Question 8

We will use properties of angles formed by parallel lines and transversals.

Step 1 — Find angle a

Angle $a$ and the $48^\circ$ angle are alternate interior angles. Alternate interior angles are always equal.

$a = \mathbf{48^\circ}$

$\boxed{a = 48^\circ}$

Step 2 — Find angle b

Angle $b$ and the $52^\circ$ angle are alternate interior angles. Alternate interior angles are always equal.

$b = \mathbf{52^\circ}$

$\boxed{b = 52^\circ}$

Step 3 — Find angle c

The $99^\circ$ angle and the angle next to it form a straight line. Angles on a straight line add up to $180^\circ$. Let us call the angle next to $99^\circ$ as $x$.

$x = 180^\circ - 99^\circ$

$= \mathbf{81^\circ}$

Angle $x$ and angle $c$ are corresponding angles. Corresponding angles are always equal.

$c = \mathbf{81^\circ}$

$\boxed{c = 81^\circ}$

Step 4 — Find angle d

Angle $d$ and the $81^\circ$ angle are consecutive interior angles. Consecutive interior angles add up to $180^\circ$.

$d + 81^\circ = 180^\circ$

$d = 180^\circ - 81^\circ$

$d = \mathbf{99^\circ}$

$\boxed{d = 99^\circ}$

Step 5 — Find angle e

Angle $e$ and the $69^\circ$ angle are alternate interior angles. Alternate interior angles are always equal.

$e = \mathbf{69^\circ}$

$\boxed{e = 69^\circ}$

Step 6 — Find angle f

Angle $f$ and the $132^\circ$ angle are consecutive interior angles. Consecutive interior angles add up to $180^\circ$.

$f + 132^\circ = 180^\circ$

$f = 180^\circ - 132^\circ$

$f = \mathbf{48^\circ}$

$\boxed{f = 48^\circ}$

Step 7 — Find angle g

Angle $g$ and the $122^\circ$ angle (top left) are vertically opposite angles. Vertically opposite angles are always equal.

$g = \mathbf{122^\circ}$

$\boxed{g = 122^\circ}$

Step 8 — Find angle h

Let us find the interior angle next to $120^\circ$. Angles on a straight line add up to $180^\circ$.

$180^\circ - 120^\circ = \mathbf{60^\circ}$

Let us find the interior angle next to $75^\circ$. Angles on a straight line add up to $180^\circ$.

$180^\circ - 75^\circ = \mathbf{105^\circ}$

These three angles form a triangle. The sum of angles in a triangle is $180^\circ$.

$h + 60^\circ + 105^\circ = 180^\circ$

$h + 165^\circ = 180^\circ$

$h = 180^\circ - 165^\circ$

$h = \mathbf{15^\circ}$

$\boxed{h = 15^\circ}$

Step 9 — Find angle i

Let us consider the triangle formed by the two transversals and the bottom parallel line. The angles of this triangle are $i$, $70^\circ$, and $56^\circ$. The sum of angles in a triangle is $180^\circ$.

$i + 70^\circ + 56^\circ = 180^\circ$

$i + 126^\circ = 180^\circ$

$i = 180^\circ - 126^\circ$

$i = \mathbf{54^\circ}$

$\boxed{i = 54^\circ}$

Step 10 — Find angle j

Angle $j$ and the $97^\circ$ angle are alternate interior angles. Alternate interior angles are always equal.

$j = \mathbf{97^\circ}$

$\boxed{j = 97^\circ}$

Answer

(a) $\mathbf{48^\circ}$ (b) $\mathbf{52^\circ}$ (c) $\mathbf{81^\circ}$ (d) $\mathbf{99^\circ}$ (e) $\mathbf{69^\circ}$ (f) $\mathbf{48^\circ}$ (g) $\mathbf{122^\circ}$ (h) $\mathbf{15^\circ}$ (i) $\mathbf{54^\circ}$ (j) $\mathbf{97^\circ}$