Measuring Segments and Angles
Apply the Segment Addition Postulate and Angle Addition Postulate to find missing measures. Use the midpoint formula and angle bisector definition. Classify angles (acute, right, obtuse, straight) precisely.
A lesson of Geometry, in Foundations of Geometry, about 10 minutes.
By the end of this lesson, you will be able to measure and compare segments using the Segment Addition Postulate, measure and classify angles using a protractor and the Angle Addition Postulate, and apply the definitions of congruent segments and congruent angles.
The Big Idea
Geometry is both abstract (relationships, proofs) and concrete (measurement). Before proving anything, you need to measure precisely. Segments have length; angles have degree measure. Both follow an addition postulate that is the backbone of most geometric proofs.
Measuring Segments
The length of a segment $\overline{AB}$ is written $AB$ (no bar). It's a positive real number.
Ruler Postulate: Points on a line can be assigned coordinates, and the distance between two points is the absolute value of the difference of their coordinates.
Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$.
Conversely, if $AB + BC = AC$, then $B$ is between $A$ and $C$.
Congruent Segments
Congruent segments have equal lengths: $\overline{AB} \cong \overline{CD}$ means $AB = CD$.
Use tick marks on a figure to indicate congruent segments. One tick = one set, two ticks = another set.
Midpoint: The point $M$ on $\overline{AB}$ such that $AM = MB$. The midpoint divides the segment into two congruent halves.
A segment bisector is any line, ray, or segment that passes through the midpoint of a segment.
Example 1 — Segment Addition
Given $A$, $B$, $C$ collinear with $B$ between $A$ and $C$. If $AB = 3x + 1$ and $BC = 2x - 4$ and $AC = 22$, find $x$ and $AB$.
$$AB + BC = AC$$ $$(3x + 1) + (2x - 4) = 22$$ $$5x - 3 = 22 \implies x = 5$$ $$AB = 3(5) + 1 = 16$$
Verify: $BC = 2(5) - 4 = 6$. $16 + 6 = 22$ ✓
Measuring Angles
An angle is formed by two rays (sides) sharing a common endpoint (vertex).
Naming: $\angle ABC$ (vertex is the middle letter), or $\angle B$ (if unambiguous), or $\angle 1$ (by number).
Protractor Postulate: Every angle has a unique measure from 0° to 180°.
Classifying Angles by Measure
| Classification | Degree measure |
|---|---|
| Acute | $0° < m\angle < 90°$ |
| Right | $m\angle = 90°$ |
| Obtuse | $90° < m\angle < 180°$ |