Back to Exercise: Prove theorems about lines

Exercises: Prove Theorems About Lines and Angles

Work through each section in order. For proof problems, show each step with its justification. Express angle measures in degrees.

Grade 9·21 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-co-c-9

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Warm-Up: Review What You Know

These problems review skills you already know.

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

$70^\circ$

$110^\circ$

$140^\circ$

$20^\circ$

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?

Corresponding angles

Alternate exterior angles

Alternate interior angles

Co-interior angles

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

Angle bisector of $\angle AMB$

Perpendicular bisector of $\overline{AB}$

Median of triangle $AMB$

Altitude of $\overline{AB}$

Fluency Practice

Apply the theorems directly. Identify angle relationships and find missing measures.

$Two intersecting lines at point O with angles 1, 2, 3, 4 labeled clockwise around the intersection.$

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?

$43^\circ$

$133^\circ$

$47^\circ$

$90^\circ$

$Two parallel lines l and m cut by a transversal, with angles 1-4 at line l and angles 5-8 at line m.$

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?

$\angle 1$ and $\angle 5$

$\angle 4$ and $\angle 5$

$\angle 4$ and $\angle 6$

$\angle 3$ and $\angle 6$

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 $m$ ?

$65^\circ$

$115^\circ$

$90^\circ$

$180^\circ$

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 $m$ ?

$122^\circ$

$32^\circ$

$58^\circ$

$116^\circ$

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