Question
The bisectors of $\angle\text{B}$ and $\angle\text{C}$ of an isosceles triangle with $AB = AC$ intersect each other at a point $O. BO$ is produced to meet $AC$ at a point $M.$ Prove that $\angle\text{MOC}=\angle\text{ABC}.$
✓
Answer
In $\triangle\text{ABC},\text{ AB = AC}$
$\Rightarrow\angle\text{ABC}=\angle\text{ACB}$
$\Rightarrow\frac{1}{2}\angle\text{ABC}=\frac{1}{2}\angle\text{ACB}$
$\angle\text{OBC}=\angle\text{OCB}...\text{(i)}$
Now, by exterior angle property,
$\angle\text{MOC}=\angle\text{OBC}+\angle\text{OCB}$
$\Rightarrow\angle\text{MOC}=2\angle\text{OBC} [$from $(i)]$
$\Rightarrow\angle\text{MOC}=\angle\text{ABC}$
$\big(OB$ is the bisector of $\angle\text{ABC}\big)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Numbers $50, 42, 35, 2x + 10, 2x - 8, 12, 11, 8$ are written in descending order and their median is $25$, find $x$.
→Obtain the mean of the following distribution:
→|
Frequency
|
Varible
|
|
$4$
$8$
$14$
$11$
$3$
|
$4$
$6$
$8$
$10$
$12$
|
Find the following products:
$(3 x-4 y+5 z)\left(9 x^2+16 y^2+25 z^2+12 x y-15 z x+20 y z\right)$
→$(3 x-4 y+5 z)\left(9 x^2+16 y^2+25 z^2+12 x y-15 z x+20 y z\right)$
Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}.$
→Evaluate the following:
$(9.9)^3$
→$(9.9)^3$
If $1176=2^\text{a}\times3^\text{b}\times7^\text{c},$ find the values of $a, b$ and $c$. Hence, compute the value of $2^\text{a}\times3^\text{b}\times7^{-\text{c}}$ as a fraction.
→If the angles of a triangle are in the ratio $1 : 2 : 3$, determine three angles.
→In a parallelogram $ABCD$, $\angle \text{D}=135^\circ$. Determine the measures of $\angle\text{A}$ and $\angle\text{B}$.
→Show that a diagonal divides a parallelogram into two triangles of equal area.
Give the geometric representations of the following equations:
$a.$ On the number line.
$b.$ On the Cartesain plane.
$y + 3 = 0$
→$a.$ On the number line.
$b.$ On the Cartesain plane.
$y + 3 = 0$