Perpendicular Lines and Their Properties
Define perpendicular lines and apply theorems about perpendicularity (perpendicular bisectors, altitudes). Understand that perpendicular lines form 90° angles and prove related theorems about right angles.
A lesson of Geometry, in Parallel and Perpendicular Lines, about 10 minutes.
By the end of this lesson, you will be able to define perpendicular lines and prove key theorems about them, apply properties of perpendicular lines in proof and problem-solving, and construct perpendiculars using geometric tools.
The Big Idea
Perpendicular lines form right angles — and right angles are the most special angles in geometry. They appear in every coordinate grid, every building, every proof involving distance. Understanding their properties unlocks a large portion of geometric reasoning.
Definition and Notation
Perpendicular lines: two lines that intersect at a right angle (90°).
Notation: $\ell \perp m$ (read "line $\ell$ is perpendicular to line $m$").
A right angle is marked with a small square at the vertex.
Perpendicular bisector: a line perpendicular to a segment at its midpoint. Every point on the perpendicular bisector of a segment is equidistant from the two endpoints.
Key Theorems
Theorem: If two lines are perpendicular, they form four right angles.
Proof: One angle is 90° (definition). Its vertical angle is also 90° (Vertical Angles Theorem). The linear pair angles are each $180° - 90° = 90°$. All four = 90°.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse: If a point is equidistant from the endpoints of a segment, it lies on the perpendicular bisector.
Example 1 — Identifying Perpendicular Angles
Lines $\ell$ and $m$ are perpendicular. One angle formed is $(2x + 10)°$. Find $x$.
All angles are 90°:
$$2x + 10 = 90 \implies x = 40$$
Example 2 — Perpendicular Bisector
$\overline{CD}$ is the perpendicular bisector of $\overline{AB}$, where $A = (0,0)$ and $B = (6, 0)$. Point $P$ is on $\overline{CD}$. Is $PA = PB$?
By the Perpendicular Bisector Theorem, yes — $PA = PB$ for any point $P$ on the perpendicular bisector. No coordinates of $P$ needed.
Example 3 — Finding the Perpendicular Bisector
Find the equation of the perpendicular bisector of the segment with endpoints $A(1, 3)$ and $B(5, 7)$.
Midpoint: $M = (3, 5)$