CBSE Class 7 Mathematics Congruence of Triangles Notes Set 02

Congruence

Two figures are congruent when one fits exactly over the other. Congruent line segments have equal length, congruent angles equal measure, congruent circles equal radii and congruent squares equal sides. Congruent figures have equal area and perimeter.

Criteria for Triangles

SSS: three sides of one triangle equal the three sides of the other. SAS: two sides and the included angle are equal. ASA: two angles and the included side are equal. RHS: in right triangles, the hypotenuse and one side are equal.
Three equal angles (AAA) do not make triangles congruent, because the sides may be in a different scale. The vertices must be named in the matching order when writing a congruence statement.

Use of CPCT

Once two triangles are congruent, all their corresponding parts (sides and angles) are equal. This is used to prove that the base angles of an isosceles triangle are equal, that the diagonal of a parallelogram divides it into two congruent triangles and that the diagonals of a rectangle are equal.

Worked Example

In an isosceles triangle ABC with AB = AC, let the bisector of angle A meet BC at D. In triangles ABD and ACD, AB = AC, angle BAD = angle CAD and AD is common, so they are congruent by SAS. By CPCT, BD = DC and angle ADB = angle ADC = 90°.