How to Identify and Prove Similar Triangles in Geometry
The concept of similar triangles plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to identify, prove, and use similar triangles can help you solve many geometry questions quickly and with confidence.
What Is Similar Triangles?
A similar triangle is defined as a triangle that has the same shape as another triangle, but possibly a different size. Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. You’ll find this concept applied in areas such as ratio and proportion, map scaling, and heights and distances in trigonometry.
Key Formula for Similar Triangles
Here’s the standard formula:
If △ABC ∼ △XYZ (read as ‘triangle ABC is similar to triangle XYZ’), then their corresponding sides have equal ratios and corresponding angles are equal.
\( \dfrac{AB}{XY} = \dfrac{BC}{YZ} = \dfrac{CA}{ZX} \)
Corresponding angles: ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z
How to Check If Triangles Are Similar?
To check if two triangles are similar, you need to use specific similarity rules or criteria. The three main rules are listed below:
| Criteria | Condition | What to Compare? |
|---|---|---|
| AA (Angle-Angle) | Any two pairs of corresponding angles are equal | Angles only |
| SSS (Side-Side-Side) | All corresponding sides are in the same ratio | All sides |
| SAS (Side-Angle-Side) | Two pairs of sides are in the same ratio, and the included angle is equal | Two sides & included angle |
Step-by-Step Illustration
-
Suppose you are given:
△ABC and △DEF, with AB = 6 cm, BC = 8 cm, AC = 10 cm and DE = 9 cm, EF = 12 cm, DF = 15 cm.
Check if they are similar by SSS rule.1. Find the side ratios:
AB/DE = 6/9 = 2/3
BC/EF = 8/12 = 2/3
AC/DF = 10/15 = 2/3
-
All three corresponding side pairs have the same ratio (2:3).
Hence, by SSS similarity rule, △ABC ∼ △DEF.
Properties of Similar Triangles
- If two triangles are similar, their corresponding angles are equal.
- The lengths of their corresponding sides are in the same proportion or ratio.
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
- All equilateral triangles are automatically similar to each other.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to find unknown side lengths in similar triangles:
Example Trick: If you know the ratio of sides and one side in a similar triangle, just multiply or divide to find the missing length.
- Suppose Triangle 1 has a side of 6 cm, and Triangle 2 (similar) has the matching side unknown, but the overall ratio is 2:3.
Set up: 6 / x = 2/3 → x = (6×3)/2 = 9 cm
Shortcuts like this make exam-time calculations quick. Vedantu’s live classes share many such tricks to help students build confidence with triangle questions.
Try These Yourself
- Identify if two triangles with sides 4 cm, 6 cm, 8 cm and 6 cm, 9 cm, 12 cm are similar. Which rule applies?
- Two triangles have corresponding angles of 50°, 60°, and 70°. Are they similar? Why?
- ABD and PQR are similar triangles, BC = 7 cm, QR = 14 cm. Find the scale factor from ABD to PQR.
- In two similar triangles, the sides are in the ratio 3:5. If the area of the smaller triangle is 27 cm², what’s the area of the larger?
Frequent Errors and Misunderstandings
- Confusing similar triangles with congruent triangles. Remember: similar means “same shape” (angles), not “same size”. Congruent is same shape and same size.
- Not matching corresponding sides or angles correctly. Always write the names in corresponding order: △ABC ∼ △DEF means A→D, B→E, C→F.
- Using the sides’ ratio without confirming the angles are equal (especially in SAS cases).
- Forgetting that the ratio of areas is (side ratio)2, not just the side ratio.
Relation to Other Concepts
The idea of similar triangles connects closely with congruence of triangles, ratio and proportion, and angle sum property of triangles. Mastering this helps with understanding trigonometry, scaling, and area relationships. For more about area ratios in similar triangles, see area of similar triangles.
Classroom Tip
A quick way to remember similar triangles: If two triangles look the same but may be larger or smaller, and you can “zoom in or out” to make them match, they are similar! Vedantu’s teachers often color-code corresponding angles or sides in diagrams to help you visualize this easily.
We explored similar triangles—from definition, formula, examples, mistakes, and their close connection to other topics. Keep practicing, and try worksheets or past questions to build speed and understanding. For more triangle practice, visit Triangle Worksheets and MCQs. Continue learning with Vedantu for exam-ready confidence and deeper Maths skills!
FAQs on Similar Triangles – Definition, Criteria, Properties & Applications
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 in the same ratio (proportion). This means one triangle is essentially a scaled version of the other. We use the symbol '∼' to denote similarity.
2. How do you prove triangles are similar?
You can prove triangle similarity using three main criteria:
• **AA (Angle-Angle):** If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
• **SSS (Side-Side-Side):** If the ratios of corresponding sides of two triangles are equal, 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, the triangles are similar.
3. What are the applications of similar triangles?
Similar triangles have many real-world applications, including:
• **Calculating heights and distances:** Using similar triangles formed by shadows or by sightlines, we can determine inaccessible heights (like that of a tall building or tree) or distances.
• **Map scaling:** Maps use similar triangles to represent large areas on a smaller scale. The ratio of distances on the map to actual distances is consistent.
• **Engineering and architecture:** Similar triangles are used in design and construction to ensure proportions and scale are maintained.
4. What is the difference between similar and congruent triangles?
Similar triangles have the same shape but can have different sizes. Congruent triangles are identical in both shape and size; they are essentially copies of each other. All congruent triangles are similar, but not all similar triangles are congruent.
5. How do you find the ratio of the areas of similar triangles?
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If the ratio of corresponding sides is k, then the ratio of their areas is k².
6. Can you explain the AA similarity criterion with an example?
The **AA similarity criterion** states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. For example, if ∠A = ∠D and ∠B = ∠E in triangles ABC and DEF, then △ABC ∼ △DEF.
7. How do I solve problems involving unknown sides in similar triangles?
Set up a proportion using the ratio of corresponding sides. If △ABC ∼ △DEF, then AB/DE = BC/EF = AC/DF. Solve the proportion for the unknown side length.
8. Are all equilateral triangles similar?
Yes, all equilateral triangles are similar because their corresponding angles are all 60°, satisfying the AA similarity criterion.
9. What are some common mistakes students make when working with similar triangles?
Common mistakes include:
• Incorrectly identifying corresponding sides or angles.
• Setting up proportions incorrectly.
• Confusing similarity with congruence.
• Not understanding the different criteria (AA, SSS, SAS) for proving similarity.
10. How is the ratio of perimeters of similar triangles related to the ratio of their corresponding sides?
The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.
11. Explain the SSS similarity criterion.
The **SSS similarity criterion** states that if the ratios of all three pairs of corresponding sides of two triangles are equal, then the triangles are similar. For example, if AB/DE = BC/EF = AC/DF in triangles ABC and DEF, then △ABC ∼ △DEF.
