Parallel lines ($\parallel$): Coplanar lines that never intersect. Denoted $\ell \parallel m$.
Perpendicular lines ($\perp$): Lines that intersect at a 90° angle. Denoted $\ell \perp m$.
Skew lines: Lines that are not coplanar — they don't intersect and aren't parallel. Only possible in 3D.
Transversal: A line that intersects two or more other lines at distinct points.
When a transversal $t$ crosses two lines $\ell$ and $m$, it creates eight angles — four at each intersection. We name them by position (interior/exterior) and side (same side/alternate sides).
Interior region: The region between the two lines. Angles 3, 4, 5, 6 are interior angles.
Exterior region: The region outside both lines. Angles 1, 2, 7, 8 are exterior angles.
Eight angles formed when transversal $t$ cuts lines $\ell$ and $m$ — drag the angle slider to highlight pairs
Figure 3.1 — Eight Angles Formed by a Transversal
When the two lines cut by the transversal are parallel, special relationships hold:
| Angle Pair | Parallel? → Relationship | Example (if ∠1 = 65°) |
|---|---|---|
| Corresponding (e.g., ∠1 and ∠5) | Congruent (equal) | ∠5 = 65° |
| Alternate Interior (e.g., ∠3 and ∠6) | Congruent (equal) | ∠6 = 65° |
| Alternate Exterior (e.g., ∠1 and ∠8) | Congruent (equal) | ∠8 = 65° |
| Co-Interior / Same-Side Interior (e.g., ∠3 and ∠5) | Supplementary (sum 180°) | ∠5 = 115° |
Memory Tip: "F-shape" → Corresponding (equal). "Z-shape" or "N-shape" → Alternate (equal). "C-shape" or "U-shape" → Co-interior (supplementary). Sketch the letter shape formed by the two parallel lines and transversal to identify the pair.
Interactive: Drag the angle slider — see corresponding, alternate, and co-interior pairs highlighted
Figure 3.2 — Angle Pairs When Parallel Lines are Cut by a Transversal
Lines $\ell \parallel m$ are cut by transversal $t$. One angle measures 72°. Find all eight angles.
Label the angles 1–4 at the upper intersection and 5–8 at the lower (∠1 = 72°).
Check: ∠3 and ∠5 are co-interior → 72° + 108° = 180° ✓
Two parallel lines are cut by a transversal. One co-interior angle measures $(3x + 20)°$ and the other measures $(2x + 10)°$. Find $x$ and both angle measures.
The converses of the angle theorems are also true — we can use angle relationships to prove that lines are parallel.
Two lines are cut by a transversal. Alternate interior angles measure 53° and 53°. What can you conclude?
Conclusion: The two lines are parallel. (Converse of Alternate Interior Angles Theorem — if alternate interior angles are congruent, the lines are parallel.)
Lines $\ell \parallel m$. Corresponding angles measure $(5x - 15)°$ and $(3x + 25)°$. Find $x$.
Corresponding angles are equal: $5x - 15 = 3x + 25$
$2x = 40 \Rightarrow x = \mathbf{20}$
Each angle = $5(20) - 15 = 85°$. Check: $3(20) + 25 = 85°$ ✓
In coordinate geometry, we use slope to determine whether lines are parallel or perpendicular.
Slope formula: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Parallel lines: Have equal slopes: $m_1 = m_2$ (and different $y$-intercepts).
Perpendicular lines: Have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.
Line $\ell$ has slope $\frac{2}{3}$.
(a) The slope of a parallel line is $m = \mathbf{\dfrac{2}{3}}$ (same slope).
(b) The slope of a perpendicular line is $m = -\dfrac{3}{2}$ (negative reciprocal, since $\frac{2}{3} \cdot (-\frac{3}{2}) = -1$).
(c) Are $y = 4x - 3$ and $y = 4x + 7$ parallel? Yes — same slope ($m=4$), different $y$-intercepts.
(d) Are $y = 2x + 1$ and $y = -\frac{1}{2}x + 5$ perpendicular? Yes — $2 \cdot (-\frac{1}{2}) = -1$ ✓.
Interactive: Adjust slope $m$ — the red line is parallel, blue line is perpendicular to the green line
Figure 3.3 — Parallel and Perpendicular Lines in the Coordinate Plane
Line $\ell$ passes through $(0, 2)$ and $(3, 8)$. (a) Find its slope. (b) Write the equation of a parallel line through $(1, -1)$. (c) Write the equation of a perpendicular line through $(0, 4)$.
Lines $\ell \parallel m$ are cut by transversal $t$. An exterior angle at the upper intersection measures 130°. Find all angles at both intersections.
Identify the relationship: $\ell \parallel m$, cut by transversal $t$. If ∠4 = 65°, find ∠6 (alternate interior) and ∠5 (co-interior with ∠4).
Two lines are cut by a transversal. Co-interior angles measure $(4x + 10)°$ and $(3x + 30)°$. Are the lines parallel? Find $x$.
Alternate exterior angles measure $(7x - 5)°$ and $(5x + 25)°$ when two lines are cut by a transversal. If the lines are parallel, find $x$ and each angle.
A transversal creates angles measuring 72° and 108° with two lines. Are the lines parallel? Justify your answer.
Are $y = \frac{3}{4}x - 2$ and $y = \frac{3}{4}x + 5$ parallel? Are $y = 3x + 1$ and $y = -\frac{1}{3}x - 2$ perpendicular? Explain.
Write a two-column proof. Given: $\ell \parallel m$, ∠1 and ∠2 are corresponding angles. Prove: ∠1 ≅ ∠2.
In a diagram, $\ell \parallel m$ and $m \parallel n$. Angle $\angle A$ at line $\ell$ and transversal $t$ measures 58°. Find the corresponding angle at line $n$. What property justifies this?
Same position at each intersection. Congruent when lines are parallel. Converse: if corresponding angles are $\cong$, lines are parallel.
Between parallel lines, on opposite sides of transversal. Congruent when lines are parallel.
Outside parallel lines, on opposite sides of transversal. Congruent when lines are parallel.
Between parallel lines, on the same side of transversal. Supplementary (sum $= 180°$) when lines are parallel.
The sum of interior angles of any triangle $= 180°$. $m\angle A + m\angle B + m\angle C = 180°$.
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.