ICSE Class 7 Maths Chapter 26 Congruency Congruent Triangles

Chapter 26: Congruency - Congruent Triangles

Meaning Of Congruency

If two geometrical figures coincide exactly, by placing one over the other, the figures are said to be congruent to each other.

1. Two lines AB and CD are said to be congruent if, on placing AB on CD, or CD on AB, the two lines AB and CD exactly coincide.

It is possible only when AB and CD are equal in length.

2. Two figures ABCD and PQRS are said to be congruent if, on placing ABCD on PQRS or PQRS on ABCD the two figures exactly coincide, i.e., A and P coincide, B and Q coincide, C and R coincide, and D and S coincide.

It is possible only when:

AB = PQ, BC = QR, CD = RS and AD = PS

Also, \(\angle A = \angle P\), \(\angle B = \angle Q\), \(\angle C = \angle R\) and \(\angle D = \angle S\).

Congruency In Triangles

Let triangle ABC is placed over triangle DEF; such that, vertex A falls on vertex D and side AB falls on side DE, then if the two triangles coincide with each other in such a way that B falls on E, C falls on F; side BC coincides with side EF and side AC coincides with side DF, then the two triangles are congruent to each other.

The symbol used for congruency is "=" or "=".

Triangle ABC is congruent to triangle DEF is written as: \(\triangle ABC \cong \triangle DEF\) or \(\triangle ABC = \triangle DEF\).

Corresponding Sides And Corresponding Angles

In case of congruent triangles ABC and DEF, as given above, the sides of the two triangles, which coincide with each other, are called corresponding sides.

Thus, the side AB and DE are corresponding sides, sides BC and EF are corresponding sides and sides AC and DF are also corresponding sides.

In the same way, the angles of the two triangles which coincide with each other, are called corresponding angles. Thus, three pairs of corresponding angles are \(\angle A\) and \(\angle D\), \(\angle B\) and \(\angle E\) and also \(\angle C\) and \(\angle F\).

The corresponding parts of congruent triangles are always equal (congruent).

(i) AB = DE, BC = EF and AC = DF, i.e., corresponding sides are equal.

Also, (ii) \(\angle A = \angle D\), \(\angle B = \angle E\) and \(\angle C = \angle F\), i.e., corresponding angles are equal.

Teacher's Note

Congruent triangles are like identical twins in geometry - they have the same shape and size, just like how two identical picture frames can fit perfectly over each other to match completely.

Conditions Of Congruency

1. Side, Side, Side (S.S.S.)

If three sides of one triangle are equal to three sides of the other triangle, each to each, then the two triangles are congruent.

This condition is known as: side, side, side and is abbreviated as S.S.S.

In triangles ABC and PQR, given alongside:

AB = PQ,

BC = QR and AC = PR

So \(\triangle ABC \cong \triangle PQR\), i.e., \(\triangle ABC \cong \triangle PQR\) by S.S.S.

Similarly, in congruent triangles, corresponding angles are equal.

\(\angle A = \angle P\), \(\angle B = \angle Q\) and \(\angle C = \angle R\)

2. Side, Angle, Side (S.A.S.)

If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, each to each, then the triangles are congruent.

This condition is known as: side, angle, side and is abbreviated as S.A.S.

In the given triangles,

AB = XZ,

BC = XY and \(\angle ABC = \angle ZXY\)

\(\triangle ABC = \triangle ZXY\) (by S.A.S.)

Triangles will be congruent by S.A.S., only when the angles included by the corresponding equal sides are equal.

The pairs of corresponding sides of these two congruent triangles are:

AB and ZX, BC and XY, AC and ZY

The pairs of corresponding angles are:

\(\angle B\) and \(\angle X\), \(\angle A\) and \(\angle Z\), \(\angle C\) and \(\angle Y\).

3. Angle, Side, Angle (A.S.A.)

If two angles and the included side of one triangle are equal to the two angles and the included side of the other triangle, then the triangles are congruent.

This condition is known as: angle, side, angle and is abbreviated as A.S.A.

In the given figure:

BC = QR,

\(\angle B = \angle Q\) and \(\angle C = \angle R\)

\(\triangle ABC = \triangle PQR\). (by A.S.A.)

4. Angle, Angle, Side (A.A.S.)

If any two angles and a side (not the included side) of one triangle are equal to two angles and the corresponding side of the other triangle; then the two triangles are congruent.

This condition is known as: angle, angle, side and is abbreviated as: A.A.S.

5. Right Angle, Hypotenuse, Side (R.H.S.)

If the hypotenuse and one side of a right angled triangle are equal to the hypotenuse and one side of another right angled triangle, then the two triangles are congruent.

This condition is known as: right angle, hypotenuse, side and is abbreviated as R.H.S.

In the given figure:

\(\angle B = \angle E = 90°\), AB = FE

and hypotenuse AC = hypotenuse FD

\(\triangle ABC = \triangle FED\) (by R.H.S.)

The corresponding angles in this case are:

\(\angle A\) and \(\angle F\), \(\angle B\) and \(\angle E\), \(\angle C\) and \(\angle D\),

and the corresponding sides are:

AB and EF, AC and FD, BC and ED.

Since the triangles are congruent, therefore all its corresponding sides are equal and corresponding angles are also equal.

\(\angle BC = ED\), \(\angle A = \angle F\) and \(\angle C = \angle D\)

If three angles of a triangle are equal to the three angles of the other triangle, then the triangles are not necessarily congruent.

For congruency at least one pair of corresponding sides must be equal.

\(\angle A.A.A.\) is not a test of congruency.

Teacher's Note

The five congruency conditions (S.S.S., S.A.S., A.S.A., A.A.S., R.H.S.) are like different ways to verify that two triangular pieces of a puzzle are identical before fitting them together.

Example 1

State, by what test the following triangles are congruent.

(i) [Two triangles shown with two equal sides and marked angles]

(ii) [Two triangles shown with two equal sides and one equal angle]

Solution

(i) Here, two angles of one triangle are equal to the two angles of the other triangle and the included sides are equal.

The two triangles are congruent by A.S.A. (Ans.)

(ii) Since, two sides of one triangle are equal to the two sides of the other triangle and the included angles are equal.

The triangles are congruent by S.A.S. (Ans.)

Example 2

State, whether or not, the following triangles are congruent.

Solution

Here, \(\angle A = 180° - (20° + 50°) = 110°\)

Also, \(\angle Z = 180° - (110° + 20°) = 50°\)

We see that in \(\triangle ABC\) and \(\triangle XYZ\),

\(\angle C = \angle Z = 50°\)

\(\angle B = \angle Y = 20°\) and BC = YZ and they are included sides.

\(\triangle ABC \cong \triangle XYZ\) by A.S.A. (Ans.)

Example 3

In the given figure, AB // CD and AB = CD. Prove that: (i) \(\triangle AOB \cong \triangle DOC\) (ii) AO = DO (iii) BO = CO

Solution

Teacher's Note

When two parallel lines are cut by two crossing lines (like roads intersecting highways), the triangles formed have special relationships, similar to how balanced structures in architecture require equal measurements on opposite sides.

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