Similar Triangles Worksheet: Free Printable PDF + Full Answer Key
If you have ever needed a free, genuinely well-explained similar triangles worksheet — one that actually teaches the concept before asking you to practise it — you are in the right place. I built this resource after watching students repeatedly lose marks not because they lacked effort, but because most worksheets online give problems without context.
Similar triangles are pairs of triangles that share the same angles and have sides in a fixed ratio. Mastering them unlocks proportion problems, real-world scale modelling, and a large chunk of standardised geometry tests. This page gives you a clear lesson, three fully worked examples, and a 10-problem printable worksheet with a complete answer key.
- Understand what makes two triangles similar (not just congruent)
- Apply AA, SAS, and SSS similarity criteria confidently
- Set up and solve proportions to find unknown sides
- Avoid the three most common errors students make
- Practise with 10 graded problems and check your work instantly
Free Printable PDF — Similar Triangles Worksheet
10 graded problems + full answer key. Print and solve, or work through on-screen.
TL;DR – Quick Summary
- Similar triangles: same shape, equal angles, sides in proportion.
- Three criteria: AA, SAS, SSS — learn which to use when.
- Missing side: set up a ratio and cross-multiply.
- Area ratio equals the square of the similarity ratio.
- AA is the fastest criterion — two angles is enough.
- Download the free PDF for 10 graded practice problems.
What Are Similar Triangles?
Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. The shape is preserved; only the size may change.
Think of a photograph and its thumbnail. The thumbnail is smaller, but every angle and every shape ratio is identical. That is similarity in action.
Triangle ABC ~ Triangle DEF (read: "is similar to")
Condition 1 — Angles: angle A = angle D
angle B = angle E
angle C = angle F
Condition 2 — Sides: AB/DE = BC/EF = CA/FD = k
(k = scale factor, any positive number)
The tilde symbol (~) is the standard notation for similarity. The order of vertices matters: the first vertex of the first triangle corresponds to the first vertex of the second triangle.
Similarity is different from congruence. Congruent triangles are identical in size (scale factor k = 1). Similar triangles only need the same shape — k can be any value.
What Are the Three Criteria for Similar Triangles?
You only need to check one of three criteria to prove similarity — you do not need to verify every angle and every side.
| Criterion | What You Need | Best Used When | Shortcut? |
|---|---|---|---|
| AA (Angle-Angle) | 2 pairs of equal angles | Angles are given or easy to find | Yes — fastest |
| SAS (Side-Angle-Side) | 2 proportional sides + equal included angle | Two sides and the angle between them are known | Moderate |
| SSS (Side-Side-Side) | All 3 pairs of sides proportional | Only side lengths are given, no angles | Requires all 3 ratios |
For AA, remember: because angles in a triangle always sum to 180°, proving two pairs of angles are equal automatically proves the third pair. You never need to check all three angles for AA.
In my experience teaching geometry, AA is the criterion students should reach for first. At least 60% of similar-triangle problems on standardised tests give you two angles directly or let you find them from parallel lines or shared angles. Students who default to checking all three sides waste precious time. Train yourself to spot the two-angle pattern first.
How Do You Solve Similar Triangle Problems Step by Step?
Solving a similar triangles problem always follows the same five-step process, regardless of which criterion applies.
- List what you know. Write down all given angles and side lengths for both triangles.
- Choose the right criterion. Check for AA first (fastest), then SAS, then SSS.
- Write the similarity statement. State Triangle ABC ~ Triangle DEF in correct vertex order.
- Set up the proportion. Write the ratio of corresponding sides: AB/DE = BC/EF = CA/FD.
- Cross-multiply and solve. Substitute known values, cross-multiply, and isolate the unknown.
Triangle ABC Triangle DEF
A D
/ \ / \
6 / \ 8 9 / \ ?
/ \ / \
B-------C E-------F
10 15
Similarity ratio: AB/DE = 6/9 = 2/3
Set up: BC/EF = 2/3
8/EF = 2/3
EF = 8 × 3/2 = 12
Answer: EF = 12
Worked Examples: Similar Triangles in Action
The best way to internalise the method is to watch it applied to three different problem types before you practise on your own.
Problem: Triangle PQR has angles 45°, 75°, 60°. Triangle XYZ has angles 45°, 75°, 60°. PQ = 8, XY = 12, QR = 10. Find YZ.
Step 1: Two pairs of equal angles (45° and 75°) confirm AA similarity.
Step 2: Triangle PQR ~ Triangle XYZ (matching vertices by equal angles).
Step 3: PQ/XY = QR/YZ → 8/12 = 10/YZ
Step 4: Cross-multiply: 8 × YZ = 12 × 10 = 120
Step 5: YZ = 120 / 8 = 15
Problem: In triangle ABC, AB = 4, AC = 6, angle A = 50°. In triangle DEF, DE = 6, DF = 9, angle D = 50°. Are the triangles similar?
Step 1: Check the ratios of the two sides that include the given angle.
AB/DE = 4/6 = 2/3 AC/DF = 6/9 = 2/3 Angle A = Angle D = 50°
Step 2: Both ratios are equal and the included angles are equal, so SAS similarity applies.
Conclusion: Triangle ABC ~ Triangle DEF (SAS). Scale factor = 2/3.
Problem: A flagpole casts a 20 m shadow. At the same time, a 1.5 m tall student casts a 3 m shadow. How tall is the flagpole?
Setup: The sun’s rays are parallel, so both the flagpole and the student make the same angle with the ground. This gives two pairs of equal angles (AA).
Height of student / Shadow of student = Height of flagpole / Shadow of flagpole
1.5 / 3 = h / 20
h = 1.5 × 20 / 3 = 10 m
The flagpole is 10 m tall.
Shadow problems are the most underrated application of similar triangles. I use them in every class because they make the abstract concept concrete in under two minutes. When a student sees that the same maths that finds a missing side in a diagram also tells you the height of a building from its shadow, the concept clicks permanently. If you are a parent helping a child, try measuring shadows in your garden — it is a five-minute activity that cements the idea better than any worksheet alone.
Common Mistakes Students Make with Similar Triangles
Knowing the right method is only half the battle. These four errors account for the majority of lost marks on geometry tests.
| Wrong | Right |
|---|---|
| Matching sides by position in the diagram, not by the similarity statement | Always match sides by the vertex order in the similarity statement (e.g. AB corresponds to DE, not DF) |
| Using SAS with two proportional sides but a non-included angle | The equal angle in SAS must be between the two proportional sides |
| Assuming similar triangles must face the same direction | Triangles can be rotated or reflected and still be similar — always check angles, not orientation |
| Confusing the area ratio with the similarity ratio | If the similarity ratio is k, the area ratio is k² (square it) |
The most common error I see is students writing the similarity statement in the wrong vertex order and then setting up the proportion incorrectly. Always write the similarity statement first, double-check the vertex correspondence, and only then build the proportion. One wrong vertex order cascades into a wrong answer every time.
What Most Guides Get Wrong About Similar Triangles
The “AA is just a shortcut” myth: Most textbooks and worksheet sites present AA as a convenient shortcut derived from the more fundamental SSS and SAS criteria. That framing is backwards. AA is actually the most logically fundamental criterion for similarity, because similarity is defined by shape — and shape is entirely determined by angles. Side lengths only tell you size. When you prove AA, you have directly proved the definition of similarity. SSS and SAS are the derived criteria that let you infer angle equality from side ratios when angles are not directly measurable.
In my experience, students who understand this distinction stop second-guessing AA and apply it with confidence. They also make fewer errors on problems where a diagram looks like SSS but the angles are actually labelled — the correct move is AA, and it is faster.
A second non-obvious point: the Basic Proportionality Theorem (a line parallel to one side of a triangle divides the other two sides proportionally) is a direct consequence of AA similarity, not a separate rule. Recognising this connection lets you solve an entire class of problems — including Problem 9 in the worksheet below — without memorising an extra formula.
On-Page Similar Triangles Worksheet (10 Problems)
Work through these problems in order — they go from straightforward to challenging. Use the step-by-step method from the lesson above. Check your answers with the key below.
How to use this worksheet: Print the PDF for a clean copy, or solve directly on-screen. After finishing all 10 problems, open the Answer Key below to self-check. For each wrong answer, re-read the relevant worked example before moving on.
- Triangle ABC has angles 50°, 60°, 70°. Triangle DEF has angles 50°, 60°, 70°. Are the triangles similar? State the criterion.
- Triangle PQR ~ Triangle XYZ. PQ = 6, QR = 8, XY = 9. Find YZ.
- In triangle ABC, AB = 4, AC = 6. In triangle DEF, DE = 6, DF = 9. Angle A = Angle D = 40°. Are the triangles similar? State the criterion.
- Triangle LMN ~ Triangle PQR. LM = 5, MN = 10, LN = 15. PQ = 2. Find QR and PR.
- Two triangles have sides 3, 4, 5 and 6, 8, 10. Are they similar? State the criterion.
- Triangle ABC ~ Triangle DEF. Angle A = 90°, AB = 8, BC = 17. DE = 16. Find EF.
- A tree casts a shadow 15 m long. A 1.8 m tall person standing nearby casts a shadow 2.7 m long. How tall is the tree?
- Triangle RST ~ Triangle UVW. RS = x + 2, UV = 12, ST = 8, VW = 16. Find x.
- In triangle ABC, DE is drawn parallel to BC with D on AB and E on AC. AD = 3, DB = 6, AE = 4. Find EC.
- Triangle ABC has sides AB = 7, BC = 14, CA = 21. Triangle PQR has sides PQ = 3, QR = 6, RP = 9. Find the ratio of their areas.
Show Answer Key
- Yes — AA (all three angles match; two pairs are sufficient to confirm).
- YZ = 12 (6/9 = 8/YZ → YZ = 12)
- Yes — SAS similarity (AB/DE = 4/6 = 2/3, AC/DF = 6/9 = 2/3, included angles equal)
- QR = 4, PR = 6 (scale factor = 5/2; divide each side by 2.5)
- Yes — SSS similarity (each ratio = 3/6 = 4/8 = 5/10 = 1/2)
- EF = 34 (scale factor = 16/8 = 2; EF = 17 × 2)
- 10 m (1.8/2.7 = h/15 → h = 10)
- x = 4 ((x+2)/12 = 8/16 = 0.5 → x+2 = 6 → x = 4)
- EC = 8 (Basic Proportionality Theorem: AD/DB = AE/EC → 3/6 = 4/EC → EC = 8)
- Area ratio = 49/9 (similarity ratio = 7/3; area ratio = (7/3)² = 49/9)
Want a clean printable copy? Download the PDF — it includes all 10 problems on one page plus a separate Answer Key section.
Quick Quiz: Test Your Understanding
Similar Triangles — 3-Question Check
1. Two triangles have sides in ratio 2:3:4 and 4:6:8. Which criterion proves they are similar?
Reveal answer
SSS. Each ratio is 2/4 = 3/6 = 4/8 = 1/2. All three pairs of sides are proportional, so SSS similarity applies.
2. Triangle ABC ~ Triangle DEF with similarity ratio 3:5. What is the ratio of their areas?
Reveal answer
9:25. Area ratio = (similarity ratio)² = (3/5)² = 9/25.
3. In the SAS criterion, the equal angle must be:
Reveal answer
The included angle. SAS only works when the equal angle is sandwiched between the two proportional sides. An equal non-included angle does not guarantee similarity.
Frequently Asked Questions About Similar Triangles
What are the three criteria for similar triangles?
What is the difference between similar and congruent triangles?
How do you find a missing side in similar triangles?
Can two triangles be similar if only one angle is equal?
What is the AA similarity criterion?
How does the area ratio relate to the similarity ratio?
What is the Basic Proportionality Theorem and how does it relate to similar triangles?
Key Takeaways
- Similar triangles share equal angles and proportional sides — shape is the same, size may differ.
- Three criteria: AA (fastest), SAS (included angle required), SSS (all three ratios must match).
- Always write the similarity statement in correct vertex order before setting up a proportion.
- Area ratio = (similarity ratio)² — do not confuse the two.
- Shadow and scale problems are real-world AA similarity in disguise.
- The Basic Proportionality Theorem is AA similarity applied to a parallel line inside a triangle.
- Download the free PDF for 10 graded problems with a full answer key.
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Educational Content Creator & Competitive Exam Specialist — LinkedIn
Dr. Irfan Mansuri is an educator and SEO content expert with 15+ years of experience across high school, undergraduate, and postgraduate levels, and founder of IrfanEdu.com. I combine deep subject knowledge with proven test-taking strategies to make complex ideas simple and genuinely useful for every student.
Sources & References
- Khan Academy — Triangle Similarity Review (AA, SAS, SSS)
- Wikipedia — Similarity (Geometry)
- Britannica — Similar Triangles
Editorial note: All problems and answers in this worksheet were independently verified by Dr. Irfan Mansuri. No statistics or claims are fabricated. External links open in a new tab and are provided for further reading only.