Question

geometry unit 2 review

1. lines l and m are parallel.

a. name three pairs of vertical angles. explain how you know these angles are vertical angles.
b. name three pairs of corresponding angles. explain how you know these are corresponding angles.
c. name three pairs of alternate - interior angles. explain how you know these angles are alternate - interior angles.
d. name three pairs of alternate exterior angles. explain how you know these angles are alternate exterior angles.
e. name five pairs of angles that form a linear pair. how do you know these angles form a linear pair?

1. for each problem below, find all missing angles. show your work.

Explanation:

1a. Vertical Angles

Vertical angles are formed when two lines intersect. They are opposite each other. For the given parallel - line intersection:

• $\angle1$ and $\angle3$: They are opposite each other at the intersection of one of the transversals with a line.
• $\angle2$ and $\angle4$: Opposite at the same intersection.
• $\angle5$ and $\angle7$: Opposite at the intersection of the other transversal with a line.

1b. Corresponding Angles

Corresponding angles are in the same relative position with respect to the parallel lines and the transversals.

• $\angle1$ and $\angle5$: Both are in the upper - left position at the intersections of the transversals with the parallel lines.
• $\angle2$ and $\angle6$: Upper - right position.
• $\angle4$ and $\angle8$: Lower - right position.

1c. Alternate Interior Angles

Alternate interior angles are between the parallel lines and on opposite sides of the transversals.

• $\angle3$ and $\angle5$: Between the parallel lines and on opposite sides of the transversal.
• $\angle4$ and $\angle6$: Also between the parallel lines and on opposite sides of the transversal.

1d. Alternate Exterior Angles

Alternate exterior angles are outside the parallel lines and on opposite sides of the transversals.

• $\angle1$ and $\angle7$: Outside the parallel lines and on opposite sides of the transversal.
• $\angle2$ and $\angle8$: Outside the parallel lines and on opposite sides of the transversal.

1e. Linear - Pair Angles

Linear - pair angles are adjacent and their non - common sides form a straight line.

• $\angle1$ and $\angle2$: Adjacent and their non - common sides form a straight line.
• $\angle2$ and $\angle3$: Adjacent with non - common sides in a straight line.
• $\angle3$ and $\angle4$: Adjacent and non - common sides form a straight line.
• $\angle5$ and $\angle6$: Adjacent with non - common sides in a straight line.
• $\angle6$ and $\angle7$: Adjacent and non - common sides form a straight line.

2. First set of parallel - line angles

Let's assume the two lines are parallel and a transversal intersects them. If one angle is $70^{\circ}$, then:

• The angle marked $x$ that is corresponding to the $70^{\circ}$ angle has $x = 70^{\circ}$ (corresponding angles are equal for parallel lines and a transversal).

2. Second set of parallel - line angles

• Let's use angle - relationships. If we consider the parallel lines and transversals:
• For the $44^{\circ}$ angle, its corresponding angle (say $y$) has $y = 44^{\circ}$.
• For the $70^{\circ}$ angle, its corresponding angle (say $z$) has $z = 70^{\circ}$.
• For the $64^{\circ}$ angle, its corresponding angle (say $w$) has $w = 64^{\circ}$.
• If we consider linear - pair and angle - sum properties, we can find other angles. For example, if an angle is $a$ and it forms a linear - pair with a known angle, we use the fact that the sum of angles in a linear - pair is $180^{\circ}$.

Answer:

1a. $\angle1$ and $\angle3$, $\angle2$ and $\angle4$, $\angle5$ and $\angle7$; formed by two - line intersection.
1b. $\angle1$ and $\angle5$, $\angle2$ and $\angle6$, $\angle4$ and $\angle8$; same relative position.
1c. $\angle3$ and $\angle5$, $\angle4$ and $\angle6$; between parallel lines, opposite transversal sides.
1d. $\angle1$ and $\angle7$, $\angle2$ and $\angle8$; outside parallel lines, opposite transversal sides.
1e. $\angle1$ and $\angle2$, $\angle2$ and $\angle3$, $\angle3$ and $\angle4$, $\angle5$ and $\angle6$, $\angle6$ and $\angle7$; adjacent, non - common sides form straight line.

1. First set: $x = 70^{\circ}$. Second set: use corresponding, linear - pair and angle - sum properties to find all angles as described above.