Exercises: Prove Theorems About Lines and Angles

Two lines intersect forming four angles. One angle measures $70^\circ$. Which angle measure could belong to an adjacent angle at the same intersection?

When a transversal crosses two parallel lines, eight angles are formed. Which term describes a pair of angles that are on opposite sides of the transversal and between the two parallel lines?

Segment $\overline{AB}$ has midpoint $M$. A line through $M$ is perpendicular to $\overline{AB}$. Which term names this line?

Two lines intersect at point $O$, forming angles $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$ arranged around $O$ (numbered clockwise). $\angle 1$ and $\angle 3$ are vertical angles; $\angle 2$ and $\angle 4$ are vertical angles.

Write the key steps of a proof that $\angle 1 \cong \angle 3$. Name the justification for each step.

Two lines intersect. One angle at the intersection measures $47^\circ$. What is the measure of the vertical angle?

Line $l$ is parallel to line $m$. A transversal crosses both lines, creating eight angles labeled $\angle 1$ through $\angle 8$ (angles $\angle 1$–$\angle 4$ at the intersection with $l$; angles $\angle 5$–$\angle 8$ at the intersection with $m$, in matching positions). Which pair is a pair of alternate interior angles?

Line $l \parallel$ line $m$, cut by transversal $t$. $\angle 1 = 115^\circ$ (at line $l$, upper-left position). What is the measure of the corresponding angle at line $m$?

Line $l \parallel$ line $m$, cut by transversal $t$. $\angle 3 = 58^\circ$ (alternate interior position at line $l$). What is the measure of its alternate interior angle at line $m$?

In a two-column proof that vertical angles are congruent, a student writes as a reason: "Vertical angles are congruent." What is wrong with this reason?

Line $l \parallel$ line $m$, cut by transversal $t$. $\angle 1 = 65^\circ$ (upper-left at line $l$). Using the parallel line theorems, what are the measures of all four interior angles (the angles between lines $l$ and $m$)?

Two lines are cut by a transversal. The alternate interior angles measure $72^\circ$ and $81^\circ$ respectively. Can you conclude that the two lines are parallel? Explain your reasoning, citing the relevant theorem or its converse.

Point $P$ lies on the perpendicular bisector of segment $\overline{AB}$. $M$ is the midpoint of $\overline{AB}$ and $PM \perp AB$. In the proof that $PA = PB$, which congruence criterion is used to show $\triangle PMA \cong \triangle PMB$?

Segment $\overline{AB}$ has endpoints $A(0, 0)$ and $B(6, 0)$. The perpendicular bisector of $\overline{AB}$ is the vertical line $x = 3$. Point $P$ is at $(3, 50)$. Without the Perpendicular Bisector Theorem, a student argues that $PA \neq PB$ because $P$ "looks too far away." Which statement correctly addresses the student's doubt?

Explain, citing the relevant theorem or its converse, whether the engineer can conclude that Beam $l$ is parallel to Beam $m$.

Using the Perpendicular Bisector Theorem, explain why $PA = PB$. Write your explanation as a step-by-step argument referencing the theorem's proof.

What is the fundamental error in the student's proof?

What error did the student make, and what is the correct relationship between $\angle 3$ and $\angle 5$?

Line $l \parallel$ line $m$, cut by transversal $t$. At line $l$, two angles are formed: $\angle 1 = (3x + 15)^\circ$ and $\angle 2 = (x + 45)^\circ$, where $\angle 1$ and $\angle 2$ are co-interior angles with the corresponding angle of $\angle 2$ at line $m$. If $\angle 1$ and the alternate interior angle of $\angle 2$ at line $m$ are supplementary, find $x$ and both angle measures.

(Hint: co-interior angles formed by parallel lines are supplementary.)

Write a complete paragraph proof (in full sentences) that when a transversal crosses two parallel lines, alternate interior angles are congruent. Use $\angle 3$ (at line $l$, lower-right interior position) and $\angle 6$ (at line $m$, upper-left interior position) as your angle labels. Cite the Corresponding Angles Postulate and the Vertical Angles Theorem in your proof.