Let's solve each problem one by one from the image provided:

Problem 31

Given: Lines $l \parallel m$. Find $x$.

In the triangle $\triangle ABC$:

• The sum of angles in a triangle is 180°.
• Given angles:
  • $\angle B = 50^\circ$
  • $\angle C = 115^\circ$

Using the angle sum property: $x + 50^\circ + 115^\circ = 180^\circ$ $x + 165^\circ = 180^\circ$ $x = 15^\circ$

Answer: $x = 15^\circ$

Problem 32

Given: $l \parallel m$ and $n$ is a transversal. Find $x$.

Since $l \parallel m$, corresponding angles are equal.

1. $3y = x + 15^\circ$
2. $2y + 25^\circ = 3y$

Solving the second equation: $2y + 25^\circ = 3y$ $25^\circ = y$

Substitute $y = 25^\circ$ into the first equation: $3(25^\circ) = x + 15^\circ$ $75^\circ = x + 15^\circ$ $x = 60^\circ$

Answer: $x = 60^\circ$, $y = 25^\circ$

Problem 33

For what value of $x$, $l \parallel m$.

Given angles in the figure:

• $50^\circ$ (corresponding angle to $x - 20^\circ$)
• $120^\circ$ (alternate angle)

Since $l \parallel m$, the sum of the interior angles on the same side of the transversal should be 180°: $50^\circ + (x - 20^\circ) = 120^\circ$ $x = 90^\circ$

Problem