The Arc Addition Postulate
Just like the Segment Addition Postulate says parts add to the whole, the Arc Addition Postulate says adjacent arcs add up to the larger arc.
Introduction
Arc AB + Arc BC = Arc AC
Past Knowledge
Central angles (10.1.2). Segment Addition (1.2.1). Arc naming.
Today's Goal
Add adjacent arcs to find larger arcs and solve for unknowns.
Future Success
Inscribed angles (10.1.4), chord angle theorem (10.2.2).
Key Concepts
Arc Addition Postulate
If B is a point on , then:
Congruent Arcs and Chords
- In the same circle (or congruent circles): congruent chords ↔ congruent arcs
- A diameter perpendicular to a chord bisects the chord AND its arc
Worked Examples
Adding Arcs
Arc AB = 70°, arc BC = 45°. Find arc AC.
Arc AC = 115°
Finding a Missing Arc
Points A, B, C, D on a circle. Arc AB = 80°, arc BC = 100°, arc CD = 70°. Find arc DA.
Arc DA = 110°
Perpendicular Diameter
Diameter AB is perpendicular to chord CD at point E. If arc AC = 55°, find arc AD, arc BC, and arc BD.
⊥ diameter bisects the chord's arc, so arc AC = arc AD (wrong sides of diameter).
Actually: ⊥ diameter bisects CD, so arc CD is bisected → arc CB = arc DB (wait, let me reconsider).
A ⊥ diameter to a chord bisects the arc. So: arc BC = arc BD.
Arc AC = 55° → arc AD = 180° − 55° = 125° (semicircle through A).
Since arc BC = arc BD: arc BC + arc BD = 360° − 180° = 180°. So arc BC = arc BD = 90°. Wait — let me use: arc AC + arc CB = semicircle = 180°. So arc CB = 180° − 55° = 125°. Since the ⊥ bisects the chord's arc: arc CB = arc DB? No. The perpendicular from center bisects the chord. Here AB is a diameter ⊥ to CD. So the two arcs of CD on the same side: arc CD on one side and arc CD on the other.
Let me re-solve cleanly: Points on circle in order A, C, B, D. Diameter AB ⊥ CD. By the perpendicular-diameter theorem, arc AC = arc AD is NOT correct. Instead: C and D are symmetric about AB. So arc AC = arc AD... Actually C and D are reflections. So arc AC = arc AD. But we're told arc AC = 55°, so arc AD = 55° too.
Arc ACB (semicircle) = 180°. Arc CB = 180° − 55° = 125°.
By symmetry, arc DB = 125°. Check: 55 + 125 + 125 + 55 = 360° ✓
arc AD = 55°, arc CB = 125°, arc BD = 125°
Common Pitfalls
Forgetting to Use 360°
All arcs around a circle must sum to 360°. This is your primary equation for finding missing arcs.
Adding Non-Adjacent Arcs
The Arc Addition Postulate only works with adjacent arcs — arcs that share exactly one endpoint. You can't add arcs that overlap or have a gap.
Real-Life Applications
Dashboard Gauges
Speedometers and fuel gauges use arc addition. The needle sweeps through adjacent arcs, and the total arc from “E” to “F” on a fuel gauge is the sum of the intermediate arcs.
Pizza Slicing
If you cut a pizza into 3 unequal slices with arcs 100°, 140°, and 120°, arc addition confirms: 100 + 140 + 120 = 360° — the full circle.
Practice Quiz
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