Geometry - How to find out if lines are parallel

Transverselines

Which answer contains all the angles (other than itself) that are congruent to Angle 1?

Angles 4, 5, and 8

Angles 2 and 4

Angles 2 and 5

Angles 8 and 6

Angles 4 and 5

Explanation

Because of the Corresponding Angles Theorem (Angle 2 and Angle 5), Alternate Exterior Angles (Angle 2 and Angle 8), and Vertical Angles (Angle 2 and Angle 4).

Transverselines

Angles 2 and 3 are congruent based on which Theorem?

Vertical Angles

Alternate Interior Angles

Corresponding Angles

Consecutie Internior Angles

Alternate Exteriors Angles

Explanation

Veritcal angles means that the angles share the same vertex. Angles 2 and 3 are a vertical pair of angles, which mean that they are congruent.

One line on the coordinate plane has its intercepts at and . A second line has its intercepts at and . Are the lines parallel, perpendicular, or neither?

Perpendicular

Parallel

Neither

Explanation

To answer this question, we must determine the slopes of both lines. If a line has as its intercepts and , its slope is

$m = -\frac{b}{a}$

The first line has as its slope

$m = -\frac{4}{7}$

The second line has as its slope

$m = -\frac{7}{-4} = \frac{7}{4}$

Two lines are parallel if and only if their slopes are equal; this is not the case.

They are perpendicular if and only if the product of their slopes is . The product of the slopes of the given lines is

$-\frac{4}{7} \cdot \frac{7}{4} = -1$ ,

so they are perpendicular.

Transverselines

If angles 2 and 6 are congruent, lines AB and CD are parallel based on which theorem?

Corresponding Angles

Alternate Interior Angles

Alternate Exterior Angles

Vertical Angles

Consecutive Interior Angles

Explanation

Angles 2 and 6 are Corresponding Angles. If each of the set of angles were taken separately, angels 2 and 6 would occupy the same place and are thus corresponding angles.

Transverselines

What is the sum of Angle 3 and Angle 5?

180 deg

90 deg

360 deg

15 deg

45 deg

Explanation

Because of the Consecutive Interior Angle theorem, the sum of Angles 3 and 5 would be 180 deg.

A line which includes the point is parallel to the line with equation

Which of these points is on that line?

Explanation

Write the given equation in slope-intercept form:

$\frac{1}{2}\cdot 2y=\frac{1}{2}\cdot \left (-4x \right )+\frac{1}{2}\cdot 15$

$y=-2x+\frac{15}{2}$

The given line has slope , so this is the slope of any line parallel to that line.

We can use the slope formula $m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}}$ , testing each of our choices.

$(9, -2):; m = \frac{4-(-2)}{6-9}}=\frac{6}{-3}=-2$

$(10, 6):; m = \frac{4-6}{6-10}}=\frac{-2}{-4}=\frac{1}{2}$

$(5,2):; m = \frac{4-2}{6-5}}=\frac{2}{1}=2$

$(6,2):; m = \frac{4-2}{6-6}}=\frac{2}{0}$ , which is undefined

$(0, 7):; m = \frac{4-7}{6-0}}=\frac{-3}{6}=-\frac{1}{2}$

The only point whose inclusion yields a line with slope is .

Transverselines

If lines AB and CD are parallel, angles 1 and 8 are congruent based on which theorem?

Alternate Exterior Angles

Vertical Angles

Alternate Interior Angles

Consecutive Interior Angles

Corresponding Angles

Explanation

Angles 1 and 8 are on the exterior of the parallel lines and are on opposite sides of the transversal. This means the Theorem is the Alternate Exterior Angle theorem.

Choose the equation that represents a line that is parallel to .

Explanation

Two lines are parallel if and only if they have the same slope. To find the slopes, we must put the equations into slope-intercept form, , where equals the slope of the line. In this case, we are looking for . To put into slope-intercept form, we must subtract from each side of the equation, giving us . We then subtract from each side, giving us . Finally, we divide both sides by , giving us , which is parallel to .