Question

8 b (7x + 1)° a 40° e c d (13x + 4)°

Explanation:

Step1: Use the property of vertical - angles

Vertical angles are equal. The angle at $E$ in $\triangle ABE$ and the angle at $E$ in $\triangle DCE$ are vertical angles. Also, in right - triangle $ABE$, the sum of the interior angles of a triangle is $180^{\circ}$. So, in $\triangle ABE$, the third angle (besides the right - angle and the $40^{\circ}$ angle) is $180^{\circ}-90^{\circ} - 40^{\circ}=50^{\circ}$.

Step2: Set up an equation based on angle relationships

We know that the angle $(7x + 1)^{\circ}$ and the $50^{\circ}$ angle are related. Since the sum of angles on a straight line is $180^{\circ}$, we can also use the fact that the vertical - angle relationship gives us an equation. The angle $(7x + 1)^{\circ}$ and the angle $(13x+4)^{\circ}$ are related as follows:
The angle adjacent to $(7x + 1)^{\circ}$ in $\triangle ABE$ (the non - right, non - $40^{\circ}$ angle) is $50^{\circ}$. And we know that $(7x + 1)+(13x + 4)=180 - 40$ (because of the linear pair and vertical - angle relationships).
Combining like terms in the equation $(7x + 1)+(13x + 4)=140$, we get $20x+5 = 140$.

Step3: Solve the equation for $x$

Subtract 5 from both sides of the equation $20x+5 = 140$:
$20x=140 - 5=135$.
Then divide both sides by 20: $x=\frac{135}{20}=\frac{27}{4}=6.75$.

Answer:

$x = 6.75$