Proving Triangles Similar
Proving that two triangles are similar is a core skill in Chapter 6 — Triangles of the CBSE Class 10 syllabus. A proof requires identifying the similarity criterion (AA, SSS, or SAS) and showing that the required conditions are met.
Unlike congruence (which requires exact equality of shape AND size), similarity requires only the same shape — corresponding angles are equal and corresponding sides are proportional. Similar triangles appear in proofs involving parallel lines, altitude-on-hypotenuse, and tangent-secant problems.
Board exam questions on similarity proofs carry 3-5 marks and require a structured, step-by-step approach with reasons for each statement.
What is Proving Triangles Similar?
Definition: Two triangles are similar if:
- Their corresponding angles are equal, AND
- Their corresponding sides are in the same ratio (proportional).
If △ABC ~ △DEF (read: triangle ABC is similar to triangle DEF), then:
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
- AB/DE = BC/EF = AC/DF
Important: The order of vertices in the similarity statement matters. △ABC ~ △DEF means A corresponds to D, B to E, and C to F.
Proving Triangles Similar Formula
Three Criteria for Similarity:
| Criterion | What to Show | Full Name |
|---|---|---|
| AA | Two pairs of corresponding angles are equal | Angle-Angle Similarity |
| SSS | All three pairs of corresponding sides are proportional | Side-Side-Side Similarity |
| SAS | Two pairs of sides are proportional AND the included angle is equal | Side-Angle-Side Similarity |
Key results used in proofs:
- BPT (Basic Proportionality Theorem): If a line is parallel to one side of a triangle, it divides the other two sides proportionally.
- Vertically opposite angles are equal.
- Angles in the same segment of a circle are equal.
- Alternate interior angles are equal when a transversal crosses parallel lines.
- Angle sum property: If two angles are equal, the third is automatically equal.
Derivation and Proof
Why does AA imply similarity?
- In any triangle, the three angles sum to 180°.
- If two angles of △ABC equal two angles of △DEF, the third pair is also equal: ∠C = 180° − ∠A − ∠B = 180° − ∠D − ∠E = ∠F.
- With all three angles equal, the triangles have the same shape.
- It can be proved (using BPT) that equal angles force the sides to be proportional.
- Therefore, two equal angle pairs are sufficient — the third pair and proportional sides follow automatically.
Why is AA the most commonly used criterion?
- Angles are easier to establish than side ratios.
- Parallel lines give equal corresponding/alternate angles immediately.
- Vertically opposite angles are always equal.
- Common angles (shared vertex) are trivially equal.
Types and Properties
Type 1: Proving similarity using parallel lines
- Parallel lines create equal alternate/corresponding angles → AA criterion.
Type 2: Proving similarity using common angle + one more angle
- Two triangles sharing a vertex have one common angle. If another pair is equal (e.g., both 90°), similarity follows by AA.
Type 3: Proving similarity in right triangles (altitude on hypotenuse)
- The altitude from the right angle to the hypotenuse creates two smaller triangles, each similar to the original and to each other.
Type 4: Proving similarity using proportional sides (SSS)
- When all three ratios of corresponding sides are computed and shown equal.
Type 5: Proving similarity using SAS
- Show two sides proportional and the included angle equal.
Type 6: Proving a result USING similarity
- First prove similarity, then use the proportional sides or equal angles to establish the required result.
Methods
Steps to write a similarity proof:
- State what is to be proved: △ABC ~ △DEF (specify the order of vertices).
- Identify the criterion you will use (AA, SSS, or SAS).
- Establish the required conditions:
- For AA: show two pairs of equal angles with reasons.
- For SSS: compute all three side ratios and show they are equal.
- For SAS: show two side ratios equal and the included angle equal.
- State the conclusion: "By AA (or SSS/SAS) criterion, △ABC ~ △DEF."
- If the problem requires a further result (e.g., a ratio), use the proportionality of corresponding sides.
Common reasons used in proofs:
- "Vertically opposite angles" (for angles at an intersection point).
- "Alternate interior angles, since AB ∥ CD" (for parallel lines).
- "Common angle" (for shared angles at a vertex).
- "Each is 90°" (for right angles).
- "Angles in the same segment" (for cyclic quadrilaterals).
- "By BPT" (for proportional sides from parallel lines).
Solved Examples
Example 1: AA Similarity with Parallel Lines
Problem: In △ABC, DE ∥ BC where D is on AB and E is on AC. Prove that △ADE ~ △ABC.
Proof:
In △ADE and △ABC:
- ∠ADE = ∠ABC (corresponding angles, since DE ∥ BC and AB is a transversal).
- ∠AED = ∠ACB (corresponding angles, since DE ∥ BC and AC is a transversal).
- ∠A = ∠A (common angle).
By AA similarity criterion:
△ADE ~ △ABC.
Consequence: AD/AB = AE/AC = DE/BC (corresponding sides are proportional).
Example 2: AA Similarity with Vertically Opposite Angles
Problem: Two chords AB and CD of a circle intersect at point P inside the circle. Prove that △APC ~ △DPB.
Proof:
In △APC and △DPB:
- ∠APC = ∠DPB (vertically opposite angles at P).
- ∠ACP = ∠DBP (angles in the same segment — both subtended by arc AD).
By AA similarity criterion:
△APC ~ △DPB.
Consequence: PA/PD = PC/PB, i.e., PA × PB = PC × PD.
Example 3: Altitude on Hypotenuse
Problem: In a right triangle ABC (right-angled at B), BD is the altitude from B to hypotenuse AC. Prove that △ADB ~ △ABC.
Proof:
In △ADB and △ABC:
- ∠ADB = ∠ABC = 90° (BD is altitude; angle B is 90°).
- ∠A = ∠A (common angle).
By AA similarity criterion:
△ADB ~ △ABC.
Similarly: △BDC ~ △ABC (using ∠BDC = 90° and common angle ∠C).
Therefore: △ADB ~ △BDC ~ △ABC.
Consequence: BD² = AD × DC (geometric mean relation).
Example 4: SSS Similarity
Problem: In △ABC, AB = 4 cm, BC = 6 cm, AC = 8 cm. In △DEF, DE = 2 cm, EF = 3 cm, DF = 4 cm. Prove that △ABC ~ △DEF.
Proof:
Computing ratios of corresponding sides:
- AB/DE = 4/2 = 2
- BC/EF = 6/3 = 2
- AC/DF = 8/4 = 2
All three ratios are equal (= 2).
By SSS similarity criterion:
△ABC ~ △DEF with scale factor 2.
Example 5: SAS Similarity
Problem: In △ABC and △DEF, AB/DE = AC/DF = 3/2 and ∠A = ∠D = 50°. Prove that the triangles are similar.
Proof:
In △ABC and △DEF:
- AB/DE = 3/2 (given).
- AC/DF = 3/2 (given).
- ∠A = ∠D = 50° (given — this is the included angle between the proportional sides).
Since two pairs of sides are proportional and the included angle is equal:
By SAS similarity criterion:
△ABC ~ △DEF.
Example 6: Proving a Ratio Using Similarity
Problem: In the figure, PQ ∥ BC. If AP = 3 cm, PB = 6 cm, and AQ = 2 cm, find QC and show that PQ = BC/3.
Solution:
Step 1: Prove similarity.
In △APQ and △ABC:
- ∠APQ = ∠ABC (corresponding angles, PQ ∥ BC).
- ∠A = ∠A (common).
By AA: △APQ ~ △ABC.
Step 2: Use proportional sides.
- AP/AB = AQ/AC = PQ/BC
- 3/9 = 2/AC = PQ/BC
- AC = 6 → QC = 6 − 2 = 4 cm
- PQ/BC = 1/3 → PQ = BC/3
Answer: QC = 4 cm. PQ = BC/3 (proved).
Example 7: Two Right Triangles with Common Hypotenuse
Problem: Triangles ABC and DBC are on the same base BC with A and D on opposite sides of BC. ∠BAC = ∠BDC = 90°. Prove that △ABD ~ △CBD and hence BD² = AD × CD... Wait, let me reconsider.
Revised Problem: △ABC and △ADE are such that ∠B = ∠D = 90° and ∠A is common. Prove △ABC ~ △ADE.
Proof:
In △ABC and △ADE:
- ∠ABC = ∠ADE = 90° (given).
- ∠A = ∠A (common angle).
By AA similarity criterion:
△ABC ~ △ADE.
Consequence: AB/AD = BC/DE = AC/AE.
Example 8: Tangent and Secant Similarity
Problem: From a point P outside a circle, a tangent PT and a secant PAB are drawn (A and B on the circle). Prove that △PTA ~ △PBT.
Proof:
In △PTA and △PBT:
- ∠P = ∠P (common angle at P).
- ∠PTA = ∠PBT (tangent-chord angle ∠PTA = angle in alternate segment = ∠PBT).
By AA similarity criterion:
△PTA ~ △PBT.
Consequence: PT/PB = PA/PT → PT² = PA × PB.
This proves the tangent-secant theorem.
Example 9: Proving BPT Using Similarity
Problem: Using the concept of similarity, prove that a line drawn parallel to one side of a triangle divides the other two sides proportionally.
Proof:
Given: In △ABC, DE ∥ BC with D on AB and E on AC.
To prove: AD/DB = AE/EC.
- In △ADE and △ABC:
- ∠ADE = ∠ABC (corresponding angles, DE ∥ BC)
- ∠A = ∠A (common)
- By AA criterion: △ADE ~ △ABC.
- Therefore: AD/AB = AE/AC ... (i)
- Since AB = AD + DB and AC = AE + EC:
- AD/(AD + DB) = AE/(AE + EC)
- Cross-multiplying: AD(AE + EC) = AE(AD + DB)
- AD × AE + AD × EC = AE × AD + AE × DB
- AD × EC = AE × DB
- AD/DB = AE/EC (proved).
Example 10: Medians and Similarity
Problem: In △ABC, D and E are midpoints of AB and AC respectively. Prove that △ADE ~ △ABC and DE = BC/2.
Proof:
In △ADE and △ABC:
- AD/AB = 1/2 (D is midpoint of AB).
- AE/AC = 1/2 (E is midpoint of AC).
- ∠A = ∠A (common angle — the included angle).
Since AD/AB = AE/AC and the included angle is equal:
By SAS similarity criterion: △ADE ~ △ABC.
Therefore: DE/BC = AD/AB = 1/2.
Hence DE = BC/2 (the line joining midpoints of two sides is half the third side).
This is the Midpoint Theorem, proved using similarity.
Real-World Applications
- Similar triangles are used to calculate heights of buildings, trees, and towers from their shadows.
- If a person and a tower cast shadows at the same time, the triangles formed are similar.
Map Scales:
- Maps use similar triangles — the map is a scaled-down version of the actual region.
Photography:
- The image on a camera sensor is similar (in the geometric sense) to the actual scene.
Engineering:
- Scale models of bridges, buildings, and machines use similarity.
- Stress analysis uses similar triangles for force resolution.
Key Points to Remember
- Two triangles are similar if corresponding angles are equal and corresponding sides are proportional.
- AA criterion: Two pairs of equal angles are sufficient to prove similarity.
- SSS criterion: All three pairs of sides in the same ratio prove similarity.
- SAS criterion: Two pairs of proportional sides with equal included angle prove similarity.
- The order of vertices in the similarity statement is critical: △ABC ~ △DEF means A↔D, B↔E, C↔F.
- Parallel lines are the most common source of equal angles for AA proofs.
- Vertically opposite angles are always equal — useful at intersection points.
- The altitude on the hypotenuse of a right triangle creates two triangles similar to the original.
- Similarity proofs must include reasons for every equal angle or proportional side claim.
- After proving similarity, use proportional sides to derive length relationships.
Practice Problems
- In △ABC, DE ∥ BC where D is on AB and E is on AC. If AD = 4 cm, DB = 8 cm, and AE = 3 cm, find EC and prove △ADE ~ △ABC.
- In a right triangle ABC (∠C = 90°), CD is the altitude to the hypotenuse AB. Prove that △ACD ~ △ABC ~ △CBD.
- Two triangles have sides 3 cm, 4 cm, 5 cm and 6 cm, 8 cm, 10 cm. Prove they are similar and state the scale factor.
- In the figure, ∠BAC = ∠CDA. Prove that △BAC ~ △CDA and hence BC × DA = AC × CD.
- ABCD is a trapezium with AB ∥ DC. If the diagonals AC and BD intersect at O, prove that △AOB ~ △COD.
- In △PQR, a line parallel to QR intersects PQ at M and PR at N. Prove that PM/MQ = PN/NR using similarity.
- Two poles of heights 6 m and 11 m stand on a horizontal ground. The distance between their feet is 12 m. Find the distance between their tops using similarity concepts.
- If the areas of two similar triangles are in the ratio 25:36, find the ratio of their corresponding sides.
Frequently Asked Questions
Q1. What is the simplest way to prove two triangles are similar?
The AA (Angle-Angle) criterion is the simplest. Show that two pairs of corresponding angles are equal. The third pair is automatically equal, and side proportionality follows.
Q2. Does the order of vertices matter in a similarity statement?
Yes. △ABC ~ △DEF means A corresponds to D, B to E, C to F. Writing △ABC ~ △EDF would mean different correspondences and would be incorrect unless A↔E, B↔D, C↔F.
Q3. What is the difference between similar and congruent triangles?
Congruent triangles are identical in shape AND size (all sides and angles equal). Similar triangles have the same shape but may differ in size (angles equal, sides proportional but not necessarily equal). All congruent triangles are similar, but not all similar triangles are congruent.
Q4. How do parallel lines help in similarity proofs?
Parallel lines create equal corresponding angles and equal alternate interior angles with a transversal. This provides the two equal angle pairs needed for the AA criterion.
Q5. Can two triangles be similar if they have one equal angle?
One equal angle is not sufficient. You need at least two equal angle pairs (AA) or specific side conditions (SSS or SAS) to prove similarity.
Q6. What is the scale factor?
The scale factor is the common ratio of corresponding sides. If △ABC ~ △DEF with AB/DE = BC/EF = AC/DF = k, then k is the scale factor. If k > 1, △ABC is larger; if k < 1, it is smaller.
Q7. How are areas of similar triangles related?
The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides. If the scale factor is k, the area ratio is k².
Q8. Do I need to show all three angle pairs for AA?
No. Showing any two pairs of equal angles is sufficient. The third pair follows from the angle sum property (all angles add to 180°).
Q9. What are the most common ways to get equal angles?
Common angle (shared vertex), vertically opposite angles, alternate/corresponding angles from parallel lines, angles in the same segment of a circle, and given right angles (90°).
Q10. How many marks do similarity proof questions carry in CBSE board exams?
Typically 3-5 marks. Full marks require stating the two triangles, identifying the criterion, providing reasons for each equal angle or proportional side, and writing the conclusion clearly.
Related Topics
- Criteria for Similarity of Triangles
- AA Similarity Criterion
- SSS Similarity Criterion
- SAS Similarity Criterion
- Angle Sum Property of Triangle
- Exterior Angle Property of Triangle
- Properties of Isosceles Triangle
- Properties of Equilateral Triangle
- Triangle Inequality Property
- Medians and Altitudes of Triangle
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
