Question

In fig.1 $ABCD$ is a parallelogram. Find the angles $x$ and $y$.

Answer 1

Hint:
Here we need to find the value of two unknown angles of a parallelogram. For that, we will use the properties of parallelogram to find the value of the first unknown angle. Then we will consider the triangle inside the parallelogram and we will use the property of the triangle to find the value of the second unknown angle.

Complete step by step solution:
We need to find the value of angles \[x\] and \[y\] of the given parallelogram \[ABCD\].
As we know the opposite sides of a parallelogram are parallel to each other.
We know that the alternate angles of a parallelogram are equal.
In parallelogram \[ABCD\], \[\angle ADB\] and \[\angle DBC\] are alternate angles.
Therefore,
\[\angle ADB = \angle DBC\] ………. \[\left( 1 \right)\]
 It is given in the question that the value of \[\angle DBC\] is equal to \[40^\circ \].Now, we will substitute the value of this angle in equation \[\left( 1 \right)\].
\[ \Rightarrow \angle ADB = 40^\circ \] ………. \[\left( 2 \right)\]
We know from the question that \[\angle ADB = y\]
Substituting this value in equation \[\left( 2 \right)\], we get
\[ \Rightarrow y = 40^\circ \]
We know that the sum of all interior angles of a triangle is equal to \[180^\circ \]
Now, we will apply this property of triangle in \[\Delta ADB\] .
In \[\Delta ADB\],
\[\angle ADB + \angle DBA + \angle BAD = 180^\circ \]
Now, substituting the value of all angle , we get
\[ \Rightarrow 40^\circ + x + 80^\circ = 180^\circ \]
On adding the terms, we get
\[ \Rightarrow x + 120^\circ = 180^\circ \]
Subtracting \[120^\circ \] from both sides, we get
\[\begin{array}{l} \Rightarrow x + 120^\circ - 120^\circ = 180^\circ - 120^\circ \\ \Rightarrow x = 60^\circ \end{array}\]

Hence, the value of angles \[x\] is \[60^\circ \] and of \[y\] is \[40^\circ \].

Note:
We have been provided with the diagram of a parallelogram. A parallelogram is a two dimensional geometric figure. To solve this question, we need to know the different properties of parallelograms.
Some properties are listed below:
1) The sides of a parallelogram that are opposite are always parallel to each other.
2) The angles of a parallelogram that are opposite are always equal to each other.
3) The two diagonals of a parallelogram always bisect each other.