Question

In the diagram if $\Delta ABC$ and $\Delta PQR$ are equilateral. The $\angle CXY$ equals

A) ${35^ \circ }$
B) ${40^ \circ }$
C) ${45^ \circ }$
D) ${50^ \circ }$

Answer 1

Hint:
Here, we have given that if $\Delta ABC$ and $\Delta PQR$ are equilateral. So, we will find the value of $\angle CXY$. First, we have to find the value of $\angle CBP,\angle XBP,\angle BYP$. Then from the $\Delta XYC$, we have to get the value of $\angle CXY$.

Complete step by step solution:
Here, we have given that if $\Delta ABC$ and $\Delta PQR$ are equilateral
So, we have to find $\angle CXY$
 $\because $ $\Delta ABC$ is an equilateral triangle
 $\therefore $ $\angle ABC$ $ = {60^ \circ }$
Now, $\angle CBP = 180 - \angle ABC - 65$ ( $\because $ Angles on a straight line)
$\angle CBP = 180 - 60 - 65$
$\angle CBP = {55^ \circ }$
Now, we have to similarly with $\angle XBP$
$\angle XBP = {45^ \circ }$
Now, In $\Delta YBP$
$\angle YBP + \angle YPB + \angle BYP = {180^ \circ }$
$\angle BYP = 180 - 45 - 55$
$\angle BYP = {80^ \circ }$
$\angle XYC = \angle BYP = {80^ \circ }$ ($\because $Vertical opposite angles)
Sum of angles of $\Delta XYC = {180^ \circ }$
$\angle XYC + \angle CXY + \angle XCY = 180$ ($\angle XCY = {60^ \circ }$)
80 + 60 + $\angle CXY$ \[ = {\text{ }}180\]

$\therefore $$\angle CXY = {40^ \circ }$

Note:
Equilateral triangle: An equilateral triangle in which all the three sides have the same length. An equilateral triangle is also an equiangular i.e. all the three internal angles are also congruent to each other and are each ${60^ \circ }$.
Vertical opposite angles: Vertically opposite angles are the angles opposite to each other when two lines cross. “Vertically” in this case means they share the same vertex (corner point), not the usual meaning of up-down.

In the above diagram, $\angle A$ and $\angle B$ are vertically opposite angles.