Question

when two lines are intersected by a third line, known as a transversal, eight angles are formed. there are two different special angle pairs that arise from these eight angles. exercise #4: given parallel lines r and s crossed by transversal line t with the eight - marked angles, state each of the following: (a) what appears to be true about all acute angles in this diagram? verify using tracing paper. they are congruent (b) what appears to be true about all obtuse angles in this diagram? verify using tracing paper. they are congruent (c) list each corresponding angle pair (same relative place where the lines cross). ∠1 corresponds to ∠6 ∠3 corresponds to ∠8 ∠2 corresponds to ∠7 ∠4 corresponds to ∠5 (d) list each alternate interior angle pair (opposite sides of transversal and inside of the parallel lines). ∠3 and ∠5 ∠4 and ∠6 (e) list each same side interior angle pair (same side of transversal and inside of parallel lines). ∠3 and ∠6 ∠4 and ∠5 (f) list each alternate exterior angle pair. ∠1 and ∠7 ∠2 and ∠8 (g) list each same side exterior angle pair. ∠1 and ∠8 ∠2 and ∠7 (h) if m∠1 = 125°, then determine, based on (a) and (b), the measures of all other seven angles. state them below and label them on the diagram.

Explanation:

Step1: Identify angle - type relationships

When two parallel lines are cut by a transversal, acute - acute, obtuse - obtuse angles are congruent. $\angle1$ is obtuse.

Step2: Find congruent angles to $\angle1$

Obtuse angles $\angle1,\angle3,\angle5,\angle7$ are congruent. So $m\angle3 = m\angle5=m\angle7 = 125^{\circ}$.

Step3: Use linear - pair relationship

Since $\angle1$ and $\angle2$ form a linear pair ($\angle1+\angle2 = 180^{\circ}$), then $m\angle2=180 - 125=55^{\circ}$.

Step4: Find congruent acute angles

Acute angles $\angle2,\angle4,\angle6,\angle8$ are congruent. So $m\angle4 = m\angle6=m\angle8 = 55^{\circ}$.

Answer:

$m\angle2 = 55^{\circ}$, $m\angle3 = 125^{\circ}$, $m\angle4 = 55^{\circ}$, $m\angle5 = 125^{\circ}$, $m\angle6 = 55^{\circ}$, $m\angle7 = 125^{\circ}$, $m\angle8 = 55^{\circ}$