Question

From the following figure find,
(a) X
(b) $\angle ABC$
(c) $\angle ACD$

Answer 1

Hint: At first consider all the interior angles of the quadrilateral then use the fact, sum of angles of the quadrilateral is $360{}^\circ $ to find x. Using x find $\angle ABC$. Then consider $\Delta ACD$ and use the fact that the sum of the triangles is $180{}^\circ $ to get what is asked.

Complete step-by-step answer:

In the question, we have been given a figure which represents quadrilateral ABCD shown below.

We know that the sum of the interior angles of a quadrilateral is 4 right angles or $4\times 90{}^\circ =360{}^\circ $ .

First, we will write what sent up the interior angle of the quadrilateral.

So, the interior angles of the quadrilateral contain angles A, B, C, D.

Here angle $A=48+x$, angle $B=4x$ , angle $C=3x$ and angle $D=4x$ .

We can say that
Angle A + angle B + angle C + angle D $=360{}^\circ $
Now substituting angle $A=48+x$ , angle $B=4x$ , angle $C=3x$ and angle $D=4x$ .

So, we get,
$48+x+4x+3x+4x=360{}^\circ $

On simplification we get,
$12x+48=360{}^\circ $

Now on subtracting $48{}^\circ $ from both sides we get,

$12x=312$

So, $x=\dfrac{312}{12}=26{}^\circ $

Hence, the value of x is $26{}^\circ $.

We know $\angle ABC=4x$ so, we will substitute $x=26{}^\circ $ hence, we get

$\angle ABC=104{}^\circ $

Let’s first find $\angle ADC$ which is $4x$ , so $\angle ADC=4\times 26{}^\circ =104{}^\circ $

Now, let’s consider $\Delta ADC$,

We will apply the sum of angles of the triangle is $180{}^\circ $ .

So, $\angle DAC+\angle ACD+\angle CDA=180{}^\circ $

We know that $\angle DAC=48{}^\circ ,\angle CDA=104{}^\circ $

$\angle ACD=180{}^\circ -48{}^\circ -104{}^\circ $

$=28{}^\circ $

Hence, the value of x is $26{}^\circ $, $\angle ABC=104{}^\circ ,\angle ACD=28{}^\circ $ .

Note: Students can also proceed in another way, they can consider first $\Delta ACD$, thus then find $\angle ACD$ in terms of x. After finding it, subtract it from 3x to get the value of $\angle ACB$. Now consider $\Delta ABC$ and use the fact, sum of angles of the triangle is $180{}^\circ $ .