Question
Horizontal and parallel lines m and n are cut by transversal k. At the intersection of lines k and m, the bottom left angle is 50 degrees. At the intersection of lines k and n, the uppercase right angle is 50 degrees.
Which theorem correctly justifies why the lines m and n are parallel when cut by transversal k
Which theorem correctly justifies why the lines m and n are parallel when cut by transversal k
Answers
GPT-4o mini
The scenario you described involves two lines, \(m\) and \(n\), cut by a transversal \(k\). The relevant angles formed are the bottom left angle at the intersection of \(k\) and \(m\) which measures \(50^\circ\), and the uppercase right angle at the intersection of \(k\) and \(n\), which also measures \(50^\circ\).
To justify that lines \(m\) and \(n\) are parallel, we can apply the **Alternate Interior Angles Theorem**. This theorem states that if a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
In this case, the angle of \(50^\circ\) at line \(m\) is an alternate interior angle to the angle of \(50^\circ\) at line \(n\) (since both are situated alternately with respect to the transversal \(k\)).
Since the alternate interior angles are congruent (both are \(50^\circ\)), we conclude that lines \(m\) and \(n\) are parallel according to the Alternate Interior Angles Theorem.
To justify that lines \(m\) and \(n\) are parallel, we can apply the **Alternate Interior Angles Theorem**. This theorem states that if a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
In this case, the angle of \(50^\circ\) at line \(m\) is an alternate interior angle to the angle of \(50^\circ\) at line \(n\) (since both are situated alternately with respect to the transversal \(k\)).
Since the alternate interior angles are congruent (both are \(50^\circ\)), we conclude that lines \(m\) and \(n\) are parallel according to the Alternate Interior Angles Theorem.