Question

when two lines are intersected by a third line, known as a transversal, eight angles are formed. there are many 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.
(b) what appears to be true about all obtuse angles in this diagram? verify using tracing paper.
(c) list each corresponding angle pair (same relative place where the lines cross).
(d) list each alternate interior angle pair (opposite sides of transversal and inside of the parallel lines).
(e) list each same side interior angle pair (same side of transversal and inside of parallel lines).
(f) list each alternate exterior angle pair.
(g) list each same side exterior angle pair.
(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.
we will work far more with parallel lines and all these different angle pairs. there is one final fact that is very important regarding parallel lines.
parallel line postulate*
given line m and point p not on line m, there exists only one line that can be drawn through point p that is parallel to line m. (*a postulate is something that is assumed to be true without proof.)
exercise #5: illustrate the parallel line postulate on the given diagram.

Explanation:

Step1: Recall angle - relationships in parallel lines cut by a transversal

When two parallel lines are cut by a transversal, corresponding angles are congruent, alternate - interior angles are congruent, alternate - exterior angles are congruent, and same - side interior angles are supplementary.

Step2: Analyze angle 1

Given \(m\angle1 = 125^{\circ}\). \(\angle1\) and \(\angle3\) are vertical angles, so \(m\angle3=m\angle1 = 125^{\circ}\) (vertical angles are congruent).

Step3: Find other angles using angle - relationships

\(\angle1\) and \(\angle5\) are corresponding angles, so \(m\angle5 = 125^{\circ}\). \(\angle3\) and \(\angle7\) are corresponding angles, so \(m\angle7 = 125^{\circ}\). \(\angle2\) and \(\angle1\) are supplementary (\(\angle2=180 - 125=55^{\circ}\)). \(\angle4\) and \(\angle1\) are vertical angles, so \(m\angle4 = 125^{\circ}\). \(\angle6\) and \(\angle2\) are corresponding angles, so \(m\angle6 = 55^{\circ}\). \(\angle8\) and \(\angle2\) are vertical angles, so \(m\angle8 = 55^{\circ}\).

Answer:

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