Section 1.5 Perpendicular Lines
This section contains an important definition, several basic theorems, and some additional work on proof.
- Definition: Perpendicular lines are two lines that form right angles.
Definitions are often expressed in if-and-only-if form, so we could restate our definition above as:
- Definition: Two lines are perpendicular if and only if they form right angles.
If $\overleftrightarrow{AB}\perp
\overleftrightarrow{CD}$, then angles 1, 2, 3, and 4 are right angles.
If any of angles 1, 2, 3, or 4 is a
right angle, then $\overleftrightarrow{AB}\perp
\overleftrightarrow{CD}$.
We indicate in a diagram that two lines are perpendicular by putting a little square in the angle, as shown to the right. Here we have $\ell\perp m$.
Once you've asserted that two lines are perpendicular, you can say that an angle is a right angle. Using the definition of a right angle, you can then say that its measure is 90. This gives the following trio of statements that should appear in proofs.
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \overleftrightarrow{AB}\perp\overleftrightarrow{CD} & 1. \text{ Given} \\
& 2. \angle 1\text{ is a right angle.} & 2. \text{ Definition of perpendicular lines} \\
& 3. m\angle 1=90 & 3. \text{ Definition of a right angle}\\
\end{array}
To prove that two lines are perpendicular, you reverse the trio.
Two important theorems involving supplements and complements are the following.
- Theorem 1.4 (The Supplement Theorem) If two angles are supplements of
congruent angles or of the same angle, then the two angles are
congruent.
- Theorem 1.5 (The Complement Theorem) If two angles are complements of
congruent angles or of the same angle, then the two angles are
congruent.
Here's a proof of the first of these.
Given: $\angle 1$ and $\angle 2$ are supplementary.
$\angle 3$ and $\angle 4$ are supplementary.
$m\angle 2=m\angle 4$
Prove: $m\angle 1= m\angle 3$
\begin{array}{ l l | l} \text{Proof:} & \text{Statements} & \text{Reasons} \\ \hline & 1.\, \angle 1\text{ and }\angle 2\text{ are supplementary and }&1. \text{ Given}\\ &\quad\angle 3 \text{ and }\angle 4\text{ are supplementary.} & \\ & 2. m\angle 1+m\angle 2=180\text{ and }m\angle 3+m\angle 4=180 & 2. \text{ Definition of supplementary angles} \\ & 3. m\angle 1+m\angle 2=m\angle 3+m\angle 4 & 3. \text{ Substitution}\\ & 4. m\angle 2=m\angle 4 & 4. \text{ Given}\\ & 5. m\angle 1=m\angle 3 & 5. \text{Addition Property of equality}\\ \end{array}
Initially, these theorems are notoriously difficult for students to get their tongues around. You might rephrase them as "Supplements of congruent angles are congruent" and "Complements of congruent angles are congruent," or just use the acronyms SCAC and CCAC.
Conclusions to be drawn from diagrams
Diagrams present certain kinds of information that can be considered to be 'given,' but students should be careful not to jump to conclusions that are neither intended nor warranted. For example, in the diagram below you can conclude that points $B$, $C$, and $D$ are collinear.
In a proof, you could write $\overline{GB}\perp\overline{AD}$ because it is marked; you could also write that $\angle ACE$ is supplementary to $\angle ECD$. Your reason in both cases is simply "given." However, even though it looks like it, you cannot state that $\overline{FC}\perp\overline{AD}$, you cannot conclude from the diagram that $\overline{HB}$ bisects $\angle ABG$, nor that $\angle ECD=60$. You may be able to prove such things, but you cannot state that they are given.
To prove that $\angle 1$ and $\angle 2$ are complementary, you do need to state that $\overline{GB} \perp \overline{AD}$ and you will need Theorem 1.8 which is proved in the problem set. Here are some of the steps that could appear in a proof using the diagram above.
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \angle ACE\text{ is supplementary to }\angle ECD & 1. \text{ Given} \\
\end{array}
or
\begin{array}{ l l | l}
\text{Proof:} & \text{Statements} & \text{Reasons} \\
\hline
& 1. \angle ACE+\angle ECD=180 & 1. \text{ Angle Addition Postulare (Part 2)} \\
\end{array}
or
\begin{array}{ l l | l} \text{Proof:} & \text{Statements} & \text{Reasons} \\ \hline & 1. \overline{GB}\perp\overline{AD} & 1. \text{ Given} \\ & 2. \angle 1\text{ and } \angle 2 \text{ are complementary.}&2. \text{ If the exterior sides of two adjacent angles are}\\ & &\quad \text{ perpendicular, then the angles are complementary.}\\ & 3. m\angle 1+m\angle 2=90 & 3. \text{ Definition of complementary angles.}\\ \end{array}
We conclude by stating four theorems whose proofs are called for in the problems for this section.
- Theorem 1.6: If two lines are perpendicular, then the adjacent angles they form are congruent.
- Theorem 1.7: If two lines form congruent adjacent angles, then the lines are perpendicular.
- Theorem 1.8: If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are complementary.
- Theorem 1.9 (The Right Angle Theorem) All right angles are congruent.