Question 10

We will use properties of parallel lines, transversals, and triangles to find the unknown angles.

Step 1 — Finding angle $x$ in the first figure

The vertical line meets the top parallel line at a right angle. This means the angle is 90 degrees. The two horizontal lines are parallel. So, the vertical line also meets the bottom parallel line at a right angle. This angle is 90 degrees.

Now, let us look at the triangle at the bottom. Its sides are the vertical line, the slanted line, and the bottom parallel line. The sum of angles inside a triangle is always 180 degrees. The angles in this triangle are $x^\circ$, $65^\circ$, and $90^\circ$.

$x^\circ + 65^\circ + 90^\circ = 180^\circ$

$x^\circ + 155^\circ = 180^\circ$

$x^\circ = 180^\circ - 155^\circ$

$\boxed{x = 25^\circ}$

Step 2 — Finding angle $b$ in the first figure

The angle $b^\circ$ is on the top parallel line. It is formed by the slanted line. Let us find the angle corresponding to $b^\circ$. This corresponding angle is on the bottom parallel line. It is formed by the slanted line.

This angle is the sum of two angles. The first angle is between the vertical line and the bottom parallel line, which is 90 degrees. The second angle is between the vertical line and the slanted line, which is 65 degrees. So, the corresponding angle is their sum.

$90^\circ + 65^\circ = 155^\circ$

Since the horizontal lines are parallel, corresponding angles are equal. The angle $b^\circ$ and the angle $155^\circ$ are corresponding angles.

$\boxed{b = 155^\circ}$

Step 3 — Finding angle $x$ in the second figure

The top and bottom lines are parallel. The left slanted line is a transversal. The $53^\circ$ angle is on the bottom line. Its alternate interior angle is on the top line. It is formed by the left slanted line. Let us call this angle $\alpha_1$. Alternate interior angles are equal.

$\alpha_1 = 53^\circ$

The right slanted line is also a transversal. The $78^\circ$ angle is on the bottom line. Its alternate interior angle is on the top line. It is formed by the right slanted line. Let us call this angle $\alpha_2$. Alternate interior angles are equal.

$\alpha_2 = 78^\circ$

Now, consider the angles below the top line. The angle between the two slanted lines is the difference between $\alpha_2$ and $\alpha_1$. Let us call this angle $\alpha_x$.

$\alpha_x = \alpha_2 - \alpha_1$

$\alpha_x = 78^\circ - 53^\circ$

$\alpha_x = 25^\circ$

The angle $x^\circ$ in the diagram is vertically opposite to $\alpha_x$. Vertically opposite angles are equal.

$\boxed{x = 25^\circ}$

Answer

(i) For the first figure, $x = \mathbf{25^\circ}$. (ii) For the first figure, $y = \mathbf{155^\circ}$ (this is angle $b$ in the diagram). (iii) For the second figure, $x = \mathbf{25^\circ}$.