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REET L1+L2 · Maths

Triangles and Congruence

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This page explains triangle angle properties, the exterior angle theorem, and the four congruence criteria (SSS, SAS, ASA, RHS) with examples, for REET. This is a regularly-tested sub-topic within the Geometry section.

  • 1 question in REET 2024
  • Class 6–7 level
  • 8 min read
Verified 17 September 2026

How REET asks this topic

REET 2024 carried a question based on a triangle's exterior angle property, which required solving two equations together. Questions that directly test recognition of congruence criteria (SSS, SAS, ASA, RHS) are also asked regularly from this section.

Other Useful Triangle Facts

Number of right/obtuse angles
Any triangle can have at most one right angle (90°) or one obtuse angle (more than 90°), because the sum of all three angles is always 180°
Remaining angles of a right triangle
In a right triangle, the other two angles are always acute and always add up to 90°
Relation between equilateral and isosceles
Every equilateral triangle is also a special case of an isosceles triangle, since all three of its sides are equal

Basic Angle Properties of a Triangle

The sum of the three interior angles of any triangle is always 180° — this is called the angle sum property of a triangle.

Exterior Angle Property: When a side of a triangle is extended, the exterior angle formed equals the sum of the two opposite (remote) interior angles.

Triangle Inequality: The sum of any two sides of a triangle is always greater than the third side. A useful consequence of this is that the difference of any two sides is always less than the third side — together, these two conditions decide whether a triangle can even be formed from three given lengths.

The Four Congruence Criteria

CriterionCondition
SSS (Side-Side-Side)All three sides of one triangle equal the corresponding three sides of another triangle
SAS (Side-Angle-Side)Two sides and the included angle between them are equal in both triangles
ASA (Angle-Side-Angle)Two angles and the included side between them are equal in both triangles
RHS (Hypotenuse-Side) — right triangles onlyThe hypotenuse and one side of one right triangle equal the hypotenuse and corresponding side of the other

The corresponding parts of congruent triangles are always equal — this is called CPCT.

Types of Triangles

BasisTypeFeature
By sidesEquilateralAll three sides equal, each angle 60°
By sidesIsoscelesTwo sides equal, and the angles opposite them are also equal
By sidesScaleneAll three sides and angles unequal
By anglesAcute-angledAll three angles less than 90°
By anglesRight-angledOne angle exactly 90°
By anglesObtuse-angledOne angle greater than 90°

Solved Examples

Example 1: A triangle's angles are in the ratio 2:3:4. Find the angles. Solution: Total parts = 9, angle sum = 180° ⇒ one part = 20°. Angles = 40°, 60°, 80°.

Example 2 (exterior angle property): A triangle's two opposite interior angles are 50° and 70°. What is the corresponding exterior angle? Solution: Exterior angle = 50°+70° = 120°.

Example 3 (congruence): In triangles PQR and XYZ, PQ=XY, QR=YZ and ∠Q=∠Y. Here, ∠Q is the angle included between PQ and QR, so these triangles are congruent by the SAS criterion.

Example 4 (order of correspondence): If we write △ABC ≅ △DEF, it means A↔D, B↔E, C↔F correspond, so AB=DE, BC=EF, CA=FD and ∠A=∠D, ∠B=∠E, ∠C=∠F. When writing a congruence statement, this exact order of vertices must be maintained, otherwise the wrong parts end up being treated as corresponding.

How to Teach This to Children

Classroom activity

Cut out two identical triangles from paper and have students superimpose one exactly on the other, so congruence becomes a physical experience. Build triangles from straws or sticks with different side-angle measurements to see when a triangle turns out unique.

Teaching aids

A geometry box (compass, protractor), a geoboard, and a kit of straws/sticks are useful. A collection of triangles cut from coloured paper also helps children recognise and classify different types of triangles.

Common misconception

Students often ignore vertex correspondence — that is, they treat sides/angles simply being equal as congruence, without checking which part corresponds to which. The difference between 'congruent' and 'similar' also often causes confusion.

Assessment idea

Give two triangles with marked equal parts and have students pick the right criterion, then write the congruence statement in the correct order of correspondence (like △ABC ≅ △DEF). Also check whether students can correctly update the corresponding parts if the order of correspondence is changed.

Memory trick / exam tip

SSA (side-side-angle) is not a valid congruence criterion — don't fall into this trap. In exterior-angle questions, always apply 'exterior angle = sum of the two opposite interior angles' directly — there's no need to first find the triangle's third angle.

REET 2024 Questions (with Solutions)

Question 119

In the given figure, if ∠x − ∠y = 10°, then the value of ∠x will be :

Figure: Triangle PQR with interior angle x at P and interior angle y at Q. Side QR is extended beyond R, forming an exterior angle of 120 degrees at R. Diagram added and verified: the exterior angle theorem gives x+y=120, and with x-y=10, x=65 degrees, matching the stored answer.

  1. A55°
  2. B65°
  3. C70°
  4. D100°

Explanation

त्रिभुज का बहिष्कोण (exterior angle) दोनों अंतराभिमुख कोणों के योग के बराबर होता है, अतः ∠x + ∠y = 120°। साथ में ∠x − ∠y = 10°। दोनों को जोड़ने पर 2∠x = 130°, अतः ∠x = 65° (और ∠y = 55°)।

Practice Questions

Question 1

Two angles of a triangle are 50° and 60°. What is the third angle?

  1. A70°
  2. B80°
  3. C60°
  4. D65°

Explanation

The angles of a triangle add up to 180°. Third angle = 180°−50°−60° = 70°.

Question 2

The exterior angle of a triangle is 100°, and one of its interior opposite angles is 40°. What is the other interior opposite angle?

  1. A60°
  2. B40°
  3. C100°
  4. D140°

Explanation

Exterior angle = sum of the two opposite interior angles ⇒ 100° = 40° + second angle ⇒ second angle = 60°.

Question 3

The angles of a triangle are in the ratio 2:3:5. What is the largest angle?

  1. A90°
  2. B100°
  3. C72°
  4. D108°

Explanation

Total parts = 10, one part = 180/10 = 18°. Largest angle = 5×18° = 90°.

Question 4

In triangles PQR and XYZ, PQ=XY, QR=YZ and ∠Q=∠Y. Which congruence criterion applies?

  1. ASAS
  2. BSSS
  3. CASA
  4. DRHS

Explanation

∠Q is the angle included between sides PQ and QR, so this is the SAS (Side-Angle-Side) criterion.

Question 5

Two right triangles have equal hypotenuse and one equal corresponding side. By which criterion are they congruent?

  1. ARHS
  2. BSAS
  3. CAAS
  4. DSSS

Explanation

When the hypotenuse and one side of two right triangles are equal, the RHS (Hypotenuse-Side) criterion applies.

Question 6

In triangles ABC and DEF, ∠A=∠D, ∠B=∠E and AB=DE. This is an example of which criterion?

  1. AASA
  2. BSAS
  3. CSSS
  4. DRHS

Explanation

AB is the side included between angle A and angle B, so this is the ASA (Angle-Side-Angle) criterion.

Question 7

Two sides of a triangle are 4 cm and 5 cm. Which value of the third side x (in cm) is NOT possible?

  1. A1 सेमी
  2. B3 सेमी
  3. C6 सेमी
  4. D8 सेमी

Explanation

By the triangle inequality, the third side must be greater than the difference and less than the sum of the given sides: 1 < x < 9. x=1 does not satisfy this (it must be greater than 1, not equal to it).

Question 8

In triangle ABC, the exterior angle at vertex C is 130°. If ∠A − ∠B = 20°, find ∠A.

  1. A75°
  2. B65°
  3. C55°
  4. D70°

Explanation

Exterior angle = ∠A+∠B = 130°. Also ∠A−∠B=20°. Adding both: 2∠A=150° ⇒ ∠A=75°.

Question 9

While proving congruence of two triangles, what does 'CPCT' fully mean?

  1. Aसर्वांगसम त्रिभुजों के संगत भाग बराबर होते हैं
  2. Bत्रिभुज की परिधि समान होती है
  3. Cत्रिभुज के कोण बराबर होते हैं
  4. Dत्रिभुज समरूप होते हैं

Explanation

CPCT = 'Corresponding Parts of Congruent Triangles' — once congruence is proved, all corresponding parts (sides and angles) are taken as equal.

Question 10

Which activity is most suitable for clarifying the concept of congruence in the classroom?

  1. Aदो त्रिभुजों को काटकर एक-दूसरे पर पूर्णतः अध्यारोपित (superimpose) कराना
  2. Bकेवल परिभाषा याद कराना
  3. Cसूत्र रटवाना
  4. Dसीधे परीक्षा प्रश्न हल कराना

Explanation

Superimposing two cut-out triangles on each other gives children a direct, hands-on experience of congruence, which simply memorising a definition can't provide.

Frequently Asked Questions

How many questions come from triangles-congruence in REET?+

Usually 1 question from this topic appears — on the exterior angle property or a congruence criterion; REET 2024 also had 1 question.

What's the difference between congruent and similar triangles?+

Congruent triangles are identical in both shape and size, while similar triangles have the same shape but can differ in size.

What does CPCT mean?+

CPCT stands for 'Corresponding Parts of Congruent Triangles are Equal'.

What are the criteria for triangle congruence?+

SSS, SAS, ASA and RHS (only for right triangles) — these are the four main criteria.

What does the triangle inequality state?+

The sum of any two sides of a triangle must always be greater than the third side, otherwise a triangle cannot be formed.

What's the special feature of an isosceles triangle?+

An isosceles triangle has two equal sides, and the angles opposite those equal sides are also equal to each other.

Can a triangle have two right angles or two obtuse angles?+

No, because the sum of all three angles is always 180°, so at most one angle can be 90° or more, and the remaining two must be acute.

How much information is enough to prove triangle congruence?+

Generally, three suitable corresponding measurements are enough (like all three sides, or two sides and an angle, or two angles and a side, or the hypotenuse and one side in a right triangle), provided they match one of SSS, SAS, ASA or RHS.

Does having all three angles equal prove two triangles are congruent?+

No, having all three angles equal only proves the triangles are similar, not congruent — congruence also requires at least one side to be equal in measure.

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Lines and Angles

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Pythagoras Theorem

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Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked