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 is the similarity of triangles?
The similarity of triangles refers to a geometric property where two triangles have the same shape but may differ in size. Two triangles are considered similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). The concept of similar triangles is widely used in geometry and is fundamental for understanding proportional relationships and solving various mathematical problems. At Vedantu, students can explore comprehensive lessons, solved examples, and interactive classes on triangle similarity to build a strong mathematical foundation.
2. What are the four rules for similar triangles?
The four main rules or criteria used to determine if two triangles are similar are:
- AAA (Angle-Angle-Angle) Criterion: If all corresponding angles of two triangles are equal, the triangles are similar.
- AA (Angle-Angle) Criterion: If any two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar.
- SAS (Side-Angle-Side) Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, the triangles are similar.
- SSS (Side-Side-Side) Criterion: If the corresponding sides of two triangles are in the same ratio, the triangles are similar.
3. How to solve similar triangles?
To solve similar triangles, follow these essential steps:
- Identify the pairs of corresponding angles and sides.
- Verify the similarity using the appropriate criteria (AAA, SSS, SAS, or AA).
- Set up the ratio of the lengths of corresponding sides, for example, $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ for triangles $ABC$ and $DEF$.
- Use these ratios to find unknown side lengths or angles.
4. What is the AAA rule for similar triangles?
The AAA (Angle-Angle-Angle) rule for similar triangles states that if the three angles of one triangle are respectively equal to the three angles of another triangle, then both triangles are similar. This means their corresponding sides are always in the same proportion, though the sizes may differ. The AAA similarity rule is a fundamental concept taught in Vedantu’s geometry classes, empowering students to understand and apply similarity principles effectively.
5. What is the difference between similar and congruent triangles?
The main difference is:
- Similar triangles have equal corresponding angles and proportional corresponding sides, but their sizes may vary.
- Congruent triangles have equal corresponding sides and angles; they are identical in shape and size.
6. How do you prove two triangles are similar using the SAS rule?
To prove two triangles are similar using the SAS (Side-Angle-Side) criterion:
- Show that a pair of corresponding angles in both triangles are equal.
- Demonstrate that the lengths of the sides including these angles are in the same ratio, i.e., $\frac{AB}{DE} = \frac{AC}{DF}$.
7. What are some real-life applications of similar triangles?
Similar triangles have many practical uses, including:
- Measuring heights or distances that are difficult to reach, like tall buildings or trees, using indirect measurement methods.
- Map-making and surveying, where scale drawings are based on similar triangles.
- Shadow problems to estimate height using the length of the shadow and proportional reasoning.
8. What is the importance of similar triangles in trigonometry?
The concept of similar triangles forms the backbone of trigonometry. Trigonometric ratios such as sine, cosine, and tangent originate from similar right triangles, where the ratio of corresponding sides remains constant for a given angle. This principle allows us to define and use trig functions across different triangle sizes. Vedantu’s courses help students bridge geometry and trigonometry for a holistic understanding of mathematics.
9. Can two triangles be similar if only two sides are in proportion?
No, two triangles cannot be confirmed as similar solely based on two sides being in proportion. A minimum of two equal angles (AA) or an included angle between proportional sides (SAS) or all sides in the same ratio (SSS) must be satisfied. Vedantu’s in-depth resources explain the necessity of angle and side criteria for establishing triangle similarity.
10. How can Vedantu help students master the similarity of triangles?
Vedantu offers a range of comprehensive math study materials, live & recorded classes, practice worksheets, and doubt-solving sessions on the similarity of triangles. These resources include:
- Step-by-step video tutorials on triangle similarity rules
- Interactive quizzes and assignments for learning reinforcement
- Experienced math tutors guiding complex concepts
- Personalized learning paths catering to individual student needs
