CBSE Class 8 Mathematics Congruence of Triangles Notes Set 05

Congruent Figures

Two figures are congruent when they have exactly the same shape and the same size, so one can be placed over the other to cover it completely. The symbol for congruence is ≅.
Two line segments are congruent if their lengths are equal, two angles are congruent if their measures are equal, two circles are congruent if their radii are equal, and two squares are congruent if their sides are equal.
Congruent figures have equal areas, but two figures with equal areas need not be congruent.

Congruent Triangles

If △ABC ≅ △DEF with A ↔ D, B ↔ E and C ↔ F, then AB = DE, BC = EF, CA = FD and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F. The order of the letters matters because it shows which vertices match.
Corresponding parts of congruent triangles are equal (often written CPCT). This is the tool used to prove that two lengths or two angles are equal once the triangles have been shown congruent.

Criteria for Congruence

SAS: two sides and the angle between them are equal to the corresponding parts of the other triangle.
ASA: two angles and the side between them are equal.
AAS: two angles and a side that is not between them are equal. This follows from ASA because the third angles must also be equal.
SSS: all three sides are equal.
RHS: in two right-angled triangles, the hypotenuse and one other side are equal.
AAA (three equal angles) and SSA (two sides and a non-included angle) are not criteria. A small triangle cut off by a line parallel to the base of a bigger triangle has the same three angles as the big triangle but is clearly not congruent to it.

Congruence is an Equivalence Relation

Every triangle is congruent to itself (reflexive). If △ABC ≅ △DEF then △DEF ≅ △ABC (symmetric). If △ABC ≅ △DEF and △DEF ≅ △PQR then △ABC ≅ △PQR (transitive).

Useful Results Proved by Congruence

Angles opposite equal sides of a triangle are equal, and sides opposite equal angles are equal. Each angle of an equilateral triangle is 60°.
If the median from a vertex is also perpendicular to the opposite side, the triangle is isosceles. In an isosceles triangle the altitude from the apex bisects the base and the vertical angle.
If the three altitudes of a triangle are equal, the triangle is equilateral. In an isosceles triangle the altitudes on the equal sides are equal.
In a right triangle the mid-point of the hypotenuse is at the same distance from all three vertices.

Inequalities in a Triangle

If two sides of a triangle are unequal, the longer side has the greater angle opposite to it, and if two angles are unequal, the greater angle has the longer side opposite to it.
The sum of any two sides of a triangle is greater than the third side, and the difference of any two sides is less than the third side.
The perpendicular from a point to a line is the shortest segment from the point to the line. In a right triangle the hypotenuse is the longest side.

Angle Facts Used with Triangles

The three angles of a triangle add up to 180°. An exterior angle equals the sum of the two interior opposite angles.
The bisectors of ∠B and ∠C of a triangle meet at an angle of 90° + ½∠A. The bisectors of the two exterior angles at B and C meet at an angle of 90° − ½∠A.

Worked Examples

Example 1: In △ABC the median AD is perpendicular to BC. Show that AB = AC. In △ABD and △ACD, BD = DC (D is the mid-point), ∠ADB = ∠ADC = 90° and AD is common. So the triangles are congruent by SAS and AB = AC.
Example 2: In a parallelogram ABCD the diagonals AC and BD are equal. Find ∠ABC. In △ABD and △BAC, AD = BC, AB is common and BD = AC, so they are congruent by SSS and ∠DAB = ∠ABC. As AD ∥ BC, ∠DAB + ∠ABC = 180°, so 2∠ABC = 180° and ∠ABC = 90°.
Example 3: AD and BC are equal perpendiculars to AB. Show that CD bisects AB. Let AB and CD meet at O. In △AOD and △BOC, ∠AOD = ∠BOC (vertically opposite), ∠OAD = ∠OBC = 90° and AD = BC. So the triangles are congruent by AAS and OA = OB.
Example 4: In △ABC, ∠B = 45° and ∠C = 55°. The bisector of ∠A meets BC at D. Then ∠A = 80°, ∠BAD = 40° and ∠ADC = ∠B + ∠BAD = 85°.