Question

question 4: find the angle x in each question below. give reasons for your answer.

Explanation:

Step1: Identify corresponding angles

In (a), since $CF\parallel GJ$, the angle corresponding to the $66^{\circ}$ angle and the angle adjacent to $x$ are equal. The angle adjacent to $x$ is $66^{\circ}$. Using the fact that angles on a straight - line sum to $180^{\circ}$, we have $x = 180^{\circ}-66^{\circ}-59^{\circ}=55^{\circ}$.

Step2: Use angle - sum properties in (b)

The angle adjacent to the $72^{\circ}$ angle on line $AC$ is $180 - 72=108^{\circ}$. Since the two vertical lines are parallel, we can use the angle - sum property of a triangle formed by the transversals. The sum of angles in a triangle formed by the transversals is $180^{\circ}$. So $x=180^{\circ}-108^{\circ}-55^{\circ}=17^{\circ}$.

Step3: Apply exterior - angle property in (c)

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So $x = 41^{\circ}+60^{\circ}=101^{\circ}$.

Step4: Use isosceles - triangle and angle - sum properties in (d)

The base angles of the isosceles triangle are equal. The angle adjacent to the $134^{\circ}$ angle is $180 - 134 = 46^{\circ}$. Since the triangle is isosceles, the other base angle is also $46^{\circ}$. Then $x=180^{\circ}-46^{\circ}=134^{\circ}$.

Step5: Use angle - sum of a triangle in (e)

The sum of angles in a triangle is $180^{\circ}$. First, find the angle adjacent to the $48^{\circ}$ angle which is $180 - 48=132^{\circ}$. Then $x=180^{\circ}-132^{\circ}-77^{\circ}= - 29^{\circ}$, but this is wrong. We should use the fact that the sum of angles in $\triangle ACD$ gives us the correct way. The angle at $C$ in $\triangle ACD$ is $48^{\circ}$, and the angle at $D$ (adjacent to the $77^{\circ}$ angle) is $180 - 77 = 103^{\circ}$. So $x=180^{\circ}-48^{\circ}-77^{\circ}=55^{\circ}$.

Step6: Use isosceles - triangle and exterior - angle property in (f)

The base angles of the isosceles triangle are equal. The angle adjacent to the $76^{\circ}$ angle is $180 - 76=104^{\circ}$. Since the triangle is isosceles, the base angles are equal. The exterior angle $x$ of the isosceles triangle is $x = 2\times(180 - 104)\div2=76^{\circ}$.

Answer:

(a) $55^{\circ}$
(b) $17^{\circ}$
(c) $101^{\circ}$
(d) $134^{\circ}$
(e) $55^{\circ}$
(f) $76^{\circ}$