Question

To explain: what can be said about the lines \overline{PQ}\ and\ \overline{RS}

Congruence
ANSWERED
asked 2021-07-28
To explain: what can be said about the lines \(\displaystyle\overline{{{P}{Q}}}\ {\quad\text{and}\quad}\ \overline{{{R}{S}}}\), it is given that \(\displaystyle\angle{1}\stackrel{\sim}{=}\angle{2}\ {\quad\text{and}\quad}\ \angle{4}\stackrel{\sim}{=}\angle{5}\).
Given:
The diagram:
image

Answers (1)

2021-07-29

Concept Used:
Theorem 2-3: Vertically opposite angles are congruent.
Transitive property of congruence: if \(\displaystyle{a}\stackrel{\sim}{=}{b}\ {\quad\text{and}\quad}\ {b}\stackrel{\sim}{=}{c}\), then \(\displaystyle{a}\stackrel{\sim}{=}{c}\).
Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
Consider the given figure:
image

In this figure, by theorem 2-3,
\(\displaystyle\angle{2}\stackrel{\sim}{=}\angle{5}\) (Vertically opposite angles)
Also, it is given that,
\(\displaystyle\angle{1}\stackrel{\sim}{=}\angle{2}\ {\quad\text{and}\quad}\ \angle{4}\stackrel{\sim}{=}\angle{5}\)
Now applying the transitive property of congruence, it can be said that
\(\displaystyle\angle{1}\stackrel{\sim}{=}\angle{4}\)
From the above figure, it can be observed that the angles \(\displaystyle\angle{1}\ {\quad\text{and}\quad}\ \angle{4}\) are the alternate interior angles which are formed when the lines \(\displaystyle\overline{{{P}{Q}}}\ {\quad\text{and}\quad}\ \overline{{{R}{S}}}\) are cut by the transversal \(\displaystyle\overline{{{Q}{R}}}\).
And since \(\displaystyle\angle{1}\stackrel{\sim}{=}\angle{4}\), so by the theorem 3-5, it can be said that the lines \(\displaystyle\overline{{{P}{Q}}}\ {\quad\text{and}\quad}\ \overline{{{R}{S}}}\) are parallel.
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