The problem asks to find the values of the angles marked with letters in two diagrams.

1. Problem Description

The problem asks to find the values of the angles marked with letters in two diagrams.

2. Solution Steps

Problem 1:

We are given that one angle is $135^{\circ}$ and another is $70^{\circ}$. We need to find $u, v, w, x, y, z$.

Since $u$ and $135^{\circ}$ form a linear pair, we have:

$u + 135^{\circ} = 180^{\circ}$

$u = 180^{\circ} - 135^{\circ} = 45^{\circ}$

Since $v$ and $70^{\circ}$ form a linear pair, we have:

$v + 70^{\circ} = 180^{\circ}$

$v = 180^{\circ} - 70^{\circ} = 110^{\circ}$

Angles $w$ and $u$ are vertically opposite angles, so $w = u = 45^{\circ}$.

Angles $x$ and $v$ are vertically opposite angles, so $x = v = 110^{\circ}$.

Angles $y$ and $70^{\circ}$ are vertically opposite angles, so $y = 70^{\circ}$.

Angles $z$ and $135^{\circ}$ are vertically opposite angles, so $z = 135^{\circ}$.

Problem 2:

We are given an angle of $57^{\circ}$ and an angle of $70^{\circ}$. We need to find $a, b, c, d, e, f$.

We are given that the lines are parallel.

Since the angle $70^{\circ}$ and $d$ form a linear pair, we have:

$d + 70^{\circ} = 180^{\circ}$

$d = 180^{\circ} - 70^{\circ} = 110^{\circ}$

Since the lines are parallel, alternate interior angles are equal. So $a = 70^{\circ}$.

Since the lines are parallel, corresponding angles are equal. So $b = 57^{\circ}$.

Since the lines are parallel, $e$ and $70^{\circ}$ are supplementary angles.

$e + 70^{\circ} = 180^{\circ}$. However, this $70^\circ$ is not angle $a$. Since line 2 and 3 are parallel, the angle vertically opposite to $57^{\circ}$ (call it $x$) and the angle $e$ are supplementary. Therefore,

$57^\circ + e = 180^{\circ}$.

$e = 180^{\circ} - 57^{\circ} = 123^{\circ}$.

$c$ and $e$ are vertically opposite, so $c = e = 123^{\circ}$.

$f$ and $57^{\circ}$ are vertically opposite, so $f = 57^{\circ}$.

3. Final Answer

Problem 1:

$u = 45^{\circ}$

$v = 110^{\circ}$

$w = 45^{\circ}$

$x = 110^{\circ}$

$y = 70^{\circ}$

$z = 135^{\circ}$

Problem 2:

$a = 70^{\circ}$

$b = 57^{\circ}$

$c = 123^{\circ}$

$d = 110^{\circ}$

$e = 123^{\circ}$

$f = 57^{\circ}$