Question

Find the angle measure of $$x$$ in the following figures:

Answer 1

Hint: Here, we need to find the value of $$x$$ in the given figures. We will use the angle sum property of a quadrilateral to find the value of $$x$$ in first figures. Then we will use the properties of linear pair angles to find the angle in the third figure. We will then use the angle sum property of a pentagon to find the value of $$x$$ in the fourth figure.

Complete step-by-step answer:
(a)
The sum of all the interior angles of a quadrilateral is equal to $${360}^{\circ }$$.
Therefore, we get
$${50}^{\circ }+{130}^{\circ }+{120}^{\circ }+x={360}^{\circ }$$
This is a linear equation in terms of $$x$$. We will solve this equation to find the value of $$x$$.
Adding the terms of the equation, we get
$$\Rightarrow {300}^{\circ }+x={360}^{\circ }$$
Subtracting $${300}^{\circ }$$ from both sides of the equation, we get
$$\Rightarrow {300}^{\circ }+x-{300}^{\circ }={360}^{\circ }-{300}^{\circ }$$
Thus, we get
$$\Rightarrow x={60}^{\circ }$$
Therefore, we get the value of $$x$$ as $${60}^{\circ }$$.
(b)
First, we will mark another angle in the figure.

The sum of all the angles lying on a line is equal to $${180}^{\circ }$$. These angles are said to form a linear pair.
From the figure, we can observe that the right angle and angle 1 form a linear pair.
Therefore, we get
$${90}^{\circ }+\mathrm{\angle }1={180}^{\circ }$$
Subtracting $${90}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow {90}^{\circ }+\mathrm{\angle }1-{90}^{\circ }={180}^{\circ }-{90}^{\circ }\\ \Rightarrow \mathrm{\angle }1={90}^{\circ }\end{array}$$
The sum of all the interior angles of a quadrilateral is equal to $${360}^{\circ }$$.
Therefore, we get
$${70}^{\circ }+{60}^{\circ }+\mathrm{\angle }1+x={360}^{\circ }$$
This is a linear equation in terms of $$x$$. We will solve this equation to find the value of $$x$$.
Substituting $$\mathrm{\angle }1={90}^{\circ }$$ in the equation, we get
$$\Rightarrow {70}^{\circ }+{60}^{\circ }+{90}^{\circ }+x={360}^{\circ }$$
Adding the terms of the equation, we get
$$\Rightarrow {220}^{\circ }+x={360}^{\circ }$$
Subtracting $${220}^{\circ }$$ from both sides of the equation, we get
$$\Rightarrow {220}^{\circ }+x-{220}^{\circ }={360}^{\circ }-{220}^{\circ }$$
Thus, we get
$$\Rightarrow x={140}^{\circ }$$
Therefore, we get the value of $$x$$ as $${140}^{\circ }$$.
(c)
First, we will mark two angles in the figure.

The sum of all the angles lying on a line is equal to $${180}^{\circ }$$. These angles are said to form a linear pair.
From the figure, we can observe that the angle measuring $${70}^{\circ }$$ and angle 1 form a linear pair.
Therefore, we get
$${70}^{\circ }+\mathrm{\angle }1={180}^{\circ }$$
Subtracting $${70}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow {70}^{\circ }+\mathrm{\angle }1-{70}^{\circ }={180}^{\circ }-{70}^{\circ }\\ \Rightarrow \mathrm{\angle }1={110}^{\circ }\end{array}$$
From the figure, we can observe that the angle measuring $${60}^{\circ }$$ and angle 2 form a linear pair.
Therefore, we get
$${60}^{\circ }+\mathrm{\angle }2={180}^{\circ }$$
Subtracting $${60}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow {60}^{\circ }+\mathrm{\angle }2-{60}^{\circ }={180}^{\circ }-{60}^{\circ }\\ \Rightarrow \mathrm{\angle }2={120}^{\circ }\end{array}$$
The sum of all the interior angles of a pentagon is equal to $${540}^{\circ }$$.
Therefore, we get
$${120}^{\circ }+x+x+\mathrm{\angle }1+\mathrm{\angle }2={540}^{\circ }$$
This is a linear equation in terms of $$x$$. We will solve this equation to find the value of $$x$$.
Substituting $$\mathrm{\angle }1={110}^{\circ }$$ and $$\mathrm{\angle }2={120}^{\circ }$$ in the equation, we get
$$\Rightarrow {120}^{\circ }+x+x+{110}^{\circ }+{120}^{\circ }={540}^{\circ }$$
Adding the terms of the equation, we get
$$\Rightarrow {350}^{\circ }+2x={540}^{\circ }$$
Subtracting $${350}^{\circ }$$ from both sides of the equation, we get
$$\begin{array}{l}\Rightarrow 2x={540}^{\circ }-{350}^{\circ }\\ \Rightarrow 2x={190}^{\circ }\end{array}$$
Dividing both sides of the equation by 2, we get
$$\Rightarrow x={95}^{\circ }$$
Therefore, we get the value of $$x$$ as $${95}^{\circ }$$.
(d)
It is shown that all sides of the pentagon are equal.
Therefore, the given pentagon is a regular pentagon.
We know that all the sides and interior angles of a regular polygon are equal.
Therefore, we get the measure of the five interior angles as $$x$$.
The sum of all the interior angles of a pentagon is equal to $${540}^{\circ }$$.
Therefore, we get
$$x+x+x+x+x={540}^{\circ }$$
This is a linear equation in terms of $$x$$. We will solve this equation to find the value of $$x$$.
Adding the terms of the equation, we get
$$\Rightarrow 5x={540}^{\circ }$$
Dividing both sides of the equation by 5, we get
$$\Rightarrow x={108}^{\circ }$$
Therefore, we get the value of $$x$$ as $${108}^{\circ }$$.

Note: We have formed linear equations in one variable in terms of $$x$$ in the solution. A linear equation in one variable is an equation that can be written in the form $$ax+b=0$$, where $$a$$ is not equal to 0, and $$a$$ and $$b$$ are real numbers. For example, $$x-100=0$$ and $$100P-566=0$$ are linear equations in one variable $$x$$ and $$P$$ respectively. A linear equation in one variable has only one solution.
We used the term ‘regular polygon’ in the solution. A polygon is a closed figure made using straight lines as sides. A regular polygon is a polygon whose interior angles and sides are equal. For example: a square is a regular polygon having 4 sides, a pentagon is a regular polygon having 5 sides, a hexagon is a regular polygon having 6 sides, etc.