Parallel Lines and Transversals: Practice

Angles 1 and 2 form a linear pair, so $2x+1+3x+19=180$.

$5x+20=180$

Thus $x=32$ and angle 1 measures 65 degrees. Corresponding angle 5 also measures 65 degrees.

$4y-7=65$

Adding 7 and dividing by 4 gives $y=18$.

Angles 3 and 6 are same-side interior angles, so $9x-4+5x+2=180$.

$14x-2=180$

This gives $x=13$. Angle 6 then measures $5(13)+2=67$ degrees. Angle 8 is vertical to angle 6, so it has the same measure.

Use corresponding congruence first for $\ell$ and $m$, then for $m$ and $n$. The northeast angle at the bottom crossing measures 136 degrees.

Angle $B$ is its adjacent linear-pair partner.

$m\angle B=180^\circ-136^\circ=44^\circ$

Along $PQ$, alternate interior congruence places 42 degrees between the leftward ray of $\ell$ and $PQ$. Along $PR$, it places 61 degrees between $PR$ and the rightward ray of $\ell$.

These two angles and $\angle QPR$ fill the straight angle below $\ell$.

$m\angle QPR=180^\circ-42^\circ-61^\circ$

Therefore angle QPR measures 77 degrees.

Corresponding angles give equal measures for angles 1 and 5. The local linear pair gives $m\angle5+m\angle8=180^\circ$.

Substituting the equal measure of angle 1 yields the required exterior sum. No equality between angles 1 and 8 has been assumed.

Changing the transversal can change each individual measure. The hypotheses of the proved same-side interior theorem still hold at every allowed position.

Therefore each of its two interior sums remains 180 degrees. Parallelism does not force eight right angles or a fixed individual measure.

At the upper crossing, angles 1 and 3 are vertical, so they have equal measures.

$m\angle3=34^\circ$

At the lower crossing, angles 5 and 7 measure 51 degrees. The upper neighboring measures are 146 degrees and the lower neighboring measures are 129 degrees, so the other proposed equalities fail.

The foundation guarantees exactly one parallel to $\ell$ through the external point $P$. Both $a$ and $b$ satisfy that same requirement.

Uniqueness means they cannot be distinct lines, so the two names describe one line.

Their total is indeed 360 degrees, as required by angle addition. But the angle postulate also applies to either side separately.

Since 177 degrees is below 180 degrees, the crossed lines must meet on that side. That contradicts the parallel hypothesis.

The corresponding pair along $t$ is congruent, so $7p+5=3p+49$. Subtracting gives $4p=44$, hence $p=11$.

The same-side exterior pair along $u$ is supplementary.

$8q+10+6q+30=180$

This gives $14q=140$, hence $q=10$. Thus $p+q=21$. The $t$ angles each measure 82 degrees and the $u$ angles each measure 90 degrees.

For parallel lines the postulate excludes sums below 180 degrees, and the total of 360 degrees fixes both interior sums. A linear pair and one of these sums have a common supplement, giving corresponding congruence.

Vertical angles connect the corresponding pair to the alternate interior pair. Each result is established before the next uses it.

Same-side interior angles 4 and 5 have sum 180 degrees.

$2z+13+3z+7=180$

Thus $5z=160$ and $z=32$. Angle 4 measures 77 degrees, and angle 5 measures 103 degrees. Angle 2 is vertical to angle 4, so it also measures 77 degrees.

$103^\circ-77^\circ=26^\circ$