Notes
Line relationships
- Parallel lines are coplanar and never meet. Perpendicular lines meet at right angles. Skew lines are not coplanar and never meet.
Transversal angle pairs
When a transversal crosses two parallel lines:
| Angle pair | Relationship |
|---|---|
| Corresponding | Congruent |
| Alternate interior | Congruent |
| Alternate exterior | Congruent |
| Same-side interior | Supplementary |
Proving lines parallel
The converses work: if corresponding, alternate interior or alternate exterior angles are congruent, or same-side interior angles are supplementary, the lines are parallel. Two lines perpendicular to the same line (in a plane) are parallel.
Slopes
- $\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$.
- Parallel lines: equal slopes. Perpendicular lines: opposite reciprocal slopes, with product $-1$.
- Point-slope form: $y - y_1 = m(x - x_1)$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Each angle is either $48^\circ$ or its supplement $180 - 48 = 132^\circ$.
$8x + 4 = 180$, so $x = 22$. The angles are $3(22) + 12 = 78^\circ$ and $5(22) - 8 = 102^\circ$.
The slope is $\displaystyle \frac{-2 - 4}{3 - (-1)} = -\frac{6}{4} = -\frac{3}{2}$. The perpendicular slope is $\displaystyle \frac{2}{3}$.
The given line has slope $-2$, so use $\displaystyle \frac{1}{2}$: $\displaystyle 1 = \frac{1}{2}(4) + b$, so $b = -1$.
Set them equal: $2x = 22$.
Two coplanar lines perpendicular to the same line are parallel.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A city planner draws a map on a coordinate grid. Main Street runs through $(0, 2)$ and $(4, 4)$. A new street, Oak Street, must be parallel to Main Street and pass through the library at $(2, -3)$.
(a) Find the slope of Main Street and write an equation for Oak Street.
Show the answer and scoring guide
Slope of Main Street: $\displaystyle \frac{4 - 2}{4 - 0} = \frac{1}{2}$. Oak Street: $\displaystyle -3 = \frac{1}{2}(2) + b$, so $b = -4$ and $\displaystyle y = \frac{1}{2}x - 4$.
Check off the points you earned:
(b) A footpath must be perpendicular to Main Street and pass through $(4, 4)$. Write its equation.
Show the answer and scoring guide
Slope $-2$: $4 = -2(4) + b$, so $b = 12$ and $y = -2x + 12$.
Check off the points you earned:
(c) Elm Avenue crosses Main Street and Oak Street. One angle Elm makes with Main Street is $(7x - 10)^\circ$. The corresponding angle at Oak Street is $(5x + 20)^\circ$. Find $x$ and the angle.
Show the answer and scoring guide
Main and Oak are parallel, so corresponding angles are congruent: $7x - 10 = 5x + 20$, so $x = 15$ and the angle is $95^\circ$.
Check off the points you earned: