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Congruent definition

Two shapes are said to be congruent if their corresponding characteristics are the same. For example, two line segments are said to be congruent if they have the same length. Similarly, two circles are said to be congruent if they have the same radius.

Congruent angles

Two angles are said to be congruent if they have the same measure.

• Vertically opposite angles formed by two intersecting lines are always congruent as they have the same measure
• When two parallel lines are intersected by a transversal, the alternate interior angles formed are congruent to each other.

In the above figure, line P is parallel to line Q and R is the transversal.

Angle 4 and angle 6 are alternate interior angles and are congruent.

Similarly, angle 3 and angle 5 are alternate interior angles and are congruent

Congruent triangles

Two triangles are congruent if any of the following conditions is true:

1. All three sides are equal. This is called the SSS rule (Side - Side - Side)
2. Two sides and the included angle are equal. This is called the SAS rule (Side - Angle - Side)
3. Two angles and one side is equal. Two triangles are congruent if two angles are equal and one similarly located side is equal. This is called AAS rule (Angle - Angle - Side)
4. Two sides in a right angled triangle are equal. If two right angled triangles have the hypotenuse and one of the sides equal then the triangles are congruent. This is called the RHS rule (Right angled - Hypotenuse - Side).
5. Two angles and the included side is equal. Two triangles are congruent if two angles are equal and the included side is also equal. This is called ASA rule (Angle - Side - Angle)

• There is no AAA (Angle- Angle - Angle) rule for congruency. If two triangles have all three angles equal, it does not mean their sides are equal and hence they need not be congruent.

Practice Questions

Q1: Identify the congruent angles in the below figure where P and Q are intersecting lines.

∠1 is equal to ∠3 as they are vertically opposite. Thus, they are congruent.

∠2 is equal to ∠4 as they are vertically opposite. They are the second congruent pair of angles.

Q2: Determine if triangle ABC is congruent to triangle PQR

In triangle ABC, one angle is 30 degrees, second angle is 50 degrees and the side between these two angles is marked as 4 cms.

In triangle PQR, one angle is 30 degrees, second angle is 50 degrees and the side between these two angles is also marked as 4 cms.

This satisfies the ASA (Angle - Side - Angle) rule and hence the two triangles are congruent.

Q3: Determine if the two triangles are congruent to each other.

In triangle 1, the measures of the three sides are 2 cms, 2.5 cms and 3 cms respectively.

In triangle 2, the measures of the three sides are also 2 cms, 2.5 cms and 3 cms respectively.

This satisfies the SSS (Side - Side - Side) rule and hence the two triangles are congruent.

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