Question

In the given figure, $$AB||CD$$ and $$CA=CE$$. Find the values of $$x$$, $$y$$, and $$z$$.

Answer 1

Hint: Here, we need to find the values of $$x$$, $$y$$, and $$z$$. We will use the alternate interior angles property, the isosceles triangle property, the exterior angle property, and the angle sum property to find the required values. The alternate interior angles on the opposite sides of a transversal, intersecting two parallel lines, are equal. In an isosceles triangle, the angles that are opposite to equal sides are equal. The exterior angle of a triangle is equal to the sum of two opposite interior angles. The angle sum property states that the sum of the three interior angles of a triangle is always $${180}^{\circ }$$.

Complete step-by-step answer:
We can observe that $$AB$$ is parallel to $$CD$$, and $$AD$$ is the transversal.
Therefore, we get the alternate interior angles
$$\mathrm{\angle }BAD=\mathrm{\angle }ADC$$
Substituting $$\mathrm{\angle }BAD=x^{\circ }$$ and $$\mathrm{\angle }ADC={36}^{\circ }$$, we get
$$\begin{array}{l}\Rightarrow x^{\circ }={36}^{\circ }\\ \Rightarrow x=36\end{array}$$
Thus, we get the value of $$x$$ as 36.
Next, it is given that $$CA=CE$$.
Therefore, $$ACE$$ is an isosceles triangle.
We know that in an isosceles triangle, the angles that are opposite to equal sides are equal.
Therefore, since $$CA=CE$$ in triangle $$ACE$$, we get
$$\mathrm{\angle }CAE=\mathrm{\angle }CEA$$
Substituting $$\mathrm{\angle }CAE=y^{\circ }$$, we get
$$\Rightarrow y^{\circ }=\mathrm{\angle }CEA$$
Now, we will use the exterior angle property of a triangle.
In triangle $$ECD$$, we get
$$\mathrm{\angle }CEA=\mathrm{\angle }ECD+\mathrm{\angle }EDC$$
Substituting $$\mathrm{\angle }CEA=y^{\circ }$$, $$\mathrm{\angle }ECD={32}^{\circ }$$, and $$\mathrm{\angle }EDC={36}^{\circ }$$ in the equation, we get
$$\Rightarrow y^{\circ }={32}^{\circ }+{36}^{\circ }$$
Adding the terms in the expression, we get
$$\begin{array}{l}\Rightarrow y^{\circ }={68}^{\circ }\\ \Rightarrow y=68\end{array}$$
Thus, we get the value of $$y$$ as 68.
Therefore, we get $$\mathrm{\angle }CAE=\mathrm{\angle }CEA={68}^{\circ }$$.
Finally, we will use the angle sum property of a triangle to get the value of $$z$$.
Using the angle sum property in triangle $$CAE$$, we get
$$\mathrm{\angle }CAE+\mathrm{\angle }CEA+\mathrm{\angle }ACE={180}^{\circ }$$
Substituting $$\mathrm{\angle }ACE=z^{\circ }$$ and $$\mathrm{\angle }CAE=\mathrm{\angle }CEA={68}^{\circ }$$ in the equation, we get
$$\Rightarrow {68}^{\circ }+{68}^{\circ }+z^{\circ }={180}^{\circ }$$
Adding the terms of the expression, we get
$$\Rightarrow {136}^{\circ }+z^{\circ }={180}^{\circ }$$
Subtracting $${136}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow {136}^{\circ }+z^{\circ }-{136}^{\circ }={180}^{\circ }-{136}^{\circ }\\ \Rightarrow z^{\circ }={44}^{\circ }\end{array}$$
Thus, we get the value of $$z$$ as 44.
Therefore, the values of $$x$$, $$y$$, and $$z$$ are 36, 68, and 44 respectively.

Note: We can also calculate the value of $$z$$ using the property of co-interior angles.
The co-interior angles on the same side of a transversal, intersecting two parallel lines, are supplementary.
We can observe that $$AB$$ is parallel to $$CD$$, and $$AD$$ is the transversal.
Therefore, we get
$$\mathrm{\angle }CAB+\mathrm{\angle }ACD={180}^{\circ }$$
Rewriting $$\mathrm{\angle }CAB=x^{\circ }+y^{\circ }$$ and $$\mathrm{\angle }ACD=z^{\circ }+{32}^{\circ }$$, we get
$$\Rightarrow x^{\circ }+y^{\circ }+z^{\circ }+{32}^{\circ }={180}^{\circ }$$
Substituting $$x^{\circ }={36}^{\circ }$$ and $$y^{\circ }={68}^{\circ }$$, we get
$$\Rightarrow {36}^{\circ }+{68}^{\circ }+z^{\circ }+{32}^{\circ }={180}^{\circ }$$
Adding the terms of the expression, we get
$$\Rightarrow {136}^{\circ }+z^{\circ }={180}^{\circ }$$
Subtracting $${136}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow {136}^{\circ }+z^{\circ }-{136}^{\circ }={180}^{\circ }-{136}^{\circ }\\ \Rightarrow z^{\circ }={44}^{\circ }\end{array}$$
Thus, we get the value of $$z$$as 44.
$$\therefore$$ The values of $$x$$, $$y$$, and $$z$$ are 36, 68, and 44 respectively.