When two angles are placed together, their measures combine in predictable ways, and where the angles sit tells you which combination to use. This lesson names the important pairs of angles — complementary, supplementary, adjacent, vertical, and linear pairs — and then chains the relationships together: find one angle from a pair, and use that result in the next relationship.
Pairs of angles have names based on how their measures combine and where they sit. Complementary angles total 90° and supplementary angles total 180°. Adjacent angles share a vertex and a side; vertical angles sit opposite each other where two lines intersect and are always congruent (equal in measure); a linear pair is both adjacent and supplementary (their measures add to 180°). Perpendicular lines are the special case of intersecting lines that form right angles.
What you are looking at: Two lines intersect at point O, forming four angles numbered 1 to 4: angle 1 is top-left, angle 2 is top-right, angle 3 is bottom-left, and angle 4 is bottom-right. The two angles of the pair being studied are filled with the gold wedge, and each one shows its measure, computed from the geometry of the two lines.
What to do: Drag the orange handle on the slanted line to change the size of the angles. Choose a pair type with the buttons under the diagram — the line directly above each group of buttons explains what that group does. The highlighted pair changes on its own every couple of seconds, and you can also step through the pairs yourself.
What to notice: Vertical angles are congruent (equal in measure) whenever two lines intersect. The angles of a linear pair are supplementary (their measures add to 180°), because together they form a straight angle. Neither fact needs the lines to be parallel.
Pair type buttons — choose which kind of pair is highlighted. Vertical Angles are opposite each other at the intersection; a Linear Pair sits side by side on one straight line.
Pair buttons — step through the pairs of the chosen type yourself. The highlighted pair also advances on its own.
What to do: Each item shows a figure with two angles filled by the gold wedge. Look at where the two angles sit and at how their measures combine, then choose the most specific name for the pair. A first miss gives a hint and another try, with the missed choice removed. A second miss names the correct choice and explains it, and the activity then moves on. Press Next Item for the next figure: the five items take turns in order, and the item number is shown above the figure.
What to do: Each problem chains two relationships: work the first relationship to find one angle, then use that result in the next relationship. Work in three steps: (1) choose the equation to use to solve the problem, (2) solve for x, and (3) report the measure the question actually asks for, using the second relationship. The gold wedge in the diagram always sits in the part the current step asks about, and a part solved in an earlier step is labeled with its found measure. Each choice you pick sweeps a wedge across the part that choice describes, whether the choice is right or wrong, so you can see what the choice means; a missed choice is shown with a dashed amber wedge. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all three steps are complete, the full worked solution appears. Press New Item for the next problem: the four problems take turns in order, and the problem number is shown with the step question.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Angle Relationships — Mr. J Geometry. All activities on this page run entirely in your browser.