Question
Find the interior angles of the following triangles:
✓
Answer
In $\triangle ABD,$
$AD = BD ....($given$)$
$\Rightarrow \angle ABD = \angle BAD ....($angles opposite to two equal sides are equal$)$
Now, $\angle ABD = 37^\circ ....($given$)$
$\Rightarrow \angle BAD = 37^\circ $
By exterior angle property,
$\angle ADC =\angle ABD + \angle BAD$
$\Rightarrow \angle ADC = 37^\circ + 37^\circ = 74^\circ $
In $\triangle ADC,$
$AC = DC ....($given$)$
$\Rightarrow \angle ADC = \angle DAC ....($angles opposite to two equal sides are equal$)$
$\Rightarrow \angle DAC = 74^\circ $
Now, $\angle BAC = \angle BAD + \angle DAC$
$\Rightarrow \angle BAC = 37^\circ + 74^\circ = 111^\circ $
In $\triangle ABC,$
$\angle BAC + \angle ABC + \angle ACB = 180^\circ $
$\Rightarrow 111^\circ + 37^\circ + \angle ACB = 180^\circ $
$\Rightarrow \angle ACB = 180^\circ - 111^\circ - 37^\circ = 32^\circ $
Hence, the interior angles of $\triangle ABC$ are $37^\circ , 111^\circ $ and $32^\circ .$
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Case Study I : One day three friends Amit (A), Binay (B) and Chanchal (C) were playing hide and seek game in the park of their society. Amit and Binay hide in the shrubs and Chanchal has to find both of them. If the position of three friends are at A, B and C respectively, as shown in the figure and forms a right angled triangle ABC such that AB = 6m BC = $2 \sqrt{3}$ m and $\angle$B = $90^{\circ}$.
Based on the above information, answer the following questions :
1. The length of AC is :
(a) 4 m (b) $8 \sqrt{3}$ m (c) $5 \sqrt{3}$ m (d) $4 \sqrt{3}$ m
2. The measure of $\angle$A is:
(a) $30^{\circ}$ (b) $45^{\circ}$ (c) $60^{\circ}$ (d) $90^{\circ}$
3. The measure of $\angle$C is:
(a) $30^{\circ}$ (b) $45^{\circ}$ (c) $60^{\circ}$ (d) $90^{\circ}$
4. cos 2A is equal to:
(a) 1 (b) $\frac{1}{2}$ (c) $\frac{\sqrt{3}}{2}$ (d) $\sqrt{3}$
5. $2 \sin \left(\frac{C}{2}\right)$ is equal to:
(a) 1 (b) $\frac{1}{2}$ (c) $\sqrt{3}$ (d) $\frac{\sqrt{3}}{2}$