Question

question 4: find the angle x in each question below. give reasons for your answer. (a) (b) (c) (d) (e) (f)

Explanation:

Step1: Identify corresponding - angles

In (a), since $CF\parallel GJ$, $\angle KDE=\angle HID = 59^{\circ}$ (corresponding - angles). Then, considering the triangle $DEI$, we know that the sum of interior angles of a triangle is $180^{\circ}$. Also, $\angle CDA=\angle KDE = 59^{\circ}$ (vertically - opposite angles), and $\angle CDA$ and the $66^{\circ}$ angle are corresponding angles.
We use the property of angles in a triangle. Let's find $\angle x$.
In triangle $DEI$, we know that $\angle x=180^{\circ}-(66^{\circ}+59^{\circ})$.

Step2: Calculate the value of $\angle x$

$\angle x = 180^{\circ}-(66^{\circ}+59^{\circ})=180^{\circ}-125^{\circ}=55^{\circ}$.

In (b), first, find the angle adjacent to $72^{\circ}$. Let's call it $\angle 1$. $\angle 1 = 180^{\circ}-72^{\circ}=108^{\circ}$.
Since the lines are parallel, we use the property of angles formed by parallel lines and a transversal.
We know that $\angle x$ and the angle adjacent to $55^{\circ}$ are corresponding angles. The angle adjacent to $55^{\circ}$ is $180 - 55=125^{\circ}$.
We consider the relationship between angles formed by parallel lines and transversals.
Let's find $\angle x$ using the fact that the sum of angles around a point where the lines intersect is $360^{\circ}$. But a simpler way is to use the property of corresponding and supplementary angles.
Since the lines are parallel, we know that $\angle x=108^{\circ}-55^{\circ}=53^{\circ}$.

In (d), since $AB\parallel CG$ and $DE\parallel FG$, we first find the angle at $E$ in the triangle. The angle adjacent to $134^{\circ}$ is $180 - 134 = 46^{\circ}$.
Since the triangle is isosceles (the two - line segments are equal as indicated by the marks), the base angles are equal.
The sum of interior angles of a triangle is $180^{\circ}$. Let the base angles be $\alpha$.
$2\alpha+46^{\circ}=180^{\circ}$, so $\alpha=\frac{180 - 46}{2}=67^{\circ}$.
$\angle x = 180^{\circ}-67^{\circ}=113^{\circ}$.

In (e), the sum of interior angles of a triangle is $180^{\circ}$.
We know that $\angle x=180^{\circ}-(48^{\circ}+77^{\circ})$.
$\angle x=180^{\circ}-125^{\circ}=55^{\circ}$.

Answer:

(a) $55^{\circ}$; (b) $53^{\circ}$; (d) $113^{\circ}$; (e) $55^{\circ}$