Question

The angle measures of $x$ and $y$ are respectively

A) ${63}^{\circ },{142}^{\circ }$
B) ${113}^{\circ },{38}^{\circ }$
C) ${117}^{\circ },{79}^{\circ }$
D) ${115}^{\circ },{75}^{\circ }$

Answer 1

Hint: To solve this question we need to know the relation between the angles formed in the triangle. The angle formed on the straight line is ${180}^{\circ }$. The property of corresponding angle is also used to solve the problem. The sum of angles of a triangle is ${180}^{\circ }$.The transversal line is the line which cuts the two parallel lines given.

Complete step by step solution:
The question asks us to find the angles $x$ and $y$ in the given triangle. On analysing the figure given below we can see that line $AB$ is parallel to line $j$. Since these two lines are parallel lines $AC$ and line $BC$ are transversal on the parallel lines. Transversal line refers to the line which cuts the two parallel lines.

We will start with finding the angle$\mathrm{\angle }Y$. We are aware of the fact that the sum of angles in the triangle is equal to ${180}^{\circ }$. On writing it mathematically we get:
$\mathrm{\angle }ABC+\mathrm{\angle }BCA+\mathrm{\angle }CAB={180}^{\circ }$ ……………(i)
We can see in the figure $\mathrm{\angle }CBA$ is not given. Instead the outer angle is given. So here we can apply the property that the angle made by straight line is ${180}^{\circ }$,so applying the same we get:
$\mathrm{\angle }ABC+\alpha ={180}^{\circ }$
$\Rightarrow \mathrm{\angle }ABC={180}^{\circ }-\alpha$
$\Rightarrow \mathrm{\angle }ABC={180}^{\circ }-{142}^{\circ }={38}^{\circ }$
On substituting the values in equation (i), we get:
$\Rightarrow {38}^{\circ }+Y+{63}^{\circ }={180}^{\circ }$
$\Rightarrow Y+{101}^{\circ }={180}^{\circ }$
$\Rightarrow Y={180}^{\circ }-{101}^{\circ }$
$\Rightarrow Y={79}^{\circ }$
Now we will find the angle $\mathrm{\angle }x$ . We know that the lines are parallel and AC is the transversal on the two parallel lines. Angle$\theta$ and angle $x$ are on the same line so their sum will result in ${180}^{\circ }$. Mathematically it would be written as:
$x+\theta ={180}^{\circ }$
To find angle $\theta$ we need to know the property of angles in case of parallel lines and transversal. So angle $\theta$ and angle $\beta$ are equal because both are corresponding angles.
$\Rightarrow \beta =\theta ={63}^{\circ }$
Substituting the value of $\theta$ we get:
$\Rightarrow x+{63}^{\circ }={180}^{\circ }$
$\Rightarrow x={180}^{\circ }-{63}^{\circ }$
$\Rightarrow {117}^{\circ }$
So, the correct answer is “Option C”.

Note: All the properties of the triangle should be known to us to solve the value for the angles. Angle $\mathrm{\angle }Y$ could also be found by applying the exterior angle property. This means :$\mathrm{\angle }CAB+\mathrm{\angle }ACB=\alpha$
On putting the values we get:
$\Rightarrow {63}^{\circ }+Y={142}^{\circ }$
$\Rightarrow Y={142}^{\circ }-{63}^{\circ }$
$\Rightarrow Y={79}^{\circ }$
So the method can also be used to find the angle of the triangle if the opposite exterior angle is given.