Geometry worksheet with exercises to find unknown angles using properties of parallel lines and transversals.

Here are the solutions for the angle problems.

2) Find angle x for each option

(a)
* The angles are on the same side of the transversal and between the parallel lines (consecutive interior angles). They add up to $180^\circ$.
* $x + 125^\circ = 180^\circ$
* $x = 180^\circ - 125^\circ$
* $x = 55^\circ$

(b)
* Angle $57^\circ$ and angle $x$ are alternate interior angles. They are equal.
* $x = 57^\circ$

(c)
* Angle $70^\circ$ and angle $x$ are consecutive interior angles. They add up to $180^\circ$.
* $x + 70^\circ = 180^\circ$
* $x = 180^\circ - 70^\circ$
* $x = 110^\circ$

(d)
* Angle $105^\circ$ and angle $x$ are alternate interior angles. They are equal.
* $x = 105^\circ$

(e)
* Angle $53^\circ$ and angle $x$ are alternate exterior angles. They are equal.
* $x = 53^\circ$

(f)
* First, find the interior angle next to $133^\circ$. They form a straight line ($180^\circ$).
* Interior angle $= 180^\circ - 133^\circ = 47^\circ$.
* Now, look at angle $x$ and this new $47^\circ$ angle. They are consecutive interior angles (same side inside), so they add to $180^\circ$.
* $x + 47^\circ = 180^\circ$
* $x = 180^\circ - 47^\circ$
* $x = 133^\circ$ *(Note: In this specific geometry, alternate exterior angles are also equal, so $x$ equals the given $133^\circ$ directly).*

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3) Find the asked angles

(a)
* Angle $112^\circ$ and angle $x$ are alternate exterior angles. They are equal.
* $x = 112^\circ$

(b)
* Angle $75^\circ$ and angle $x$ are consecutive interior angles. They add up to $180^\circ$.
* $x + 75^\circ = 180^\circ$
* $x = 180^\circ - 75^\circ$
* $x = 105^\circ$

(c)
* Find x: Angle $150^\circ$ and angle $x$ are on a straight line.
* $x + 150^\circ = 180^\circ$
* $x = 30^\circ$
* Find y: Angle $y$ and angle $x$ are consecutive interior angles (add to $180^\circ$).
* $y + 30^\circ = 180^\circ$
* $y = 150^\circ$

(d)
* Find x: Angle $99^\circ$ and angle $x$ are on a straight line.
* $x + 99^\circ = 180^\circ$
* $x = 81^\circ$
* Find y: Since the vertical lines are parallel, angle $x$ and angle $y$ are alternate interior angles.
* $y = 81^\circ$
* Find z: Angle $y$ and angle $z$ are on a straight line.
* $z + 81^\circ = 180^\circ$
* $z = 99^\circ$

(e)
* Find x: Angle $74^\circ$ and angle $x$ are alternate interior angles.
* $x = 74^\circ$
* Find y: Angle $x$ and angle $y$ are consecutive interior angles (add to $180^\circ$).
* $y + 74^\circ = 180^\circ$
* $y = 106^\circ$

(f)
* Find x: Look at the top-left angle $123^\circ$. The angle vertically opposite to it (inside the parallel lines) is also $123^\circ$. Angle $x$ and that $123^\circ$ angle are consecutive interior angles.
* $x + 123^\circ = 180^\circ$
* $x = 57^\circ$
* Find y: Look at the bottom-right angle $110^\circ$. The angle vertically opposite to it (inside the parallel lines) is also $110^\circ$. Angle $y$ and that $110^\circ$ angle are consecutive interior angles.
* $y + 110^\circ = 180^\circ$
* $y = 70^\circ$

Final Answer:
2)
(a) $x = 55^\circ$
(b) $x = 57^\circ$
(c) $x = 110^\circ$
(d) $x = 105^\circ$
(e) $x = 53^\circ$
(f) $x = 133^\circ$

3)
(a) $x = 112^\circ$
(b) $x = 105^\circ$
(c) $x = 30^\circ$, $y = 150^\circ$
(d) $x = 81^\circ$, $y = 81^\circ$, $z = 99^\circ$
(e) $x = 74^\circ$, $y = 106^\circ$
(f) $x = 57^\circ$, $y = 70^\circ$