Triangles — Class 9 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Triangles" — 8 important questions with detailed answers for CBSE board exam preparation.

✓ 100% Free ✓ No Login Needed ✓ NCERT / CBSE Aligned ✓ Download as PDF

TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Triangles" — 8 important questions with detailed answers for CBSE board exam prepar…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

🤖 Stuck on any question? Ask Syllab's free AI Tutor for a step-by-step explanation — instant, unlimited, no login.

Key Questions Covered:

State the SSS (Side-Side-Side) congruence criterion and apply it: If in trian…
Apply the SAS (Side-Angle-Side) congruence criterion: In triangles ABD and CB…
Apply the ASA (Angle-Side-Angle) congruence criterion: In triangles PQR and X…
Apply the RHS (Right angle-Hypotenuse-Side) congruence criterion: In right tr…
In triangle ABC, if AB = AC, prove that the angles opposite to these sides ar…
In triangle ABC, angle A = 60°, angle B = 80°. Calculate angle C. What is the…
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
State the SSS (Side-Side-Side) congruence criterion and a…	✓ Solved
Apply the SAS (Side-Angle-Side) congruence criterion: In …	✓ Solved
Apply the ASA (Angle-Side-Angle) congruence criterion: In…	✓ Solved
Apply the RHS (Right angle-Hypotenuse-Side) congruence cr…	✓ Solved
In triangle ABC, if AB = AC, prove that the angles opposi…	✓ Solved
In triangle ABC, angle A = 60°, angle B = 80°. Calculate …	✓ Solved

Showing 6 of 8 questions

Q1: State the SSS (Side-Side-Side) congruence criterion and apply it: If in triangle ABC, AB = 3 cm, BC = 4 cm, CA = 5 cm and in triangle PQR, PQ = 3 cm, QR = 4 cm, RP = 5 cm, prove that triangle ABC ≅ triangle PQR.

Step 1: SSS congruence criterion states that if all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent. Step 2: In triangle ABC: AB = 3 cm BC = 4 cm CA = 5 cm Step 3: In triangle PQR: PQ = 3 cm QR = 4 cm RP = 5 cm Step 4: Comparing corresponding sides: AB = PQ = 3 cm BC = QR = 4 cm CA = RP = 5 cm Step 5: Since all three sides of triangle ABC are equal to corresponding sides of triangle PQR, by SSS criterion, triang...

Q2: Apply the SAS (Side-Angle-Side) congruence criterion: In triangles ABD and CBD, AB = CB, angle ABD = angle CBD, and BD is common. Prove that triangle ABD ≅ triangle CBD.

Step 1: SAS congruence criterion states that if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent. Step 2: In triangle ABD: AB = given BD = common side angle ABD = given Step 3: In triangle CBD: CB = given BD = common side angle CBD = given Step 4: Comparing the triangles: AB = CB (given) angle ABD = angle CBD (given) BD = BD (common side) Step 5: The included angle between si...

Q3: Apply the ASA (Angle-Side-Angle) congruence criterion: In triangles PQR and XYZ, angle Q = angle Y, QR = YZ, and angle R = angle Z. Prove that triangle PQR ≅ triangle XYZ.

Step 1: ASA congruence criterion states that if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent. Step 2: In triangle PQR: angle Q = given QR = given (included side between angles Q and R) angle R = given Step 3: In triangle XYZ: angle Y = given YZ = given (included side between angles Y and Z) angle Z = given Step 4: Comparing the triangles: angle Q = angle Y (given) QR = YZ ...

Q4: Apply the RHS (Right angle-Hypotenuse-Side) congruence criterion: In right triangle ABC with right angle at B and right triangle DEF with right angle at E, if AC = DF (hypotenuses) and AB = DE (one side), prove that triangle ABC ≅ triangle DEF.

Step 1: RHS congruence criterion states that if the hypotenuse and one side of a right triangle are equal to the hypotenuse and corresponding side of another right triangle, then the triangles are congruent. Step 2: In triangle ABC: Angle B = 90° (given) AC = hypotenuse AB = one side Step 3: In triangle DEF: Angle E = 90° (given) DF = hypotenuse DE = one side Step 4: Comparing the triangles: Angle B = Angle E = 90° (both right angles) AC = DF (given hypotenuses are equal) AB = DE (given, corr...

Q5: In triangle ABC, if AB = AC, prove that the angles opposite to these sides are equal, i.e., angle C = angle B.

Step 1: Given: AB = AC in triangle ABC. To prove: angle C = angle B (angles opposite equal sides are equal). Step 2: Draw the angle bisector of angle A, and let it meet BC at point D. Step 3: In triangle ABD and triangle ACD: AB = AC (given) AD = AD (common side) angle BAD = angle CAD (AD is angle bisector of angle A) Step 4: By SAS congruence criterion, triangle ABD ≅ triangle ACD. Step 5: From the congruence, corresponding angles are equal: angle ABD = angle ACD angle ADB = angle ADC BD = ...

Q6: In triangle ABC, angle A = 60°, angle B = 80°. Calculate angle C. What is the sum of all angles in the triangle?

Step 1: By the angle sum property of a triangle, the sum of all three angles equals 180°. Step 2: Given: angle A = 60° angle B = 80° angle C = ? Step 3: Using angle sum property: angle A + angle B + angle C = 180° Step 4: Substitute known values: 60° + 80° + angle C = 180° 140° + angle C = 180° Step 5: Solve for angle C: angle C = 180° - 140° angle C = 40° Step 6: Verify: 60° + 80° + 40° = 180° ✓ Answer: angle C = 40°; Sum of all angles in the triangle = 180°.

Showing 6 of 8 questions. Visit the full page for complete solutions.