the 4.
- Step 1
He finds the scale factor from the known pair: 10 ÷ 4 = 2.5.
Step 4: Your Turn
Practice makes it stick.
The Similar Pair
Problem 1 of 2
Similar triangles. 3 matches 12, and another side is 5. What is its match?
The Angles
Problem 2 of 2
Two similar triangles. One angle is 63 degrees. What is the matching angle?
degrees
Use the Ratio
1 of 4
Similar figures. 2 matches 8. Another side is 5. Its match?
2 of 4
Similar figures. 6 matches 9. Another side is 4. Its match?
3 of 4
Two triangles share angles of 50 and 60. Are they similar?
4 of 4
Similar figures. 5 matches 15. Another side is 7. Its match?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Similar figures. 4 matches 10. Another side is 6. Its match?
Question 2 of 2
What is true of similar figures?
What You Learned
- Similar means rigid motions plus a dilation carry one onto the other.
- Matching angles are equal and matching sides share one ratio.
- Two equal angles make triangles similar.