Question

In the following figure, \[\angle ABD = 3\angle DAB\,and\,\angle ADC = {120^ \circ }\]then find \[\angle ABD\].

Answer 1

Hint: According to the question, we need to see what is given and what we have to find out. We can take the help of the theorem which is the angle sum property. It tells that the sum of all the three interior angles of a triangle is equal to \[{180^ \circ }\]. After that we can solve, and get the answer.

Formula used:
\[\angle ABD + \angle BDA + \angle DAB = {180^ \circ }\]

Complete step by step solution:
From the question, we know that \[\angle ADC = {120^ \circ }\].
Now, we know that the \[\angle D = {180^ \circ }\], and we also know that:
 \[\angle BDC = \angle BDA + \angle ADC\]
Now, if \[\angle D = {180^ \circ }\], then \[\angle BDC = {180^ \circ }\]. So, when we put the values, we get:
 \[ \Rightarrow {180^ \circ } = \angle BDA + {120^ \circ }\]
 \[ \Rightarrow \angle BDA = {180^ \circ } - {120^ \circ }\]
 \[ \Rightarrow \angle BDA = {60^ \circ }\]
Now, we know that by the angle sum property. This theorem says that the sum of all the three interior angles of a triangle is equal to \[{180^ \circ }\]. From that we get that in \[\Delta ABD\]:
 \[\angle ABD + \angle BDA + \angle DAB = {180^ \circ }\]
We know that \[\angle BDA = {60^ \circ }\] (solved above). We know that \[\angle ABD = 3\angle DAB\] (given in question).
Now, we will put the values, and we get:
 \[ \Rightarrow 3\angle DAB + {60^ \circ } + \angle DAB = {180^ \circ }\]
Now, we will simplify the terms, and we get:
 \[ \Rightarrow 4\angle DAB + {60^ \circ } = {180^ \circ }\]
Now, we will try to make \[\angle DAB\]alone. We will shift \[{60^ \circ }\]on the other side of the equation, and we get:
 \[ \Rightarrow 4\angle DAB = {180^ \circ } - {60^ \circ }\]
Now, we will simplify the right-hand side of the equation, and we get:
 \[ \Rightarrow 4\angle DAB = {120^ \circ }\]
Now, we will shift the 4 to the other side of the equation. It gets divided on the other side of the equation, and we get:
 \[ \Rightarrow \angle DAB = dfrac{{{{120}^ \circ }}}{4}\]
When we divide \[{120^ \circ }\]by 4, we get:
 \[ \Rightarrow \angle DAB = {30^ \circ }\]
Our question was to find \[\angle ABD\], and in the question given was that \[\angle ABD = 3\angle DAB\]. So, here we will put the value of \[\angle DAB\], and we get:
 \[ \Rightarrow \angle ABD = 3 \times {30^ \circ }\]
 \[ \Rightarrow \angle ABD = {90^ \circ }\]
Therefore, we got the answer as \[\angle ABD = {90^ \circ }\]
So, the correct answer is “ \[\angle ABD = {90^ \circ }\] ”.

Note: The above theorem which we used was for internal angles. There is another theorem for external angles. This theorem says that if any side of a triangle is getting extended from one side, the exterior angle which is getting formed is equal to the sum of the opposite interior angles of that triangle.