Congruence – GCSE Maths

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Introduction

  • Congruent shapes are those which have the same shape and size. These shapes can be rotated to look different but remain congruent.
  • The Congruent shapes if are reflections (mirror image) of each other they might look very different.

Examples of congruent and non-congruent shapes including stars, pentagons, and congruent triangles with marked sides

Congruent Triangles

  • Congruent Triangles are those which are same in size and shape but their orientation maybe different.

Examples:

Three examples showing triangle congruence and non-congruence based on shape, size, and orientation

Congruent Triangle Rules

We cannot always tell by just seeing the Triangles that they are congruent or not. Thus we have following Rules and by learning them we can easily detect the Congruent Triangles.

  • S – Side
  • A – Angle
  • R – Right Angle
  • H – Hypotenuse

1. SSS

  • All sides are equal in length.

Two equilateral triangles with equal sides and 60-degree angles showing congruence

2. SAS

  • When two sides and the angle between them is same in both the triangles.

Two triangles with 3 cm and 2 cm sides and a 120-degree angle showing congruence using SAS rule

3. AAS

  • Two Angles are same and one side is equal in both triangles at the same place with respect to these two equal angles.

Two right-angled triangles with 4 cm hypotenuse and 45-degree angles showing congruence using RHS rule

4. RHS

  • When one of the angles is Right angle and Hypotenuse lengths are same along with one equal side.

Two right-angled triangles with 4 cm and 6 cm sides shown in different orientations, demonstrating congruence using the RHS rule

certified Physics and Maths tutorSolved Example

Problem: Detect the congruent triangles among the following triangles?

(a)

Four triangles with different side lengths for identifying congruent pairs based on SSS rule

Answer: 2nd and 4th Triangles are Congruent (by SSS).

(b)

Four triangles with pairs of equal sides and angles for comparing congruence using the ASA rule

Answer: 1st and 3rd Triangles are Congruent (by SAS).

(c)

Four triangles with side lengths and right angles shown for checking congruence using the RHS rule

Answer: 1st and 3rd Triangles are Congruent (by RHS) their orientation is different but still size and shape is same.

(d)

Four triangles showing combinations of side lengths and angles for comparing congruence using ASA rule

Answer: 1st and 4th Triangles are Congruent (by AAS) their orientation is different but still size and shape is same.

certified Physics and Maths tutorSolved Example

Problem: Given that the following Triangles are congruent, then find out the missing side in the triangle B?

Two right-angled triangles with matching 5 cm sides, one with missing side x to determine congruence

Solution: 

The above triangles are congruent by the rule RHS as both the triangles are right angle triangles and two sides are equal one of them being hypotenuse. the second triangle’s hypotenuse is equal to the hypotenuse of the first triangle.

Thus, x = 7 cm

Final Answer: x = 7 cm

certified Physics and Maths tutorSolved Example

Problem: Given that the following Triangles are congruent, then find out the missing side in the triangle A?

Two triangles with two matching sides labeled, one with missing side x to test for congruence using SSS rule

Solution: 

The triangles are congruent by the rule SSS means the sides are equal.

Thus, x = 3 cm.

Final Answer: x = 3 cm

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