Question

If any two sides of a triangle are produced beyond its base and the exterior angles thus obtained are bisected, then these bisectors will include an angle equal to:

• A. half the sum of the base angles
• B. sum of the base angles
• C. half the difference of the base angles
• D. difference of the base angles

Answer 1

The correct option is A half the sum of the base angles

Let the triangle be with angles a, b, c with base angles b and c.
The triangle formed by the given bisectors has the angles x, y and p.
P is the angle formed by the bisector.
Now 2x $=a+c$
Therefore x $=\frac{a}{2}+\frac{c}{2}$
similarly y $=\frac{a}{2}+\frac{b}{2}$
again P $={180}^{\circ }-(x+y)$
substituting for x and y we get P $={180}^{\circ }-(\frac{a}{2}+\frac{c}{2})-(\frac{a}{2}+\frac{b}{2})$
$=a+b+c-\frac{a}{2}-\frac{c}{2}-\frac{a}{2}-\frac{b}{2}$
$=\frac{b}{2}+\frac{c}{2}$
$=\frac{1}{2}$ ( sum of the base angles )