Two angles that finish a right angle, two that finish a straight line — and how to point with angles.
Point 4 of 5 in this lesson: 13.7.4 Bearings: pointing with angles
Some pairs of angles are made for each other. Lay two angles side by side and sometimes they snap together to fill a perfect corner — a right angle, exactly 90°. Other times they open all the way out and fill a flat, straight line — exactly 180°. Mathematicians gave these two partnerships names you will use for the rest of geometry: complementary and supplementary. And once you can split a turn into named pieces, you can do something practical with it — point at things. A sailor or pilot names a direction with an angle measured from north, called a bearing. These three ideas close out our first stage of geometry.
Picture a square corner — the right angle you met in 13.5, marked with a small square. Now slice that corner with a single ray. You have split one 90° angle into two smaller angles, and no matter where the cut falls, those two pieces must add back up to the whole corner.
Two angles whose measures add to 90° are called complementary angles. Each one is called the complement of the other. So if ∠1 measures 35°, its complement measures 90° − 35° = 55°; together they rebuild the right angle.
Complementary ⇒ the two angles add to 90°. To find one angle's complement, subtract from 90: complement of θ = 90° − θ. (Only angles smaller than 90° have a complement.)
In this mode the two angles always fill a right angle. As ∠1 grows, ∠2 = 90 − ∠1 shrinks to match.
Try setting the slider to exactly 45°. Then ∠2 = 90 − 45 = 45° too — an angle that is its own complement. There is exactly one such angle.
Now open the corner all the way until the two outer sides line up into one straight line. That whole sweep is a straight angle, exactly 180° (also from 13.5). Cut it with a ray and you again get two pieces that must rebuild the whole — but now the whole is 180°.
Two angles whose measures add to 180° are supplementary angles; each is the supplement of the other. The supplement of 110° is 180° − 110° = 70°. Switch the widget above to Supplementary mode and slide ∠1 across the full 0–180 range to feel it.
Complementary → 90°. Think "C for Corner" — a complement finishes the square corner. Supplementary → 180°; an extra piece "supplied" to fill out the straight line. Alphabetical helps too: C comes before S, and 90 comes before 180.
| Complementary | Supplementary | |
|---|---|---|
| Sum | 90° | 180° |
| Fills a… | right angle (corner) | straight angle (line) |
| Find the other | 90° − θ | 180° − θ |
| Equal to itself when | θ = 45° | θ = 90° |
The two partnerships, side by side.
Here is a small fact that looks obvious but turns out to be a workhorse. Suppose two different angles, ∠1 and ∠2, are both complements of the same angle ∠α. What can we say about ∠1 and ∠2?
∠1 + ∠α = 90° so ∠1 = 90° − ∠α
∠2 + ∠α = 90° so ∠2 = 90° − ∠α
∴ ∠1 = ∠2
Both equal the same quantity, 90° − ∠α, so they equal each other. In words:
If two angles are complements of the same angle (or of equal angles), then the two angles are equal. The same reasoning works for supplements: supplements of the same angle are equal.
This is the engine behind many proofs you will meet next stage. When two lines cross, the matching angles turn out to be supplements of the same angle — so they are equal. That single line of reasoning is how we will prove the famous vertical-angle fact in Stage 14. Lock the idea in now: equal complements (or supplements) are equal angles.
Angles do useful work the moment you point them at something. A bearing names a direction by how far it sits from the north–south line. The form N30°E reads "start facing North, then turn 30° toward the East." A bearing of S40°W means "face South, turn 40° toward the West."
So a bearing always has three parts: a start (N or S, whichever is nearer), an angle between 0° and 90°, and a turn direction (E or W). The four pure directions are just N, E (= N90°E), S, and W (= N90°W).
The slider is the clockwise turn from North (0°). Watch the readout switch to N/S and E/W as the needle crosses each quarter.
The angle in a bearing is measured from the north–south line, never from east or west, and it is always kept between 0° and 90°. A heading 120° clockwise from north is not "N120°E" — past 90° you switch your start to South, giving S60°E.
Look how far one little dot has travelled. Back in 13.2 a moving point drew a line, a moving line swept a surface, a moving surface built a solid. We zoomed back onto the straight pieces — lines, rays, segments — learned to measure and combine length, then met the angle and learned to measure turning. Every move we made on a segment we made again on an angle: compare, add, subtract, and bisect. Today we named two special partnerships — complementary (90°) and supplementary (180°) — and pointed an angle at the world with bearings.
So far our angles have been built by hand. Next, in Stage 14, we let geometry build them for us: cross two straight lines and four angles spring up at once. Two of them face each other (vertical angles); each sits next to a partner that completes a straight line (adjacent supplementary angles). The equal-supplements rule you proved in 13.7.3 will tell us, in one breath, that vertical angles are equal. The partnerships you named today are the gears of everything that follows.
• Complementary angles add to 90°; complement of θ = 90° − θ. Only angles below 90° have one.
• Supplementary angles add to 180°; supplement of θ = 180° − θ.
• Equal complements / equal supplements: two angles that complete the same angle are equal.
• A bearing like N30°E = start at N or S, turn an angle (0°–90°) toward E or W.
• Self-pairs: 45° is its own complement; 90° is its own supplement.
Find the complement of each angle: (a) 35° (b) 72° (c) 18°.
Subtract from 90°. (a) 90 − 35 = 55° (b) 90 − 72 = 18° (c) 90 − 18 = 72°. Notice (b) and (c) are each other's complements.
Find the supplement of each angle: (a) 110° (b) 90° (c) 1°.
Subtract from 180°. (a) 180 − 110 = 70° (b) 180 − 90 = 90° (its own supplement) (c) 180 − 1 = 179°.
Two angles are complementary. One of them measures 62°. How big is the other?
Complementary means they sum to 90°, so the other is 90 − 62 = 28°.
∠1 and ∠2 are both supplements of a 40° angle. Which is bigger, ∠1 or ∠2? Explain.
Neither — they are equal. Each one is 180 − 40 = 140°. Supplements of the same angle are equal (the equal-supplements property).
Write the bearing for a direction that points 30° east of due north. Then give the bearing for due West.
Start at North, turn 30° toward East: N30°E. Due West is 90° from north toward the west: N90°W (often just written W).
Find an angle that equals its own complement, and one that equals its own supplement.
Own complement: θ = 90 − θ ⇒ 2θ = 90 ⇒ θ = 45°. Own supplement: θ = 180 − θ ⇒ 2θ = 180 ⇒ θ = 90°.
Six questions to lock it in. Tap the answer you think is right.
The big idea. Complementary (90°) and supplementary (180°) are just named sums. The reliable move is always the same subtraction — 90 − θ or 180 − θ — and the figures here show why: each pair simply rebuilds a corner or a straight line. The "find the other angle" subtraction is the whole skill; the names are vocabulary attached to it.
The misconception to watch. Learners swap the two words constantly. Anchor it physically: a complement completes a square corner (90°); a supplement supplies the rest of a straight line (180°). A second trap is bearings — students measure the angle from East or West, or let it exceed 90°. Insist that the angle is always taken from the north–south line and kept between 0° and 90°, switching the N/S start as the direction crosses due-East or due-West.
Where this lands in the standards. This lesson aligns with Common Core 7.G.B.5 — using facts about supplementary, complementary, vertical, and adjacent angles to write and solve simple equations for an unknown angle — and supports the high-school definitions and reasoning in G-CO.A. The equal-complements/equal-supplements argument in 13.7.3 is the same deduction students will reuse to prove vertical angles congruent.