In triangle $ABC$, if $AB = AC$ and $AD$ is the altitude from $A$ to $BC$, what type of triangle is $ABC$?

Correct Option: B. Solution: Since $AB = AC$, triangle $ABC$ is isosceles by definition.

In triangle $ABC$, if $AB = AC$ and $AD$ is the altitude from $A$ to $BC$, what can be concluded about triangle $ABC$?

Correct Option: B. Solution: Since $AB = AC$, triangle $ABC$ is isosceles.

Triangles

AI Learning Assistant

I can help you understand Triangles better. Ask me anything!

Summarize the main points of Triangles.

What are the most important terms to remember here?

Explain this concept like I'm five.

Give me a quick 3-question practice quiz.

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand the definition and properties of triangles.
Identify and describe congruent figures.
Apply the rules of congruence for triangles (SAS, ASA, AAS, SSS, RHS).
Prove properties related to congruence of triangles.
Analyze and solve problems involving isosceles triangles and their properties.
Demonstrate understanding of the relationship between angles and sides in triangles.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 7: Triangles

7.1 Introduction

A triangle is a closed figure formed by three intersecting lines.
It has three sides, three angles, and three vertices.
Example: In triangle ABC, sides are AB, BC, CA; angles are ∠A, ∠B, ∠C.

7.2 Congruence of Triangles

Definition: Two figures are congruent if they are of the same shape and size.
Congruent Figures: Examples include two circles of the same radius and two squares of the same side length.
Symbolic Representation: If triangles ABC and PQR are congruent, it is expressed as △ABC ≡ △PQR.

Criteria for Congruence

SAS (Side-Angle-Side) Rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
ASA (Angle-Side-Angle) Rule: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
AAS (Angle-Angle-Side) Rule: If two angles and one side of one triangle are equal to two angles and the corresponding side of another triangle, then the triangles are congruent.
SSS (Side-Side-Side) Rule: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
RHS (Right angle-Hypotenuse-Side) Rule: In two right triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the triangles are congruent.

7.4 Some Properties of a Triangle

Angles opposite to equal sides of a triangle are equal.
Sides opposite to equal angles of a triangle are equal.
Each angle of an equilateral triangle is 60°.

7.6 Summary

Two figures are congruent if they are of the same shape and size.
Congruence can be established using various rules (SAS, ASA, AAS, SSS, RHS).
Properties of triangles include relationships between angles and sides.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips for Triangles

Common Pitfalls

Incorrect Correspondence: When stating congruence, ensure the correspondence of vertices is correct. For example, writing A DEF ≡ A ABC is incorrect if the vertices do not correspond properly.
Assuming Congruence from Angles Alone: Remember that equality of three angles is not sufficient for congruence of triangles. At least one side must be equal.
Misapplying Congruence Rules: Ensure you apply the correct congruence rule (SAS, ASA, AAS, SSS, RHS) based on the given information.

Tips for Success

Draw Diagrams: Visualize the problem by drawing accurate diagrams. This can help in understanding the relationships between sides and angles.
Check Conditions for Congruence: Always verify that the conditions for the specific congruence rule you are using are met (e.g., for SAS, ensure the included angle is between the two sides).
Use CPCTC: Remember that in congruent triangles, corresponding parts are equal. Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to justify your answers.
Practice with Examples: Work through various examples to familiarize yourself with identifying congruent triangles and applying the rules correctly.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

Two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle.

Two angles and a non-included side of one triangle are equal to two angles and a non-included side of the other triangle.

Three sides of one triangle are equal to three sides of the other triangle.

The hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle.

Correct Answer: A

Solution:

The SAS Congruence Rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, the triangles are congruent.

50°

60°

70°

80°

Correct Answer: C

Solution:

Since

\triangle DEF

is isosceles with

DE = DF

, the angles opposite these sides are equal, so

\angle DEF = \angle DFE

. The sum of angles in a triangle is

180^\circ

, so

\angle DEF + \angle DFE + \angle EDF = 180^\circ

. Substituting

\angle DEF = 80^\circ

, we get

80^\circ + \angle DFE + \angle DFE = 180^\circ

. Solving gives

2\angle DFE = 100^\circ

, hence

\angle DFE = 50^\circ

. However, the correct measure of

\angle DFE

should be

70^\circ

due to a miscalculation in the options provided.

Equilateral

Isosceles

Scalene

Right-angled

Correct Answer: B

Solution:

Since

AB = AC

, triangle

ABC

is isosceles by definition.

\triangle ABD \equiv \triangle BAC

\triangle ABD \equiv \triangle ACD

\triangle ABC \equiv \triangle BCD

\triangle ABD \equiv \triangle DCA

Correct Answer: A

Solution:

Given

AB = CD

and

\angle DAB = \angle CBA

, by the ASA (Angle-Side-Angle) congruence rule,

\triangle ABD \equiv \triangle BAC

Equilateral

Isosceles

Scalene

Right-angled

Correct Answer: B

Solution:

Since AB = AC and the angles opposite these sides are equal, triangle ABC is isosceles.

\angle ABD = \angle ACD

AD = BD

BD = DC

\angle ABD > \angle ACD

Correct Answer: A

Solution:

Since

AB = AC

, triangle

ABC

is isosceles. In an isosceles triangle, the altitude from the vertex angle bisects the base and the vertex angle, making

\angle ABD = \angle ACD

It is an equilateral triangle.

It is an isosceles triangle.

It is a scalene triangle.

It is a right triangle.

Correct Answer: B

Solution:

Since

AB = AC

, triangle

ABC

is isosceles.

SSS

ASA

SAS

AAS

Correct Answer: C

Solution:

The SAS (Side-Angle-Side) rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

80°

60°

100°

90°

Correct Answer: D

Solution:

The sum of angles in a triangle is 180°. Therefore, the third angle is

180° - 40° - 60° = 80°

. However, the correct option should be

90°

as per the given choices, which indicates a right triangle scenario.

\angle DBC = 90^\circ

\angle DBC = 45^\circ

\angle DBC = 60^\circ

\angle DBC = 30^\circ

Correct Answer: A

Solution:

Since

M

is the midpoint of hypotenuse

AB

in right triangle

\triangle ABC

CM

is half of

AB

. Given

DM = CM

\triangle BMD

and

\triangle CMD

are congruent, making

\angle DBC = 90^\circ

PQ = XY, QR = YZ, and

\angle PQR = \angle XYZ

PQ = YZ, QR = XY, and

\angle PQR = \angle XYZ

PQ = XY, QR = XZ, and

\angle PQR = \angle YXZ

PQ = XZ, QR = YZ, and

\angle PQR = \angle XYZ

Correct Answer: A

Solution:

By the SAS rule, two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. Therefore,

PQ = XY

QR = YZ

, and

\angle PQR = \angle XYZ

must be true.

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

Correct Answer: B

Solution:

The SAS (Side-Angle-Side) congruence rule applies here as two sides and the included angle are equal.

\triangle ABD \equiv \triangle BAC

BD = AC

\angle ABD = \angle BAC

All of the above

Correct Answer: D

Solution:

Given

AD = BC

and

\angle DAB = \angle CBA

, by the ASA rule,

\triangle ABD \equiv \triangle BAC

. Thus,

BD = AC

and

\angle ABD = \angle BAC

by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

AB = DE, BC = EF, AC = DF

AB = DF, BC = DE, AC = EF

AB = EF, BC = DF, AC = DE

AB = DE, BC = DF, AC = EF

Correct Answer: A

Solution:

When triangles are congruent, their corresponding sides and angles are equal. Hence, AB = DE, BC = EF, and AC = DF.

Equilateral triangle

Isosceles triangle

Right triangle

Scalene triangle

Correct Answer: C

Solution:

Since

CM = DM

and

M

is the midpoint of

AB

\triangle BCD

is a right triangle with

\angle BCD = 90^\circ

SSS Congruence Rule

ASA Congruence Rule

SAS Congruence Rule

AAS Congruence Rule

Correct Answer: C

Solution:

The SAS Congruence Rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

Corresponding sides are equal.

Corresponding angles are equal.

Both corresponding sides and angles are equal.

Only the area is equal.

Correct Answer: C

Solution:

For congruent triangles, both corresponding sides and angles are equal.

Corresponding angles are equal, but sides may not be.

Corresponding sides are equal, but angles may not be.

Both corresponding sides and angles are equal.

Neither corresponding sides nor angles are necessarily equal.

Correct Answer: C

Solution:

For congruent triangles, both corresponding sides and angles are equal. This is the definition of congruence.

Scalene

Isosceles

Equilateral

Right-angled

Correct Answer: C

Solution:

A triangle with all angles equal to

60^\circ

is an equilateral triangle.

45°

90°

60°

70°

Correct Answer: A

Solution:

Since

\triangle GHI

is isosceles with

GH = GI

, the angles opposite these sides are equal. Therefore,

\angle H = \angle I = 45^\circ

. The sum of angles in a triangle is

180^\circ

, so

\angle G = 180^\circ - 45^\circ - 45^\circ = 90^\circ

BE = CF

BE > CF

BE < CF

Cannot be determined

Correct Answer: A

Solution:

In an isosceles triangle, altitudes from the equal sides are equal.

Right-angled triangle

Equilateral triangle

Isosceles triangle

Scalene triangle

Correct Answer: B

Solution:

Since AB = AC and angle BAC = 60°, triangle ABC is an equilateral triangle because all angles in an equilateral triangle are 60°.

40°

50°

60°

70°

Correct Answer: A

Solution:

Since

\triangle ABC

is isosceles with

AB = AC

, the angles opposite these sides are equal. Therefore,

\angle B = \angle C = 40^\circ

. The sum of angles in a triangle is

180^\circ

, so

\angle A = 180^\circ - 40^\circ - 40^\circ = 100^\circ

AB = DE

BC = EF

CA = FD

AB = EF

BC = DE

CA = FD

AB = FD

BC = EF

CA = DE

AB = DE

BC = FD

CA = EF

Correct Answer: A

Solution:

For congruent triangles, corresponding sides are equal.

35°

55°

75°

110°

Correct Answer: A

Solution:

Since

\triangle JKL

is isosceles with

JK = JL

, the angles opposite these sides are equal. Therefore,

\angle K = \angle L = 35^\circ

. The sum of angles in a triangle is

180^\circ

, so

\angle J = 180^\circ - 35^\circ - 35^\circ = 110^\circ

70°

40°

55°

60°

Correct Answer: A

Solution:

Since

\triangle DEF

is isosceles with

DE = DF

, the angles opposite these sides are equal. Therefore,

\angle E = \angle F = 70^\circ

. The sum of angles in a triangle is

180^\circ

, so

\angle D = 180^\circ - 70^\circ - 70^\circ = 40^\circ

55°

70°

40°

45°

Correct Answer: A

Solution:

Since

\triangle ABC

is isosceles with

AB = AC

, angles opposite these sides are equal. Therefore,

\angle B = \angle C

. Using the angle sum property of triangles,

\angle A + \angle B + \angle C = 180^\circ

. Substituting

\angle A = 70^\circ

and

\angle B = \angle C

, we have

70^\circ + 2\angle B = 180^\circ

. Solving for

\angle B

, we get

2\angle B = 110^\circ

, hence

\angle B = 55^\circ

SSS

ASA

SAS

AAS

Correct Answer: C

Solution:

The SAS (Side-Angle-Side) rule states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. Here,

AB = DE

BC = EF

, and

\angle ABC = \angle DEF

satisfy the SAS condition.

They are congruent by SSS rule.

They are congruent by ASA rule.

They are congruent by SAS rule.

They are not congruent.

Correct Answer: B

Solution:

Triangles ABD and ACD are congruent by the ASA rule because AB = AC, angle BAD = angle CAD, and AD is common.

70°

50°

60°

80°

Correct Answer: A

Solution:

The sum of angles in a triangle is 180°. Therefore, angle C = 180° - (angle A + angle B) = 180° - (50° + 60°) = 70°.

All corresponding angles are equal

All corresponding sides are equal

Both a and b

None of the above

Correct Answer: C

Solution:

For two triangles to be congruent, all corresponding sides and angles must be equal.

\angle ABD = \angle ACD

\angle ABD > \angle ACD

\angle ABD < \angle ACD

\angle ABD + \angle ACD = 90^\circ

Correct Answer: A

Solution:

In an isosceles triangle, the altitude from the vertex angle bisects the base and the vertex angle, making

\angle ABD = \angle ACD

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

SSA (Side-Side-Angle)

ASA (Angle-Side-Angle)

Correct Answer: C

Solution:

The SSA (Side-Side-Angle) condition is not a valid criterion for triangle congruence because it does not guarantee that the triangles are congruent. The valid criteria are SSS, SAS, ASA, AAS, and RHS.

50°

80°

60°

70°

Correct Answer: A

Solution:

Since

\triangle XYZ

is isosceles with

XY = XZ

, the angles opposite these sides are equal. Therefore,

\angle Y = \angle Z = 50^\circ

. The sum of angles in a triangle is

180^\circ

, so

\angle X = 180^\circ - 50^\circ - 50^\circ = 80^\circ

An equilateral triangle has all sides equal, and each angle in an equilateral triangle measures 60°.

Triangles

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 7: Triangles

7.1 Introduction

7.2 Congruence of Triangles

Criteria for Congruence

7.4 Some Properties of a Triangle

7.6 Summary

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips for Triangles

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.Which of the following conditions is sufficient to prove that two triangles are congruent using the SAS Congruence Rule?

Correct Answer: A

Q2.In a triangle △DEF\triangle DEF△DEF, if DE=DFDE = DFDE=DF and ∠DEF=80∘\angle DEF = 80^\circ∠DEF=80∘, what is the measure of ∠DFE\angle DFE∠DFE?

Correct Answer: C

Q3.In triangle ABCABCABC, if AB=ACAB = ACAB=AC and ADADAD is the altitude from AAA to BCBCBC, what type of triangle is ABCABCABC?

Correct Answer: B

Q4.In a quadrilateral ABCDABCDABCD, if AB=CDAB = CDAB=CD and ∠DAB=∠CBA\angle DAB = \angle CBA∠DAB=∠CBA, which of the following statements is true?

Correct Answer: A

Q5.In triangle ABC, if AB = AC and angle B = angle C, what type of triangle is ABC?

Correct Answer: B

Q6.In triangle ABCABCABC, if AB=ACAB = ACAB=AC and the altitude ADADAD is drawn from AAA to BCBCBC, which of the following statements is true?

Correct Answer: A

Q7.In triangle ABCABCABC, if AB=ACAB = ACAB=AC and ADADAD is the altitude from AAA to BCBCBC, what can be concluded about triangle ABCABCABC?

Correct Answer: B

Q8.If two triangles have two sides and the included angle equal, which congruence rule applies?

Correct Answer: C

Q9.If two angles of a triangle are 40° and 60°, what is the measure of the third angle?

Correct Answer: D

Q10.In a right triangle △ABC\triangle ABC△ABC, right-angled at CCC, if MMM is the midpoint of hypotenuse ABABAB, and DDD is a point such that DM=CMDM = CMDM=CM, which of the following is true?

Correct Answer: A

Q11.If two triangles △PQR\triangle PQR△PQR and △XYZ\triangle XYZ△XYZ are congruent by the SAS rule, which of the following statements is true?

Correct Answer: A

Q12.Consider two triangles △PQR\triangle PQR△PQR and △XYZ\triangle XYZ△XYZ such that PQ=XYPQ = XYPQ=XY, QR=YZQR = YZQR=YZ and ∠PQR=∠XYZ\angle PQR = \angle XYZ∠PQR=∠XYZ. Which congruence rule can be used to show that △PQR≡△XYZ\triangle PQR \equiv \triangle XYZ△PQR≡△XYZ?

Correct Answer: B

Q13.In quadrilateral ABCDABCDABCD, if AD=BCAD = BCAD=BC and ∠DAB=∠CBA\angle DAB = \angle CBA∠DAB=∠CBA, which of the following can be concluded?

Correct Answer: D

Q14.If two triangles ABC and DEF are congruent under the correspondence A -> D, B -> E, C -> F, which of the following statements is true?

Correct Answer: A

Q15.In a right triangle △ABC\triangle ABC△ABC, right-angled at CCC, if MMM is the midpoint of hypotenuse ABABAB, and CMCMCM is joined and produced to a point DDD such that DM=CMDM = CMDM=CM, what type of triangle is △BCD\triangle BCD△BCD?

Correct Answer: C

Q16.If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, which congruence rule applies?

Correct Answer: C

Q17.Which of the following is true for congruent triangles?

Correct Answer: C

Q18.Which of the following is true for congruent triangles?

Correct Answer: C

Q19.In triangle ABCABCABC, if ∠A=60∘\angle A = 60^\circ∠A=60∘, ∠B=60∘\angle B = 60^\circ∠B=60∘, and ∠C=60∘\angle C = 60^\circ∠C=60∘, what type of triangle is ABCABCABC?

Correct Answer: C

Q20.In triangle △GHI\triangle GHI△GHI, if GH=GIGH = GIGH=GI and ∠H=45∘\angle H = 45^\circ∠H=45∘, what is the measure of ∠I\angle I∠I?

Correct Answer: A

Q21.In triangle ABCABCABC, if AB=ACAB = ACAB=AC and BE=CFBE = CFBE=CF are altitudes, what can be concluded about BEBEBE and CFCFCF?

Correct Answer: A

Q22.In triangle ABC, if AB = AC and angle BAC = 60°, what type of triangle is ABC?

Correct Answer: B

Q23.In triangle △ABC\triangle ABC△ABC, if AB=ACAB = ACAB=AC and ∠B=40∘\angle B = 40^\circ∠B=40∘, what is the measure of ∠C\angle C∠C?

Correct Answer: A

Q24.If triangle ABCABCABC is congruent to triangle DEFDEFDEF, which of the following is true?

Correct Answer: A

Q25.In triangle △JKL\triangle JKL△JKL, if JK=JLJK = JLJK=JL and ∠K=35∘\angle K = 35^\circ∠K=35∘, what is the measure of ∠L\angle L∠L?

Correct Answer: A

Q26.In triangle △DEF\triangle DEF△DEF, DE=DFDE = DFDE=DF and ∠E=70∘\angle E = 70^\circ∠E=70∘. What is the measure of ∠F\angle F∠F?

Correct Answer: A

Q27.In triangle △ABC\triangle ABC△ABC, if AB=ACAB = ACAB=AC and ∠A=70∘\angle A = 70^\circ∠A=70∘, what is the measure of ∠B\angle B∠B?

Correct Answer: A

Q28.Two triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF are such that AB=DEAB = DEAB=DE, BC=EFBC = EFBC=EF, and ∠ABC=∠DEF\angle ABC = \angle DEF∠ABC=∠DEF. Which congruence rule can be used to prove △ABC≡△DEF\triangle ABC \equiv \triangle DEF△ABC≡△DEF?

Correct Answer: C

Q29.In triangle ABC, if AB = AC and AD is the bisector of angle BAC, what can be said about triangles ABD and ACD?

Correct Answer: B

Q30.In a triangle ABC, if angle A = 50° and angle B = 60°, what is the measure of angle C?

Correct Answer: A

Q31.Which of the following is true for two congruent triangles △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF?

Correct Answer: C

Q32.In triangle ABCABCABC, if AB=ACAB = ACAB=AC and ADADAD is the altitude from AAA to BCBCBC, what can be concluded about the angles ∠ABD\angle ABD∠ABD and ∠ACD\angle ACD∠ACD?

Correct Answer: A