Chapter 3: Parallel and Perpendicular Lines¶

3.1: Parallel Lines and Transversal¶

Vocabulary

Parallel lines never intersect.
A transversal goes though two parallel lines.
Corresponding angles are both on the same side of separate parallel lines and on the same side of the transversal.
Vertical angle is when opposite angles formed when two lines intersect, always congruent.
Alternate exterior angles are on the oposite side of separate parallel lines and on the oposite side of the transversal. They are always on the outside.
Alternate interior angles are on the oposite side of separate parallel lines and on the oposite side of the transversal. They are always on the inside.
Same side interior angles are on the inside angles of separate parallel lines and on the same side of the transversal.
Same side exterior angles are on the outside angles of separate parallel lines and on the same side of the transversal.

\(\angle 1\) and \(\angle 5\) are corresponding angles. \(\angle 1\) and \(\angle 4\) are vertical angles. \(\angle 1\) and \(\angle 8\) are alternate exterior angles. \(\angle 4\) and \(\angle 5\) are alternate interior angles. The previous angles were congruent. The following are supplementary. \(\angle 3\) and \(\angle 5\) are same side interior angles. \(\angle 1\) and \(\angle 7\) are same side exterior angles.

3.2: Angles and parallel lines¶

Vocabulary

The corresponding angle theorem states that if a transversal goes though two parallel lines, then it it’s corresponding angles are congruent.
The converse corresponding angle theorem states if 2 lines and a transversal form corresponding angles, then the lines are parallel.
The transitive property of congruence states that a line segment that is congruent to another can be replaced by that segment.
The transitive property of equality states that sates that if two equations contain a common variable, they can be joined by the common variable.

3.3: Slopes of lines¶

Vocabulary

The slope of a line is the steepness of the line.
Parallel lines are two lines that never meet and have the same slope.
Perpendicular lines intersect at \(90^{\circ}\) angle and have oposite reciprocal slopes.
The oposite reciprocal is when you multiply the reciprocal of a number by \(-1\).

Note

Any number multiplied by it oposite reciprocal will result in \(-1\)

Finding the Slope of a Perpendicular Line

Example: If \(y = \frac{1}{2} x - 123\) then the perpendicular line slope would be \(-\frac{2}{1}\) or \(-2\).

3.4: Equations of lines¶

Vocabulary

Slope intercept form is when a liner equation is written in \(y=mx+b\)

Point slope form is when a liner equation is written in \(y - y_1 = m(x - x_1)\)

3.5: Proving lines parallel¶

See: 3.2: Parallel Lines and Transversal

3.6: Perpendiculars and Distance¶

Finding Distence Between Parallel Lines (long)

Find a perpendicular line to the parallel lines
Find the point the transversal intersects each parallel line using a system of equations
Find the distence beween the the points with the distance formula

Finding Distence Between Parallel Lines (shortcut)

The shortcut is:

\[\frac{|c - c1|}{\sqrt(a^2 + b^2)}\]

When seen in \(ax + bx = c\)

When seen in \(y = mx +b\):

\[d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}}\]

Finding Distence Between a Line and a Point (long)

Find a perpendicular line to the parallel lines that passes though the point
Find the point the perpendicular line intersects the line using a system of equations
Find the distence beween the the points with the distance formula

Finding Distence Between a Line and a Point (shortcut)

\[\frac{ax + by + c}{\sqrt{a^2 + b^2}}\]

x and y are the coordinates of point.

This works for \(ax + bx = c\)