Question

$y=38^\circ$

Answer 1

The statement $y=38^\circ$ is correct.

Answer 2

The problem presents a geometric figure with two parallel lines and two transversals. We are given three angles: $122^\circ$, $2y$, and $134^\circ$. The question asks to verify if $y=38^\circ$ is consistent with the given angles.

Let's label the points in the diagram to clarify the setup.
Let the top parallel line be $L_1$ and the bottom parallel line be $L_2$.
Let the left point where the transversal meets $L_1$ be $A$.
Let the right point where the transversal meets $L_1$ be $C$.
Let the common vertex where the angle $2y$ is located be $B$.
So, the figure forms a triangle $ABC$, where the base $AC$ lies on the line $L_1$, and the vertex $B$ lies on the line $L_2$.

Now, let's determine the interior angles of triangle $ABC$:

1. Angle at A: The angle given at point $A$ is $122^\circ$. This is an exterior angle formed by the line $L_1$ and the segment $AB$. The interior angle $\angle BAC$ of the triangle is supplementary to this exterior angle.
   $\angle BAC = 180^\circ - 122^\circ = 58^\circ$.

2. Angle at C: The angle given at point $C$ is $134^\circ$. This is an exterior angle formed by the line $L_1$ and the segment $BC$. The interior angle $\angle BCA$ of the triangle is supplementary to this exterior angle.
   $\angle BCA = 180^\circ - 134^\circ = 46^\circ$.

3. Angle at B: The angle given at point $B$ is $2y$. This is the interior angle $\angle ABC$ of the triangle.

Now, we use the property that the sum of interior angles in a triangle is $180^\circ$:
$\angle BAC + \angle BCA + \angle ABC = 180^\circ$
Substitute the values we found:
$58^\circ + 46^\circ + 2y = 180^\circ$
$104^\circ + 2y = 180^\circ$

Now, solve for $y$:
$2y = 180^\circ - 104^\circ$
$2y = 76^\circ$
$y = \frac{76^\circ}{2}$
$y = 38^\circ$

The calculated value of $y$ is $38^\circ$, which matches the value given in the question ($y=38^\circ$). Therefore, the statement is consistent with the diagram.

The final answer is $\boxed{y=38^\circ}$.

Explanation:

1. Identify the geometric figure as a triangle with its base on one parallel line and the opposite vertex on the other parallel line.
2. Calculate the interior angles of the triangle at the base using the supplementary angle property ($180^\circ - \text{exterior angle}$).
3. Apply the triangle angle sum theorem (sum of interior angles is $180^\circ$).
4. Solve the resulting equation for $y$.
5. Compare the calculated value of $y$ with the given value.

Answer: The statement $y=38^\circ$ is correct.