Geometry · Step-by-step guide
Similar Triangles
Prove triangles similar and use proportions to find missing side lengths.
Start with the central idea
Similar triangles have equal corresponding angles and proportional corresponding sides. Prove similarity with AA, SAS similarity, or SSS similarity, then write proportions in consistent corresponding order.
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Similarity preserves shape
Corresponding angles match while all corresponding side lengths share one scale factor.
Order matters
The order in a similarity statement determines every valid side correspondence.
Use one scale factor
New length divided by original length is constant for every corresponding pair.
See the structure
A picture makes the relationships easier to remember.
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
Two triangles each contain a 40° angle and a 70° angle.
- Two angle pairs are congruent.
- The third pair must also match.
- AA is sufficient.
A 3-4-5 triangle is scaled so the shortest side is 9.
- The scale factor is .
- Multiply every original side by 3.
- The other sides are 12 and 15.
A 6-ft person casts a 4-ft shadow; a tree casts a 14-ft shadow.
- The sun creates equal acute angles.
- Write .
- Cross-multiply and solve.
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Pairing noncorresponding sides in a proportion.
Adding the scale factor instead of multiplying.
Using congruence language when sizes differ.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Similar Triangles
Practice similarity proofs, scale factors, and missing lengths.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Use the angle measures in the diagram to prove the triangles similar.
Show hint
Two matching angle pairs are enough.
Show worked solution
- The 40° angles correspond, and the 60° angles correspond.
- Two corresponding angle pairs are congruent.
Answer: Similar by AA.
Use the side lengths in the diagram to prove the triangles similar.
Show hint
Compare all three pairs of corresponding sides.
Show worked solution
- , , and .
- Every corresponding side has the same scale factor.
Answer: Similar by SSS; scale factor 2.
The triangles in the diagram are similar. Find x.
Show hint
Match the side labeled 6 with 9 and the side labeled 10 with x.
Show worked solution
- .
- Cross-multiply: .
- .
Answer:
Use the similar shadow triangles in the diagram to find the tree height h.
Show hint
Match height with height and shadow with shadow.
Show worked solution
- .
- Cross-multiply: .
- .
Answer: 25 feet.
Use the marked sides and angles to prove the triangles similar.
Show hint
Compare the sides surrounding the equal included angles.
Show worked solution
- and .
- The included angles are congruent and the adjacent side pairs are proportional.
Answer: Similar by SAS.
Frequently asked questions
Quick answers before you move on.
What should I remember first?
Similar triangles have equal corresponding angles and proportional corresponding sides. Prove similarity with AA, SAS similarity, or SSS similarity, then write proportions in consistent corresponding order.
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.