Visualgebra

Vertically opposite angles are equal

Two crossing lines make two matching pairs of angles, and here is why.

The last two lessons were both about angles adding up to a known total. This one is different: it asks whether two angles that are not obviously related have to be equal, and answers yes.

Here are two lines crossing at O: one through P and R, the other through Q and S. Dragging any of the four outer points moves where the lines cross and changes all four angles at O.

Two lines crossing at O: one through P and R, the other through Q and S. Dragging any of P, Q, R or S moves the crossing point and the four angles it makes. This needs JavaScript switched on.

Drag P, Q, R or S and watch angle POQ keep matching angle ROS, and angle QOR keep matching angle SOP.

Two things never change, however the lines are dragged: angle POQ and angle ROS, directly across O from each other, always read the same number as each other, and angle QOR and angle SOP, the other pair directly across O, always match each other too — usually a different number from the first pair, but always matching within their own pair.

Two angles that sit directly across a crossing point from each other, like angle POQ and angle ROS, are called vertically opposite angles. Every crossing of two lines makes two such pairs.

Proving it from a fact already known

Nothing new is needed here, only the fact from two lessons ago, used twice, on two different lines through the same point O:

Angle POQ and angle QOR sit on a straight line, P-O-R, so by that fact,

angle POQ+angle QOR=180°\text{angle POQ} + \text{angle QOR} = 180°

Angle QOR and angle ROS sit on the other straight line, Q-O-S, so by the same fact again,

angle QOR+angle ROS=180°\text{angle QOR} + \text{angle ROS} = 180°

Both left-hand sides equal 180, so they equal each other:

angle POQ+angle QOR=angle QOR+angle ROS\text{angle POQ} + \text{angle QOR} = \text{angle QOR} + \text{angle ROS}

Subtract angle QOR from both sides and it cancels out completely, leaving

angle POQ=angle ROS\text{angle POQ} = \text{angle ROS}

which is exactly the thing the drag showed but did not, by itself, prove.

This is the first fact in the course that takes two separate lines of reasoning, joined by a shared angle that cancels out, to reach a conclusion that was not obvious in advance. Angle POQ and angle ROS are not adjacent, do not share an arm, and are not part of the same straight line — there was no shortcut available. The only way there was to know they must be equal was to reason it through, which is what makes this the course’s first real theorem rather than a definition or a restatement of something already agreed.

One to watch out for.

This fact is what the next lessons, about parallel lines, will lean on: a transversal crossing two parallel lines makes vertically opposite pairs at each crossing, and comparing those pairs across the two crossings is how alternate angles get shown to be equal.

Angle POQ measures 72°. What is angle ROS?

72°72°

because angle POQ and angle ROS are a vertically opposite pair.

Angle QOR measures 108°. What is angle SOP?

108°108°

because angle QOR and angle SOP are a vertically opposite pair.

Angle POQ measures 72° and angle QOR measures 108°. What is angle SOP, using the straight-line fact from two lessons ago twice, rather than the vertically-opposite fact directly?

Line Q-O-S is straight, so angle QOR and angle ROS sum to 180:

180−108=72° (angle ROS)180 - 108 = 72°\ \text{(angle ROS)}

Line P-O-R is straight, so angle ROS and angle SOP sum to 180:

180−72=108°180 - 72 = 108°

which matches angle QOR, confirming the vertically-opposite pair by the same two-step reasoning used above, not by assuming it.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.