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 \(r\) and \(s\) are cut by a transversal \(t\), we use the properties of corresponding, alternate - interior, alternate - exterior, same - side interior and same - side exterior angles.

Step2: Answer part (a)

All acute angles are congruent. This is because corresponding angles are congruent and vertical angles are congruent. Acute angles are either corresponding or vertical to each other in this setup.

Step3: Answer part (b)

All obtuse angles are congruent. Similar to part (a), obtuse angles are either corresponding or vertical to each other.

Step4: Answer part (c)

Corresponding angle pairs: \(\angle1\) and \(\angle5\), \(\angle2\) and \(\angle6\), \(\angle3\) and \(\angle7\), \(\angle4\) and \(\angle8\).

Step5: Answer part (d)

Alternate - interior angle pairs: \(\angle3\) and \(\angle5\), \(\angle4\) and \(\angle6\).

Step6: Answer part (e)

Same - side interior angle pairs: \(\angle3\) and \(\angle6\), \(\angle4\) and \(\angle5\).

Step7: Answer part (f)

Alternate - exterior angle pairs: \(\angle1\) and \(\angle7\), \(\angle2\) and \(\angle8\).

Step8: Answer part (g)

Same - side exterior angle pairs: \(\angle1\) and \(\angle8\), \(\angle2\) and \(\angle7\).

Step9: Answer part (h)

If \(m\angle1 = 125^{\circ}\), then:

• \(\angle1\) and \(\angle3\) are vertical, so \(m\angle3=125^{\circ}\).
• \(\angle1\) and \(\angle5\) are corresponding, so \(m\angle5 = 125^{\circ}\).
• \(\angle1\) and \(\angle7\) are alternate - exterior, so \(m\angle7=125^{\circ}\).
• \(\angle2\) and \(\angle1\) are supplementary (\(\angle2 = 180 - 125=55^{\circ}\)).
• \(\angle4\) and \(\angle1\) are vertical, so \(m\angle4 = 125^{\circ}\), then \(\angle6\) (corresponding to \(\angle4\)) has \(m\angle6 = 125^{\circ}\) and \(\angle8\) (vertical to \(\angle6\)) has \(m\angle8 = 125^{\circ}\), and \(\angle2\) (vertical to \(\angle8\)) has \(m\angle2 = 55^{\circ}\), \(\angle4\) (corresponding to \(\angle8\)) has \(m\angle4 = 125^{\circ}\), \(\angle6\) (vertical to \(\angle4\)) has \(m\angle6 = 125^{\circ}\). So \(m\angle2=m\angle4=m\angle6=m\angle8 = 55^{\circ}\) and \(m\angle1=m\angle3=m\angle5=m\angle7 = 125^{\circ}\).

Answer:

(a) All acute angles are congruent.
(b) All obtuse angles are congruent.
(c) \(\angle1\) and \(\angle5\), \(\angle2\) and \(\angle6\), \(\angle3\) and \(\angle7\), \(\angle4\) and \(\angle8\)
(d) \(\angle3\) and \(\angle5\), \(\angle4\) and \(\angle6\)
(e) \(\angle3\) and \(\angle6\), \(\angle4\) and \(\angle5\)
(f) \(\angle1\) and \(\angle7\), \(\angle2\) and \(\angle8\)
(g) \(\angle1\) and \(\angle8\), \(\angle2\) and \(\angle7\)
(h) \(m\angle2 = 55^{\circ}\), \(m\angle3 = 125^{\circ}\), \(m\angle4 = 55^{\circ}\), \(m\angle5 = 125^{\circ}\), \(m\angle6 = 55^{\circ}\), \(m\angle7 = 125^{\circ}\), \(m\angle8 = 55^{\circ}\)