Question
In the figure, given below, $A B=A C$.Prove that: $\angle B O C=\angle A C D$.
✓
Answer
Let $\angle A B O=\angle O B C=x$ and $\angle A C O=\angle O C B=y$ In $\triangle \mathrm{ABC}$
$\angle B A C=180^{\circ}-2 x-2 y\ldots. (i)$
Since, $\angle B=\angle C\ldots[\mathrm{AB}=\mathrm{AC}]$
$\frac{1}{2} \mathrm{~B}=\frac{1}{2} \mathrm{C}$
$ \Rightarrow \mathrm{x}=\mathrm{y}$
Now,
$\angle \mathrm{ACD}=2 \mathrm{x}+\angle \mathrm{BAC} \ldots[$ Exterior angle is equal to sum of opp. interior angles $]$
$\angle \mathrm{ACD}=2 \mathrm{x}+180^{\circ}-2 \mathrm{x}-2 \mathrm{y} \ldots[$ From$(i)]$
$\angle \mathrm{ACD}=180^{\circ}-2 \mathrm{y}\dots ....(i)$
In $\triangle O B C_1$
$\angle B O C=180^{\circ}-x-y$
$\Rightarrow \angle B O C=180^{\circ}-\mathrm{y}-\mathrm{y} \ldots[$ Already proved $]$
$\Rightarrow \angle \mathrm{BOC}=180^{\circ}-2 \mathrm{y}\dots ...(ii)$
From $(i)$ and $(ii)$
$\angle B O C=\angle A C D$
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
If $x + y + z = 12$ and $xy + yz + zx = 27$; find $x^2 + y^2 + z^2$.
→At what rate per cent per annum compound interest will ₹ 6250 amount to ₹ 7290 in 2 years?
In rectangle $\text{ABCD},$ diagonal $\text{BD} = 26 \ cm$ and cotangent of $\angle \text{ABD} = 1.5.$ Find the area and the perimeter of the rectangle $\text{ABCD}.$
→The volume of a cuboid is $14400 cm^3$ and its height is 15 cm . The cross-section of the cuboid is a rectangle having its sides in the ratio $5: 3$. Find the perimeter of the cross-section.
→$90\%$ acid solution$ (90\%$ pure acid and $10\%$ water$)$ and $97\%$ acid solution are mixed to obtain $21$ litres of $95\%$ acid solution. How manylitresof each solution are mixed.
→Construct a cumulative frequency distribution table from the frequency table given below:
→| Class Interval | Frequency |
| $1 - 10$ | $12$ |
| $11 - 20$ | $18$ |
| $21 - 30$ | $23$ |
| $31 - 40$ | $15$ |
| $41 - 50$ | $10$ |
Out of $10$ students, who appeared in a test, three secured less than $30$ marks and $3$ secured more than $75$ marks. The marks secured by the remaining $4$ students are $35, 48, 66$ and $40$. Find the median score of the whole group.
→Two years ago, the population of a village was 4000. During next year it increased by 6% but due to an epidemic, it decreased by 5% in the following year. What is its population now?
If $x^{\frac{1}{3}}+y^{\frac{1}{3}}+z^{\frac{1}{3}}=0$, prove that $(x+y+z)^3=27\text{xyz}$
→A chord of length $24 \ cm$ is at a distance of $5 \ cm$ from the center of the circle. Find the length of the chord of the same circle which is at a distance of $12 \ cm$ from the center.
→