Question

In the given figure (not drawn to scale), AD is parallel to BC. JDK, GHCI, EABF are straight and parallel lines. Find: 1) $\mathrm{\angle }GHD-\mathrm{\angle }HDC$ 2) $\mathrm{\angle }BCI+\mathrm{\angle }HAB$

A. 58°, 150°
B. 48°, 150°
C. 58°, 180°
D. 48°, 180°

Answer 1

Hint: Here, from the given figure, we know that AD||BC and JK||GI||EF. $\mathrm{\angle }CBF$ = ${60}^{\circ }$, $$\mathrm{\angle }HAB$$ is corresponding angle of $\mathrm{\angle }CBF$, hence equal to 60°. Similarly, equate all unknown angles with $\mathrm{\angle }CBF$ as an alternate angle, corresponding angles, etc. We will also use the concept of the straight line that a straight line always forms ${180}^{\circ }$. By using these concepts, we will solve this question.

Complete step by step answer:
Given, line GI is parallel to the DH is transversal.
JK || GI and DH are transversal.
$\Rightarrow \mathrm{\angle }GHD=\mathrm{\angle }HDK$(Alternate angles)
$\Rightarrow \mathrm{\angle }GHD=\mathrm{\angle }HDC+{48}^{\circ }$($\mathrm{\angle }HDK=\mathrm{\angle }HDC+{48}^{\circ }$)
$\Rightarrow \mathrm{\angle }GHD-\mathrm{\angle }HDC={48}^{\circ }$
Thus, $\mathrm{\angle }GHD-\mathrm{\angle }HDC={48}^{\circ }$
Also given,
GI || EF and BC are transversal.
$\Rightarrow \mathrm{\angle }BCI+\mathrm{\angle }CBF={180}^{\circ }$ (Co-interior angles)
$\Rightarrow \mathrm{\angle }BCI+{66}^{\circ }={180}^{\circ }$
$\Rightarrow \mathrm{\angle }BCI={180}^{\circ }-{66}^{\circ }={114}^{\circ }$
As HA || CB and AB are transversal
$\Rightarrow \mathrm{\angle }CBF=\mathrm{\angle }HAB$(Corresponding angles)
$\Rightarrow \mathrm{\angle }HAB={66}^{\circ }$
Now,
$Rightarrow\mathrm{\angle }BCI+\mathrm{\angle }HAB={114}^{\circ }+{66}^{\circ }={180}^{\circ }$
Thus, $\mathrm{\angle }BCI+\mathrm{\angle }HAB={180}^{\circ }$

$\therefore$ (1) $\mathrm{\angle }GHD-\mathrm{\angle }HDC={48}^{\circ }$ and (2) $\mathrm{\angle }BCI+\mathrm{\angle }HAB={180}^{\circ }$. Hence, option (D) is correct.

Note:
In these types of questions, from the figure, find the values of all corresponding angles, alternate angles with the help of given angle, and find whatever asked in the question.
For these types of questions, we need to keep some concepts in mind.
- Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Both the lines are said to be parallel.
- When parallel lines get crossed by another line is called a transversal.
- When two lines are parallel and a transversal pass through both the parallel lines then the alternate interior angle is always equal.
- When two lines are parallel and a transversal pass through both the parallel lines then the alternate exterior angle is always equal.
- When two lines are parallel and a transversal pass through both the parallel lines then the Corresponding angles are always equal.
- When two lines are parallel and a transversal pass through both the parallel lines then the sum of Co-interior angles is ${180}^{\circ }$.