Two figures are similar when a dilation, possibly with rigid motions, carries one onto the other.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What follows
Similar figures have equal corresponding angles and proportional corresponding sides.
The AA criterion
Two pairs of equal angles prove triangles similar. The third pair follows from the 180° total, so it need not be checked.
SSS and SAS for similarity
Three proportional sides, or two proportional sides with the included angle equal, also prove similarity.
Setting up the proportion
Match corresponding sides carefully, then set the two ratios equal and cross-multiply.
The pairing error
Most wrong answers come from pairing the wrong sides, not from the algebra. Identify the correspondence first.
Similar means reachable by motions and a dilation
Two figures are similar if rigid motions plus a dilation carry one onto the other. Congruence is the special case where the scale factor is 1, so similarity is the more general relation.
AA suffices for triangles
Two equal angles force the third to match, and the triangles are similar. This works only for triangles: a square and a long rectangle share all angles and are not similar.
Setting up the proportion
Write every ratio in the same direction — new over old throughout, or old over new throughout. Mixing directions within one proportion gives the reciprocal of the correct answer.
Identify corresponding parts first
The longest side matches the longest, and equal angles match. On rotated or reflected figures the correspondence is not visually obvious, and getting it wrong invalidates every ratio.
Step 2: Try It Yourself
Tap and try it out.
Before: 4
After: 8
The factor is more than 1, so the bar got longer.
Step 3: Watch an Example
One step at a time.
Watch Sana Find a Missing Side
Two similar triangles have corresponding sides 6 and 9. Another side of the first is 8.
- Step 1
The scale factor is 9 ÷ 6 = 1.5.
Step 4: Your Turn
Practice makes it stick.
The Factor
Problem 1 of 2
Corresponding sides are 6 and 9. What is the scale factor?
The Missing Side
Problem 2 of 2
Scale factor 1.5 applied to a side of 8. What is the corresponding side?
Use the Proportion
1 of 8
Corresponding sides 4 and 12. Scale factor?
2 of 8
Scale factor 3 on a side of 7. Corresponding side?
3 of 8
How many equal angle pairs does AA need?
4 of 8
Corresponding sides 10 and 5. Scale factor?
5 of 8
A 3-4-5 triangle scaled to have shortest side 9. What is the longest side?
6 of 8
Do similar figures have equal angles? 1 yes, 0 no.
7 of 8
Sort each criterion by whether it proves similarity or congruence.
Tap something to move it.
- Empty
- Empty
8 of 8
Scale factor 2.5 on a side of 6. Corresponding side?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Corresponding sides 5 and 20. What is the scale factor?
Question 2 of 2
Why does AA prove similarity?
What You Learned
- Similar figures have equal angles and proportional sides.
- AA is enough for triangles, because the third angle follows.
- Identify corresponding sides before writing any proportion.