Chapter 1: Points, Lines, and Planes
Learning Objectives
- Identify and describe points, lines, line segments, rays, and planes using correct notation
- Apply the Segment Addition Postulate and the Angle Addition Postulate
- Classify angles as acute, right, obtuse, or straight
- Identify complementary, supplementary, vertical, and adjacent angle pairs
- Use an angle bisector to find unknown angle measures
- Apply properties of perpendicular lines
- Chapter Summary
- Key Terms
1.1 Undefined Terms: Points, Lines, and Planes
Geometry begins with three undefined terms — concepts that are fundamental but cannot be precisely defined using simpler terms. Instead, we describe them intuitively.
The Three Undefined Terms
- A point has no size — it indicates a location. Labeled with a capital letter: point $A$.
- A line has no thickness and extends infinitely in both directions. Named by two points on it: $\overleftrightarrow{AB}$, or by a lowercase letter $\ell$.
- A plane has no thickness and extends infinitely in all directions. Named by three non-collinear points or an uppercase letter: plane $ABC$ or plane $P$.
From these undefined terms we build defined terms:
Segments and Rays
- A line segment $\overline{AB}$ consists of endpoints $A$ and $B$ and all points between them. Its length is $AB$ (no bar).
- A ray $\overrightarrow{AB}$ starts at endpoint $A$, passes through $B$, and extends infinitely in one direction.
- Two rays that share an endpoint and point in opposite directions form a line.
| Object | Symbol | Read as |
|---|---|---|
| Point A | $A$ | "point A" |
| Line through A and B | $\overleftrightarrow{AB}$ | "line AB" |
| Segment from A to B | $\overline{AB}$ | "segment AB" |
| Length of AB | $AB$ | "the distance AB" |
| Ray from A through B | $\overrightarrow{AB}$ | "ray AB" |
Points A and B form a segment, a line, and a ray. Drag the points to explore.
Figure 1.1 — Points, Line Segment, Ray, and Line
The Segment Addition Postulate
Postulate 1.1 — 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 $C$.
Example 1.1 — Using the Segment Addition Postulate
Points $A$, $B$, $C$ are collinear with $B$ between $A$ and $C$. If $AB = 3x + 1$, $BC = 2x - 4$, and $AC = 22$, find $x$ and $BC$.
Step 1: Apply the postulate: $AB + BC = AC$
$(3x + 1) + (2x - 4) = 22$
$5x - 3 = 22$
$5x = 25 \implies x = 5$
Step 2: $BC = 2(5) - 4 = \mathbf{6}$
$P$ is between $Q$ and $R$. $QP = 2x + 3$, $PR = x + 7$, $QR = 25$. Find $x$ and $PR$.
Show Answer
$(2x + 3) + (x + 7) = 25$
$3x + 10 = 25 \implies 3x = 15 \implies x = 5$
$PR = 5 + 7 = \mathbf{12}$
1.2 Angles and Angle Measurement
An angle is formed by two rays (called sides) that share a common endpoint called the vertex. We write $\angle ABC$ where $B$ is the vertex, or simply $\angle B$ if there is no ambiguity.
Classifying Angles by Measure
- Acute angle: $0° < m\angle A < 90°$
- Right angle: $m\angle A = 90°$ (indicated by a small square at the vertex)
- Obtuse angle: $90° < m\angle A < 180°$
- Straight angle: $m\angle A = 180°$ (a straight line)
Drag the slider to change the angle measure. Watch how the angle type changes.
Figure 1.2 — Angle Classification by Measure
The Angle Addition Postulate
Postulate 1.2 — Angle Addition Postulate
If point $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
Example 1.2 — Applying the Angle Addition Postulate
Ray $BD$ is in the interior of $\angle ABC$. If $m\angle ABD = (2x + 10)°$, $m\angle DBC = (3x - 5)°$, and $m\angle ABC = 90°$, find $x$.
$(2x + 10) + (3x - 5) = 90$
$5x + 5 = 90 \implies 5x = 85 \implies x = 17$
Check: $m\angle ABD = 44°$, $m\angle DBC = 46°$, and $44 + 46 = 90°$ ✓
1.3 Angle Pairs
Special Angle Pairs
- Complementary angles: two angles whose measures sum to $90°$
- Supplementary angles: two angles whose measures sum to $180°$
- Adjacent angles: two angles that share a vertex and a side, with no overlap
- Vertical angles: opposite angles formed when two lines intersect; vertical angles are always congruent
- Linear pair: adjacent angles that form a straight line; their measures sum to $180°$
Two intersecting lines form vertical angles (equal) and linear pairs (supplementary). Drag to explore.
Figure 1.3 — Vertical Angles and Linear Pairs
Theorem 1.1 — Vertical Angles Theorem
Vertical angles are congruent: if two lines intersect, the opposite (vertical) angles are equal in measure.
Example 1.3 — Finding Angles with Vertical Angles
Two lines intersect forming four angles. One angle measures $(5x + 12)°$ and its vertical angle measures $(8x - 15)°$. Find the measure of each angle.
Since vertical angles are equal: $5x + 12 = 8x - 15$
$27 = 3x \implies x = 9$
Angle measure = $5(9) + 12 = \mathbf{57°}$
The adjacent supplementary angle = $180° - 57° = \mathbf{123°}$
Two angles are supplementary. One measures $(3x + 20)°$ and the other measures $(x + 40)°$. Find both angles.
Show Answer
$4x + 60 = 180 \implies 4x = 120 \implies x = 30$
First angle: $3(30)+20 = \mathbf{110°}$ Second angle: $30+40 = \mathbf{70°}$
Check: $110 + 70 = 180°$ ✓
1.4 Angle Bisectors
Definition: Angle Bisector
An angle bisector is a ray that divides an angle into two congruent angles. If $\overrightarrow{BD}$ bisects $\angle ABC$, then $m\angle ABD = m\angle DBC = \frac{1}{2} m\angle ABC$.
Example 1.4 — Angle Bisector Problem
$\overrightarrow{BD}$ bisects $\angle ABC$. If $m\angle ABD = (4x - 8)°$ and $m\angle DBC = (2x + 16)°$, find $m\angle ABC$.
Since $\overrightarrow{BD}$ bisects, the two angles are equal:
$4x - 8 = 2x + 16 \implies 2x = 24 \implies x = 12$
$m\angle ABD = 4(12) - 8 = 40°$
$m\angle ABC = 2 \times 40° = \mathbf{80°}$
Geometry Tip: When an angle bisector splits an angle, the two resulting angles are always congruent (equal measure). Set the two expressions equal to each other, then solve — don't sum them to get the total angle.
Practice Problems
$M$ is between $L$ and $N$. $LM = 4x - 2$, $MN = 2x + 8$, $LN = 42$. Find $x$, $LM$, and $MN$.
Show Solution
$(4x - 2) + (2x + 8) = 42$
$6x + 6 = 42 \implies 6x = 36 \implies x = 6$
$LM = 4(6) - 2 = \mathbf{22}$, $MN = 2(6) + 8 = \mathbf{20}$
Check: $22 + 20 = 42$ ✓
Classify each angle as acute, right, obtuse, or straight:
(a) $m\angle A = 35°$ (b) $m\angle B = 90°$ (c) $m\angle C = 127°$ (d) $m\angle D = 180°$
Show Solution
(b) 90° — Right
(c) 127° — Obtuse (between 90° and 180°)
(d) 180° — Straight
Two angles are complementary. One angle is 14° more than three times the other. Find both angles.
Show Solution
Complementary: $x + (3x + 14) = 90$
$4x + 14 = 90 \implies 4x = 76 \implies x = 19$
Angles: 19° and 71°
Check: $19 + 71 = 90°$ ✓
Two lines intersect. One of the four angles formed measures $(6x - 30)°$ and an adjacent angle measures $(2x + 10)°$. These two angles form a linear pair. Find all four angle measures.
Show Solution
$8x - 20 = 180 \implies 8x = 200 \implies x = 25$
First angle: $6(25) - 30 = \mathbf{120°}$
Adjacent angle: $2(25) + 10 = \mathbf{60°}$
By vertical angles: the other two angles are also 120° and 60°.
$\overrightarrow{KM}$ bisects $\angle JKL$. If $m\angle JKL = (10x - 20)°$ and $m\angle MKL = (3x + 12)°$, find $m\angle JKL$.
Show Solution
Since $m\angle JKL = 2 \cdot m\angle MKL$:
$10x - 20 = 2(3x + 12)$
$10x - 20 = 6x + 24$
$4x = 44 \implies x = 11$
$m\angle JKL = 10(11) - 20 = \mathbf{90°}$
Check: $m\angle MKL = 3(11)+12 = 45°$; $2 \times 45° = 90°$ ✓
Name each figure using correct geometric notation:
(a) A line passing through points $P$ and $Q$
(b) A segment with endpoints $X$ and $Y$
(c) A ray starting at $A$ passing through $B$
(d) The length of segment $CD$
Show Solution
(b) $\overline{XY}$ (segment XY, bar on top)
(c) $\overrightarrow{AB}$ (ray AB, arrow on one end pointing from A through B)
(d) $CD$ (no bar — just the letters represent the numerical length)
Ray $\overrightarrow{EG}$ is in the interior of $\angle DEF$. $m\angle DEF = 124°$ and $m\angle DEG = (3x + 7)°$. If $\overrightarrow{EG}$ bisects $\angle DEF$, find $x$.
Show Solution
$3x + 7 = 62$
$3x = 55$
$x = \frac{55}{3} \approx 18.3$
(If the problem expects a whole-number answer, check whether the bisector condition or the expression is slightly different in your version.)
Angle $A$ and angle $B$ are supplementary. Angle $B$ and angle $C$ are complementary. If $m\angle A = 110°$, find $m\angle B$ and $m\angle C$.
Show Solution
$A$ and $B$ are supplementary: $m\angle A + m\angle B = 180°$
$110° + m\angle B = 180°$
$m\angle B = \mathbf{70°}$
Step 2 — Find angle C:
$B$ and $C$ are complementary: $m\angle B + m\angle C = 90°$
$70° + m\angle C = 90°$
$m\angle C = \mathbf{20°}$
📋 Chapter Summary
Basic Elements
A point has no dimension. A line extends infinitely in two directions. A plane is a flat surface extending infinitely. These are undefined terms in geometry.
A segment $\overline{AB}$ has two endpoints. A ray $\overrightarrow{AB}$ starts at $A$ and extends through $B$ infinitely. Midpoint divides a segment into two equal parts.
Acute: $0° < m\angle < 90°$. Right: $90°$. Obtuse: $90° < m\angle < 180°$. Straight: $180°$.
Complementary: sum $= 90°$. Supplementary: sum $= 180°$. Vertical angles: formed by intersecting lines, always congruent.
Key Postulates
- Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$
- Angle Addition Postulate: If $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$
- Unique Line Postulate: Through any two points, exactly one line exists