Lesson 4 — Bisection
Big idea: To bisect is to cut into two equal parts. A midpoint splits a segment; an angle bisector splits an angle.
Words to know
| Word | Meaning |
|---|---|
| Bisect | Divide into two equal parts. |
| Midpoint | A point that divides a segment into two equal segments. |
| Angle bisector | A line that divides an angle into two equal angles. |
| Congruent | Same size and shape; coincides when superimposed. |
| Corollary | A theorem that follows easily from a postulate or another theorem. |
Concepts to know
- Segments are congruent iff they have equal lengths.
- Angles are congruent iff they have equal measures.
- A midpoint is unique: > Corollary to the Ruler Postulate: A line segment has exactly one midpoint.
- An angle has exactly one bisecting ray: > Corollary to the Protractor Postulate: An angle has exactly one ray that bisects it.
- Bisection can be done by folding, by measuring, or by straightedge-and-compass construction.
Rules
| Situation | Conclusion |
|---|---|
| M is the midpoint of AB | AM = MB |
| CD bisects ∠ACB | ∠ACD = ∠DCB |
| Segments are congruent | Their lengths are equal. |
| Angles are congruent | Their measures are equal. |
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a midpoint | Divide the total length by 2. |
| Find a missing half | Subtract the known half from the whole, or use the equal half. |
| Bisect an angle | Divide the angle measure by 2. |
| Use a midpoint in a proof | Write the two equal segments first. |
| Use an angle bisector in a proof | Write the two equal angles first. |
| Explain why a construction works | Point to the equal radii and congruent triangles. |
| Use a fold | The crease marks the midpoint or angle bisector. |
Worked example
A segment AB is 4.2 cm long.
midpoint distance from each end = 4.2 ÷ 2 = 2.1 cmAn angle of 42° is bisected.
each half = 42° ÷ 2 = 21°Watch out
- Any point on a segment is not necessarily the midpoint.
- An angle bisector must pass through the vertex.
- “Equal” and “congruent” are related but not identical: congruent figures coincide; equal numbers have the same value.
- A midpoint is unique; there cannot be two different midpoints of the same segment.