The angle bisector: an interior ray OC that splits ∠AOB into two equal halves — each marked with a single matching tick. It is to an angle what the midpoint is to a segment.
Back in Lesson 13.4 a segment had one number attached to it — its length — and the moment a shape carried a number, all of your arithmetic came alive: you could compare two segments, lay them end to end to add, snip one to subtract, and find the midpoint that cuts a segment exactly in half. An angle carries a number too — its measure in degrees — so the very same playbook returns. Set two angles side by side and the wider opening wins. Tuck one ray inside another angle and the two pieces add. And every angle has its own "midpoint": a ray that slices it into two equal halves, called the bisector.
13.6.1 Comparing angles
How do you tell which of two angles is bigger? The trick is the same one you used for segments: line them up. Slide one angle on top of the other so their vertices coincide and one pair of sides lies exactly on top of each other. Now just look at where the second sides land. Whichever angle's free side sweeps farther out has the bigger opening.
Two angles stacked at one vertex, sharing the lower side. ∠1 (amber) opens wider than ∠2 (slate), so ∠1 > ∠2. There are exactly three outcomes: ∠1 > ∠2, ∠1 = ∠2, or ∠1 < ∠2.
Here is the single most important idea in the whole lesson, and the one students get wrong most often. The size of an angle depends only on the opening between its sides — never on how long you draw those sides. A wide angle drawn with stubby little sides is still wide; a narrow angle drawn with sides a mile long is still narrow. Length of side is decoration. Only the turn between the rays counts.
Watch out
"This angle has longer sides, so it must be bigger." No. Lengthening the sides of an angle does not change the angle at all — the rays just reach farther in the same two directions. Cover up the outer ends of the sides and the opening is unchanged.
Key idea
To compare two angles, place vertex on vertex and one side on one side. The angle whose other side falls outside is the larger. Equal openings make ∠1 = ∠2 — and equal angles are marked with the same number of arc ticks, exactly as equal segments share the same hatch marks.
13.6.2 Adding and subtracting angles
Put two angles next to each other so they share a side and a vertex, and their measures simply add. If a ray OB lies inside∠AOC, it cuts that angle into two adjacent pieces, and the whole is the sum of its parts:
∠AOC = ∠AOB + ∠BOC
Read backward, that same picture gives subtraction. If you know the whole and one piece, the other piece is the difference: ∠AOB = ∠AOC − ∠BOC. This is exactly the segment rule AC = AB + BC from 13.4, with "opening" in place of "length."
Try it Build ∠AOC out of two pieces
Set the two openings with the steppers. Ray OB sits between OA and OC, so the parts add to the whole.
∠AOB = α40°
∠BOC = β30°
Example
A door swings open 35°, then you push it another 20°. From its closed position it has now turned 35° + 20° = 55°. If instead the door stands open at 55° and you know the second push was 20°, the first push was 55° − 20° = 35°.
13.6.3 The angle bisector
A segment's midpoint cuts it into two equal halves. An angle has the very same thing — except the "cut" is a ray, not a point. The angle bisector of ∠AOB is the ray OC from the vertex that divides the angle into two equal angles:
∠AOC = ∠COB = 12∠AOB
"Bisect" just means "cut into two equal parts" (bi = two). Because the two halves are equal, we mark each with a single matching tick on its arc — the angle version of equal-length hatch marks. Drag the slider below to open the angle; the bisector always lands exactly halfway, and the two halves stay equal.
Try it Halve any angle with its bisector
Open ∠AOB to any size. The green ray OC is the bisector; the two equal halves carry the same single tick.
∠AOB = θ
Key idea
If OC bisects ∠AOB, then each half is ½∠AOB. Read this two ways: halving — given the whole 70°, each half is 35°; and doubling — given one half 25°, the whole is 50°.
13.6.4 Bisecting with compass and straightedge
You can find the bisector exactly, with no protractor at all — just a compass and a straightedge. This is one of the oldest constructions in geometry, and it is beautifully simple. Follow the four steps in the figure:
Bisecting ∠AOB. ① From vertex O, swing one arc crossing both sides at P and Q. ②–③ With the same radius, swing an arc from P and another from Q; they meet at R. ④ Draw ray OR — it bisects the angle. The two halves (single ticks) are equal.
Why does it work? Points P and Q are the same distance from O (one arc), and R is the same distance from P and from Q (the equal arcs). That symmetry forces ray OR to split the opening exactly in half. You will prove this carefully in Stage 14; for now, trust the symmetry and enjoy that it needs no measuring.
Good to know
The closely related job of copying an angle onto a fresh ray uses the very same compass moves — span the opening with an arc, then transfer that span to the new ray. Comparing, adding, and copying angles can all be done with compass and straightedge, never touching a protractor.
13.6.5 Segments and angles — one playbook
Step back and look at the whole arc of Lessons 13.4 and 13.6 together. Every single move you learned on a segment has a twin on an angle. The objects are different — one measures how long, the other how wide — but the moves line up one for one:
Segment (length)
Angle (opening)
The measure
length AB
measure ∠AOB
Compare
AB > = < CD
∠1 > = < ∠2
Add
AC = AB + BC
∠AOC = ∠AOB + ∠BOC
Subtract
AB = AC − BC
∠AOB = ∠AOC − ∠BOC
Cut in half
midpoint M: AM = MB
bisector OC: ∠AOC = ∠COB
One playbook, two columns. A midpoint halves a segment; a bisector halves an angle. Once you see the parallel, you already know angles — you learned them the day you learned segments.
★ Recap
The big ideas to carry forward:
Comparing. Stack vertex on vertex and one side on one side; the wider opening is the larger angle. Side length never matters — only the opening.
Adding & subtracting. If OB is inside ∠AOC, then ∠AOC = ∠AOB + ∠BOC, and any one piece is the whole minus the other.
Bisector. The ray OC with ∠AOC = ∠COB = ½∠AOB cuts the angle into two equal halves — an angle's "midpoint."
Construction. Compass-and-straightedge bisects (and copies) an angle exactly, with no protractor.
The analogy. Compare, add, subtract, halve — the same four moves you made on segments, now on angles.
✎ Exercises 13.6
Two angles are drawn. The first measures 40° but is drawn with very long sides; the second measures 55° but is drawn with short stubby sides. Which angle is bigger?
Answer
The 55° angle. Side length is irrelevant — only the opening counts, and 55° > 40°.
Ray OB lies inside ∠AOC. If ∠AOC = 80° and ∠AOB = 50°, find ∠BOC.
Answer
∠BOC = ∠AOC − ∠AOB = 80° − 50° = 30°.
Ray OC bisects ∠AOB, and ∠AOB = 70°. Find ∠AOC.
Answer
A bisector halves the angle: ∠AOC = ½ × 70° = 35°.
You take an angle and redraw it with sides twice as long, leaving the vertex and directions the same. Does its measure change?
Answer
No. The rays point the same two ways, so the opening — and therefore the measure — is unchanged. Lengthening sides changes nothing.
Ray OM bisects ∠AOB. If one half, ∠AOM, measures 25°, what is ∠AOB?
Answer
The bisector makes two equal halves, so the whole is double one half: ∠AOB = 2 × 25° = 50°.
In the compass-and-straightedge bisection, an arc from the vertex meets the two sides at P and Q, and equal arcs from P and Q meet at R. Why is ray OR the bisector?
Answer
Because OP = OQ (same arc) and RP = RQ (equal arcs), the figure is symmetric across OR. That mirror symmetry makes the two halves of the angle equal, so OR bisects it.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The big idea
This lesson is deliberately a mirror of segment arithmetic (13.4). The payoff is conceptual economy: students who internalize "compare, add, subtract, bisect" once on segments meet no genuinely new ideas here, only a new object. Drawing the parallel out loud — midpoint ↔ bisector, AC = AB + BC ↔ ∠AOC = ∠AOB + ∠BOC — is the whole pedagogical move.
The misconception to hunt
The number-one error is believing an angle gets bigger when its sides are drawn longer. It is worth provoking directly: draw a small wide angle and a large narrow one and ask which is bigger. Insist that students articulate "the opening, not the sides." The interactive bisector and the compare figure are built to make this visible.
Common Core alignment. Builds on 4.MD.C.5–7 (angles as turn, additive angle measure: when a ray divides an angle, the parts sum to the whole) and reaches toward 7.G.A.2 and HS G-CO.A.1 / G-CO.D.12 (precise definitions and compass-and-straightedge constructions, including bisecting an angle).