Question

let $overline{pq} parallel overline{ab}$ and $overline{rq} parallel overline{ac}$. find the measures of $angle 1$, $angle 2$, ..., $angle 8$. explain how you found your answers

angle

measure

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1

$square^circ$

2

$square^circ$

3

$61^circ$

4

$41^circ$

5

$square^circ$

6

$78^circ$

7

$square^circ$

8

$61^circ$

because $\boldsymbol{

abla}$ the measure of $angle 1$ can be found first.$angle b$ is congruent to angle(s) $square$(use a comma to separate answers as needed )$angle c$ is congruent to angle(s) $square$(use a comma to separate answers as needed )$angle 1$ is congruent to angle(s) $square$(use a comma to separate answers as needed )$angle 1$ is supplementary to angle(s) $square$(use a comma to separate answers as needed )

Explanation:

Step1: Find ∠1 via triangle sum

In $\triangle ABC$, $\angle A = 180^\circ - 78^\circ - 41^\circ = 61^\circ$. $\angle 1 = \angle A = 61^\circ$ (corresponding angles, $PQ \parallel AB$)

Step2: Find ∠2 via supplementary to ∠1

$\angle 2 = 180^\circ - \angle 1 = 180^\circ - 61^\circ = 119^\circ$

Step3: Find ∠5 via congruent to ∠4

$\angle 5 = \angle 4 = 41^\circ$ (alternate interior angles, $RQ \parallel AC$)

Step4: Find ∠7 via congruent to ∠1

$\angle 7 = \angle 1 = 61^\circ$ (corresponding angles, $RQ \parallel AC$)

Step5: Find ∠8 via supplementary to ∠7

$\angle 8 = 180^\circ - \angle 7 = 180^\circ - 61^\circ = 119^\circ$

Step6: Find ∠6 via supplementary to ∠5

$\angle 6 = 180^\circ - \angle 5 = 180^\circ - 41^\circ = 139^\circ$

Answer:

$\angle 1 = 61^\circ$, $\angle 2 = 119^\circ$, $\angle 3 = 78^\circ$, $\angle 4 = 41^\circ$, $\angle 5 = 41^\circ$, $\angle 6 = 139^\circ$, $\angle 7 = 61^\circ$, $\angle 8 = 119^\circ$

Congruence/Supplementary Relationships:

$\angle B$ is congruent to angle(s) $\angle 5$
$\angle C$ is congruent to angle(s) $\angle 3$
$\angle 1$ is congruent to angle(s) $\angle 7, \angle 8$'s supplement partner $\angle 1$ (corrected: $\angle 1$ is congruent to $\angle 7, \angle 3$'s triangle partner $\angle A = \angle1$; strictly parallel-based: $\angle1 \cong \angle7$, $\angle1 \cong \angle$ (corresponding on $PQ\parallel AB$))
$\angle 1$ is supplementary to angle(s) $\angle 2, \angle 6$

Initial Justification:

Because $\boldsymbol{PQ \parallel AB, RQ \parallel AC}$ (given parallel lines create congruent corresponding/alternate interior angles, supplementary same-side interior angles) the measure of $\angle 1$ can be found first.