Question

1. find the value of y

(2x + 40°)
(3x - 10)
(4y + 10)
a. 30
b. 100
c. 22.5
d. 80

Explanation:

Step1: Set equal the vertical - angle expressions

Vertical angles are equal. So, \(2x + 40=3x - 10\).
Solve for \(x\):
\[

$$\begin{align*} 2x+40&=3x - 10\\ 40 + 10&=3x-2x\\ x&=50 \end{align*}$$

\]

Step2: Find the measure of one of the angles

Substitute \(x = 50\) into \(2x + 40\): \(2\times50+40=100 + 40=140\).
The other pair of vertical angles are also equal. So, \(4y+10\) is equal to the angle we just found or its vertical - angle equivalent.
Set \(4y + 10=140\).

Step3: Solve for \(y\)

\[

$$\begin{align*} 4y+10&=140\\ 4y&=140 - 10\\ 4y&=130\\ y&=\frac{130}{4}=32.5 \end{align*}$$

\]
It seems there is a mistake in the problem - setup or answer choices as the correct value of \(y\) is \(32.5\) which is not in the given choices. If we assume the problem is about finding the value of the angle \(4y + 10\) instead of \(y\) itself:
\[

$$\begin{align*} 4y+10&=140 \end{align*}$$

\]

Answer:

If we assume the problem is mis - worded and we want the value of the angle \(4y + 10\) (which is equal to the vertical angle \(2x + 40\) or \(3x-10\) when \(x = 50\)), the value is \(140\). But if we strictly go by finding \(y\) from the equation \(4y+10\) based on vertical - angle equality, and since the choices are wrong for \(y\) value, there is no correct answer among A, B, C, D. If we assume we want the value of the angle expression \(4y + 10\) (equal to vertical angles), we can re - solve as follows:

Step1: Set vertical angles equal

\(2x + 40=3x - 10\), \(x = 50\).

Step2: Use the other vertical - angle pair

The angle \(4y+10\) is equal to \(2x + 40\) (vertical angles). Substitute \(x = 50\) into \(2x+40\) gives \(140\). So \(4y+10 = 140\).

Step3: Solve for the angle value

We already know the angle \(4y + 10\) (vertical to \(2x + 40\) or \(3x - 10\)) is \(140\). But if we want to solve for \(y\) from \(4y+10=140\), \(4y=130\), \(y = 32.5\). Since \(y = 32.5\) is not in the choices, if we consider the value of the angle \(4y + 10\):