Transversals and Parallel Lines

1. Calculate: Determine the measures of all angles one through four.

Download SVG · 4.2 KBClose

Solution:

Angle 2 corresponds with the 115° angle, and must also be 115° because of the Corresponding Angles Postulate. Therefore, $m\angle 2 = 115°$

Angle 2 and angle 3 are vertical angles, so they are also congruent. $m\angle 3 = 115°$

Angles 1 and 2 form a linear pair, so we know $m\angle 1 = 180°-115°$ which equals 65°. So $m\angle 1 = 65°$

Angles 1 and 4 are vertical angles, so we know they're congruent. $m\angle 4 = 65°$ You could have also figured this out by solving with angles 2 and 4 being a linear pair.

2. Definition: What theorem or postulate would you use to show angles 1 and 8 are congruent. What about angles 2 and 6?

Download SVG · 4.7 KBClose

Solution:

Angles 1 and 8 are congruent by the alternate exterior angles theorem. Angles 2 and 6 are congruent by the corresponding angles postulate.

3. Calculate: Determine 𝑚∠𝐸⁢𝐷⁢𝐹 and the value of x.

Download SVG · 54 KBClose

Solution:

There are several angle pairs we could use to solve this. I have used corresponding angles to determine that $m\angle CDF$ is congruent to the 75° angle, so $m\angle CDF=75°$

$\angle CDF$ and $\angle EDF$ form a linear pair, so $m\angle CDF\ + m\angle EDF = 180°$

$m\angle EDF = 180°-75°=105°$

Solve for x

$(3x+15)°=105°$

$3x=90$

$x=30$

4. Determine: Assume this diagram was not drawn to scale. Are lines l and m parallel? Why?

Download SVG · 16 KBClose

Solution:

They are not parallel. The converse of same-side interior angles theorem says that the two same-side interior angles must be supplementary (add up to 180°) for the lines to be parallel.

115° and 75° add  up to 190° so lines l and m cannot be parallel.

5. Identify: What are the transversals of $\overleftrightarrow{AB}$ and $\overleftrightarrow{BD}$

Download SVG · 6.0 KBClose

Solution:

The transversals of $\overleftrightarrow{AB}$ are $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$

The transversals of $\overleftrightarrow{BD}$ are $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$

6. Calculate: Given $m\angle 1=70°$, determine the measures of all other fifteen angles in this diagram using whatever theorems or postulates you would like.

Download SVG · 8.9 KBClose

Solution:

There are multiple ways to solve this. Here is one example:

$\angle 1 \cong \angle 4$ because they're vertical angles. So $m\angle 4=70°$

$\angle 1$ and $\angle 2$ form a linear pair, so $m\angle 2 = 180°-70° = 110°$

Angles 2 and 3 are vertical angles, so $m\angle 3=110°$

Now that we know the measures of angles 1 through 4, we can use the corresponding angles postulate to determine the measures of angles 5 through 8 and 9 through 12.

We can use the corresponding angles postulate again to determine the measures of the remaining angles 13 through 16.

So angles 1, 4, 5, 8, 9, 12, 13, and 16 are all 70°

The remaining angles are all 110°