# Illustrative Examples Based on Congruence of Triangle

In Î”ABC, AB = AC = 8 cm and âˆ BAC = 110Â°, find âˆ B and âˆ C.

In Î”ABC, since AB = ACÂ

... âˆ B = âˆ C ...... (Angles opposite to equal sides)

Â

Now, âˆ A + âˆ B + âˆ C = 180^{o} ...... (Sum of three angles of a triangle)

= 110Â° + âˆ B + âˆ B = 180Â°

= 2 âˆ B = 180Â° - 110Â°

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 70Â°

= âˆ B = 35Â°

...âˆ B = âˆ C = 35Â°

Â

Â

In the figure, AB = AC and MB = MC. Prove that âˆ ABM = âˆ ACM

In Î”ABC, AB = AC (given)

...Â Â Â Â Â Â âˆ ABC = âˆ ACB ...... (i) (Angles opposite to equal sides)

Now in Î”MBC, MB = MC

...Â Â Â Â Â Â âˆ MBC = âˆ MCB ...... (ii)

Subtracting eqn (ii) from eqn (i), we get

Â Â Â Â Â âˆ ABC - âˆ MBC = âˆ ACB - âˆ MCB

...Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ ABM = âˆ ACM

Â

Prove that Î”ABC is isosceles if

(i) Altitude AD bisects BC

(ii) Median AD is perpendicular to BC

Â

(i) Since AD âŠ¥ BC, âˆ ADB = âˆ ADC = 90Â°

Now in Î”ABD and Î”ACD, we have

âˆ ADB = âˆ ADC ... (each 90Â°)

AD = AD ... (common)

BD = DC ... (given)

Î”ADB â‰… Î”ADC

AB = ACÂ

Î”ABC is an isosceles triangle

(ii) Solve as in (i)

(As âˆ ADB = âˆ ADC =180Â°/2 = 90Â°)