GCSE Maths

Congruence

Edexcel

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Introduction Congruent Triangles Congruent Triangle Rules Solved Examples

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.

Congruent Triangles

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

Examples:

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

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Solved Examples

Solved Example

Detect the congruent triangles among the following triangles? (Part a)

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

SOLUTION

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

Solved Example

Detect the congruent triangles among the following triangles? (Part b)

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

SOLUTION

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

Solved Example

Detect the congruent triangles among the following triangles? (Part c)

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

SOLUTION

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

Solved Example

Detect the congruent triangles among the following triangles? (Part d)

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

SOLUTION

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

Solved Example

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

Solved Example

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