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
In $\triangle\text{PQR},\angle\text{A}=\angle\text{Q}$ and $\angle\text{B}=\angle\text{R}.$ Which side of should be equal to side AB of so, that the two triangles are congruent? Give reason for your answer.
→Write the following in ascending order of magnitude.
$\sqrt[6]{6},\ \sqrt[3]{7},\ \sqrt[4]{8}.$
→$\sqrt[6]{6},\ \sqrt[3]{7},\ \sqrt[4]{8}.$
Find the measure of an angle which is $30^\circ $ less than its supplement.
→You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how?
→If $27^\text{x}=\frac{9}{3^\text{x}},$ find $x$.
→Find the value of $k$ for which the point $(1, -2)$ lies on the graph of the linear equation $x - 2y + k = 0$.
→If $x = 3$ and $y = -1$, find the values of the following using in identity: $\Big(\frac{5}{\text{x}}+5\text{x}\Big)\Big(\frac{25}{\text{x}^2}-25+25\text{x}^2\Big)$
→If $a + b = 8$ and $ab = 6$, find the value of $a^3 + b^3.$
→Factorize:
$x\left(x^3-y^3\right)+3 x y(x-y)$
→$x\left(x^3-y^3\right)+3 x y(x-y)$
In a parallelogram $ABCD$, $E$ and $F$ are the mid-points of sides $AB$ and $CD$ respectively. Show that the line segments $AF$ and $EC$ trisect the diagonal $BD.$
→