How to Prove Two Triangles are Similar: Step-by-Step Criteria & Examples
The concept of Similarity of Triangles is a crucial topic in Maths, allowing you to compare the shape, side lengths, and angle measurements of two triangles. It has practical uses in geometry, map-reading, architecture, and problem solving for exams such as CBSE Class 9, 10, JEE, and more.
What Is Similarity of Triangles?
Similarity of triangles means two triangles have exactly the same shape, but not necessarily the same size. In detail, triangles are similar if their corresponding angles are equal, and their corresponding sides are in the same proportion. This concept is widely used in identifying scale drawings, solving geometry problems, and understanding the relationship between similar and congruent figures.
Key Formula for Similarity of Triangles
The key formula for similarity of triangles is:
If △ABC ∼ △DEF, then
\[
\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
\]
and
\[
\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F
\]
Criteria for Similarity of Triangles (AA, SSS, SAS)
To quickly test if two triangles are similar, check these three main rules:
| Criterion | What to Check | How to Apply |
|---|---|---|
| AA (Angle-Angle) | Two pairs of corresponding angles are equal. | If ∠A=∠D and ∠B=∠E, then the triangles are similar. |
| SSS (Side-Side-Side) | All three pairs of corresponding sides are in the same ratio. | If \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \), then the triangles are similar. |
| SAS (Side-Angle-Side) | Two pairs of sides are in the same ratio and the included angle is equal. | If \( \frac{AB}{DE} = \frac{AC}{DF} \) and ∠A=∠D, then the triangles are similar. |
Quick Trick: Remember "AA, SSS, SAS" — focus on angles and proportional sides!
Proving Similarity – Step-by-Step Example
Example: Show that triangles ABC and DEF (where AB=6 cm, BC=8 cm, CA=10 cm, DE=9 cm, EF=12 cm, FD=15 cm) are similar.
\( \frac{AB}{DE} = \frac{6}{9} = \frac{2}{3} \)
\( \frac{BC}{EF} = \frac{8}{12} = \frac{2}{3} \)
\( \frac{CA}{FD} = \frac{10}{15} = \frac{2}{3} \)
2. Since all three side ratios are equal, by the SSS criterion, triangles ABC and DEF are similar.
Properties of Similar Triangles
- Corresponding angles are equal.
- Corresponding sides are in the same ratio (proportional).
- The area ratio of two similar triangles equals the square of the scale factor for corresponding sides.
- If one triangle is congruent to another, it is also similar.
Difference Between Similar and Congruent Triangles
| Similar Triangles | Congruent Triangles |
|---|---|
| Same shape, can have different sizes. | Same shape and same size. |
| All corresponding angles equal; sides proportional. | All corresponding angles and sides equal. |
| Symbol: ∼ (e.g., △ABC ∼ △DEF) | Symbol: ≅ (e.g., △ABC ≅ △DEF) |
Classroom Tip
A handy mnemonic: “AA, SSS, SAS” helps you remember the similarity of triangles rules. Pair it with a simple diagram in class or revision notes! Vedantu teachers often draw colored triangles side-by-side to help you spot similarities faster.
Try These Yourself
- Are triangles with angles 65°, 55°, 60° and 65°, 60°, 55° similar?
- The sides of a triangle are 5 cm, 12 cm, 13 cm. Another triangle has sides 10 cm, 24 cm, 26 cm. Are they similar?
- Find the value of x if two similar triangles have corresponding sides of length 4 cm and 6 cm, 6 cm and x cm.
Frequent Errors and Misunderstandings
- Forgetting to check all angle pairs or side ratios.
- Mixing up similarity and congruence (not every similar triangle is congruent).
- Not matching the correct order of corresponding vertices.
Real-Life Applications
- Creating maps and scale models in geography.
- Designing ramps, roofs, and art using geometric patterns in architecture.
- Measuring the height of big objects using shadows (indirect measurement).
- Solving image enlargement/shrinking problems in computer graphics.
Relation to Other Concepts
The topic of Similarity of Triangles links directly to Triangle Theorems, and broader Polygons and Their Properties. Mastering it helps in both coordinate and practical geometry chapters.
We explored Similarity of Triangles, its rules, formulas, solved examples, and connections to real-world situations. Keep practicing with Vedantu to become a triangle similarity pro!
FAQs on Similarity of Triangles: Rules, Properties & Problems
1. What are similar triangles in Maths?
Similar triangles are triangles that have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional. This means the ratios of the lengths of corresponding sides are equal. We use the symbol ‘∼’ to denote similarity.
2. What are the criteria for proving triangle similarity?
There are three main criteria to prove that two triangles are similar:
• **AA (Angle-Angle):** If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• **SAS (Side-Angle-Side):** If two sides of one triangle are proportional to two sides of another triangle, and the included angle between those sides is congruent in both triangles, then the triangles are similar.
• **SSS (Side-Side-Side):** If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.
3. What is the difference between similar and congruent triangles?
Similar triangles have the same shape but may differ in size; congruent triangles are identical in both shape and size. Similar triangles have equal corresponding angles and proportional corresponding sides; congruent triangles have equal corresponding angles and sides. Congruent triangles are always similar, but similar triangles are not always congruent.
4. How can I use similar triangles to solve for an unknown side?
If two triangles are similar, you can set up a proportion using the ratios of corresponding sides. If you know the lengths of some sides in both triangles, you can solve for any unknown side length by cross-multiplication and solving the resulting equation. Remember to correctly identify corresponding sides.
5. What are some easy ways to remember the AA, SSS, and SAS similarity rules?
Use mnemonics or visual aids. For example, think of 'AA' as 'Angle-Angle', 'SSS' as 'Side-Side-Side', and 'SAS' as 'Side-Angle-Side'. Drawing diagrams to represent each rule can also help.
6. Why doesn't AAA directly prove similarity?
While AAA (Angle-Angle-Angle) shows that triangles are equiangular, it doesn't guarantee similarity because triangles could be different sizes (enlarged or reduced versions of one another). Two angles being equal implies the third angle is also equal (angles in a triangle add up to 180°), hence AA is sufficient for similarity.
7. Are all equilateral triangles similar?
Yes, all equilateral triangles are similar because all their angles are equal (60° each), satisfying the AA similarity criterion. However, they may not be congruent unless their side lengths are also equal.
8. Can two triangles have equal areas but not be similar?
Yes. Area alone doesn't determine similarity. Two triangles can have the same area but different shapes (angles and side ratios) and thus not be similar.
9. What are some common mistakes to avoid when proving triangle similarity?
Common mistakes include: incorrectly identifying corresponding angles or sides; not showing sufficient reasoning or justification in proof; misapplying similarity criteria; and failing to state the conclusion clearly (using the correct similarity notation).
10. How are similar triangles used in real-world applications?
Similar triangles have many real-world applications, including: surveying (measuring distances indirectly); map-making (creating scaled representations); architecture (designing scaled models); and engineering (scaling down complex structures for analysis).
11. What is the Basic Proportionality Theorem (Thales Theorem)?
The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally. This theorem is fundamental to proving the similarity of triangles and solving problems involving proportional relationships.
