Question 9

We will use properties of parallel lines and angles on a straight line to find the value of $a$ in each diagram.

Step 1 — Finding $a$ in the first diagram

Let the top parallel line be $L_1$ and the middle parallel line be $L_2$. A transversal line $T$ cuts them. The given angle is 42°. It is above $L_1$, to the right of $T$. Let us find the angle corresponding to 42° on $L_2$. This angle is also above $L_2$, to the right of $T$. Since $L_1$ and $L_2$ are parallel, corresponding angles are equal. So this angle is 42°. Angle $a^\circ$ and this 42° angle form a linear pair on $L_2$. Angles in a linear pair add up to 180°.

$a^\circ + 42^\circ = 180^\circ$

$a^\circ = 180^\circ - 42^\circ$

$\boxed{a = 138^\circ}$

Step 2 — Finding $a$ in the second diagram

Let the top horizontal line be $H_1$ and the bottom horizontal line be $H_2$. Let the left vertical line be $V_1$ and the right vertical line be $V_2$. The lines $H_1$ and $H_2$ are parallel. The lines $V_1$ and $V_2$ are parallel. The given angle is 62°. It is below $H_1$, to the left of $V_2$. Let us find the angle adjacent to 62° on the straight line $H_1$. This angle is above $H_1$, to the left of $V_2$. Angles on a straight line add up to 180°. Let us call this angle $\angle X$.

$\angle X = 180^\circ - 62^\circ$

$= 118^\circ$

Now, consider $H_1$ and $H_2$ as parallel lines, and $V_2$ as a transversal. Angle $\angle X$ (above $H_1$, left of $V_2$) and the angle above $H_2$, left of $V_2$ are corresponding angles. Since $H_1$ and $H_2$ are parallel, corresponding angles are equal. So, the angle above $H_2$, left of $V_2$ is 118°. Let us call this angle $\angle Y$. Now, consider $V_1$ and $V_2$ as parallel lines, and $H_2$ as a transversal. Angle $\angle Y$ (above $H_2$, left of $V_2$) and angle $a^\circ$ (below $H_2$, right of $V_1$) are alternate interior angles. Since $V_1$ and $V_2$ are parallel, alternate interior angles are equal.

$a^\circ = \angle Y$

$a^\circ = 118^\circ$

$\boxed{a = 118^\circ}$

Step 3 — Finding $a$ in the third diagram

Let the four parallel horizontal lines be $L_1, L_2, L_3, L_4$ from top to bottom. Let the transversal be $T$. The given angle is 110°. It is above $L_1$, to the left of $T$. Let us find the angle vertically opposite to the 110° angle. This angle is below $L_1$, to the right of $T$. Vertically opposite angles are equal. So this angle is 110°. Let us call this $\angle X$. Now, consider $L_1$ and $L_2$ as parallel lines, and $T$ as a transversal. Angle $\angle X$ (below $L_1$, right of $T$) and the angle below $L_2$, right of $T$ are corresponding angles. Since $L_1$ and $L_2$ are parallel, corresponding angles are equal. So the angle below $L_2$, right of $T$ is 110°. Let us call this $\angle Y$. The diagram shows a 35° angle. This is the angle between $L_2$ and $L_3$, to the right of $T$. The angle below $L_3$, to the right of $T$ (let us call it $\angle Z$) can be found by subtracting the 35° angle from $\angle Y$.

$\angle Z = \angle Y - 35^\circ$

$\angle Z = 110^\circ - 35^\circ$

$= 75^\circ$

Now, consider $L_3$ and $L_4$ as parallel lines, and $T$ as a transversal. Angle $\angle Z$ (below $L_3$, right of $T$) and the angle below $L_4$, right of $T$ are corresponding angles. Since $L_3$ and $L_4$ are parallel, corresponding angles are equal. So the angle below $L_4$, right of $T$ is 75°. Let us call this $\angle W$. Angle $a^\circ$ and $\angle W$ form a linear pair on $L_4$. Angles in a linear pair add up to 180°.

$a^\circ + \angle W = 180^\circ$

$a^\circ + 75^\circ = 180^\circ$

$a^\circ = 180^\circ - 75^\circ$

$\boxed{a = 105^\circ}$

Step 4 — Finding $a$ in the fourth diagram

Let the two slanted parallel lines be $L_1$ (left) and $L_2$ (right). Let the horizontal line be $H$ and the vertical line be $V$. The line $V$ is perpendicular to $H$, so the angle between them is 90°. The angles on the straight line $H$ add up to 180°. We have the angle 67° (between $L_1$ and $H$, to the left of $L_1$). We have the angle 90° (between $V$ and $H$, to the right of $V$). Let us find the angle between $L_1$ and $V$. Let us call this angle $\angle X$.

$67^\circ + \angle X + 90^\circ = 180^\circ$

$\angle X + 157^\circ = 180^\circ$

$\angle X = 180^\circ - 157^\circ$

$= 23^\circ$

So, the angle between $L_1$ and $V$ is 23°. This angle is to the right of $L_1$ and to the left of $V$. Now, consider $L_1$ and $L_2$ as parallel lines, and $V$ as a transversal. Angle $\angle X$ (between $L_1$ and $V$) and angle $a^\circ$ (between $L_2$ and $V$) are alternate interior angles. Since $L_1$ and $L_2$ are parallel, alternate interior angles are equal.

$a^\circ = \angle X$

$a^\circ = 23^\circ$

$\boxed{a = 23^\circ}$

Answer

(i) $a = 138^\circ$ (ii) $a = 118^\circ$ (iii) $a = 105^\circ$ (iv) $a = 23^\circ$