Question

The opposite angles of a parallelogram are (3x-2) and (x+48). Find the measure of each angle of the parallelogram.

Answer 1

Hint: We know the property of a parallelogram that opposite angles of a parallelogram are equal to each other. And it is given that opposite angles of the parallelogram are (3x-2) and (x+48). We also know the property that the sum of adjacent angles of a parallelogram is equal to $${180}^0$$ . Put the value of x in (3x-2) and get its measure. Now, using this angle, get the measure of adjacent angles.

Complete step-by-step answer:

We have a parallelogram ABCD whose opposite angles are (3x-2) and (x+48).
We know the property of a parallelogram that the measure of opposite angles is equal to each other.
Here, $$\mathrm{\angle }DAB$$ and $$\mathrm{\angle }BCD$$ are opposite to each other. So, as per the above property, we can say that $$\mathrm{\angle }DAB$$ and $$\mathrm{\angle }BCD$$ are equal to each other.
$$\mathrm{\angle }DAB$$ = $$\mathrm{\angle }BCD$$ …………………(1)
According to the question, it is given that the opposite angles are (3x-2) and (x+48).
Let us assume,
$$\mathrm{\angle }DAB=(3x-2)$$ ………………(2)
$$\mathrm{\angle }BCD=(x+48)$$ …………….(3)
From equation (1), equation (2), and equation (3), we get
$$(3x-2)=(x+48)$$
$$\begin{array}{rl}& \Rightarrow 3x-x=48+2\\ & \Rightarrow 2x=50\\ & \Rightarrow x=25\end{array}$$
Now, putting the value of x in equation (2) and equation (3), we get
$$\mathrm{\angle }DAB={(3x-2)}^0={(3.25-2)}^0={73}^0$$ ……………….(4)
 $$\mathrm{\angle }BCD={(x+48)}^0={(25+48)}^0={73}^0$$ ………………..(5)
We also know the property of a parallelogram that the sum of adjacent angles of a parallelogram is equal to $${180}^0$$ .
Here, the angle $$\mathrm{\angle }ADC$$ is adjacent to $$\mathrm{\angle }DAB$$ . So, as per the above property, we can say that the sum of the $$\mathrm{\angle }ADC$$ and $$\mathrm{\angle }DAB$$ is equal to $${180}^0$$ .
$$\mathrm{\angle }ADC+\mathrm{\angle }DAB={180}^0$$ ……………..(5)
From equation (4) and equation (5), we get
$$\mathrm{\angle }ADC+\mathrm{\angle }DAB={180}^0$$
$$\begin{array}{rl}& \Rightarrow \mathrm{\angle }ADC+{73}^0={180}^0\\ & \Rightarrow \mathrm{\angle }ADC={180}^0-{73}^0\end{array}$$
$$\Rightarrow \mathrm{\angle }ADC={107}^0$$ …………………….(6)
We have the property that in a parallelogram the measure of opposite angles is equal to each other.
Here, $$\mathrm{\angle }ADC$$and $$\mathrm{\angle }ABC$$ are opposite to each other. So, as per the above property, we can say that $$\mathrm{\angle }ADC$$ and $$\mathrm{\angle }ABC$$ are equal to each other.
$$\mathrm{\angle }ADC=\mathrm{\angle }ABC$$ ……………………(7)
From equation (6) and equation (7), we have
$$\mathrm{\angle }ADC=\mathrm{\angle }ABC={107}^0$$ ………………..(8)
From equation (4), equation (5), and equation (8), we have
$$\mathrm{\angle }DAB={73}^0$$ , $$\mathrm{\angle }BCD={73}^0$$ , and $$\mathrm{\angle }ADC=\mathrm{\angle }ABC={107}^0$$ .
Hence, the angles are $${73}^0$$ , $${107}^0$$ , $${73}^0$$ , and $${107}^0$$ .

Note: In this question, one might do a mistake in the property. One can think that the sum of the opposite angles of a parallelogram is equal to $${180}^0$$ . Also, one can think that the measure of the adjacent angles is equal which is wrong. So, we have to keep the properties of the parallelogram in mind.