Question

In the figure (not drawn to scale), ABC and DEF are two triangles, CA is parallel to FD and CF, BE is a straight line. Find the value of \[x + y\].

A) \[185^\circ \]
B) \[134^\circ \]
C) \[158^\circ \]
D) \[176^\circ \]

Answer 1

Hint:
Here we will first find the value of \[x\] by using the concept of the corresponding angles. Then we will find the value of \[y\] by using the exterior angle property of the triangle. We will then add the value of \[x\] and \[y\] to get the value of \[x + y\].

Complete step by step solution:
It is given that the line CA is parallel to line FD and CF, BE is a straight line.
As the line CA is parallel to FD, we can clearly see that angle \[\angle ACF\] is equal to the angle \[\angle DFB\] by the property of the corresponding angles. Therefore, we get
\[\angle ACF = \angle DFB\]
From the diagram, we know that the value of angle \[\angle ACF = x\] and \[\angle DFB = 51^\circ \] .
Substituting the values of angle in the above equation, we get
\[x = 51^\circ \]………………..\[\left( 1 \right)\]
Now we will find the value of \[y\]. Now we will use the exterior angle property of the triangle. Therefore, we get
\[y = \angle ACF + \angle CAB\]
Substituting \[\angle ACF = 51^\circ \] and \[\angle CAB = 134^\circ \] in the above equation, we get
\[ \Rightarrow y = 51^\circ + 83^\circ = 134^\circ \]…………………….\[\left( 2 \right)\]
Now we will add the equation \[\left( 1 \right)\] and equation \[\left( 2 \right)\] to get the value of \[x + y\]. Therefore, we get
\[x + y = 51^\circ + 134^\circ = 185^\circ \]
Hence, the value of \[x + y\] is \[185^\circ \].

So, option A is the correct option.

Note:
The corresponding angles are the angles which are formed at the corresponding corners when the transversal line is passed through the two parallel lines. We need to keep in mind the exterior angle property of the triangle which states that an exterior angle of a triangle is always equal to the sum of the two opposite interior angles of the triangle. In addition to this, vertically Opposite Angles (vertical angles) are the angles opposite each other when two lines intersect or cross each other and pairs of vertically opposite angles (vertical angles) are always equal to each other.