Question

triangle pqr and lines j and k are shown in the figure below.
lines j and k are parallel.

• the intersections of the lines with the vertices of the triangle create seven

numbered angles as shown.

• the figure is used to prove that the sum of the measures of the interior angles of a

triangle is always 180°.

• in the figure, angles 2, 5, and 6 are the interior angles of triangle pqr.

part a
explain how you know that angle 1 is congruent to angle 5 and that angle 3 is
congruent to angle 6.
enter your explanation in the box provided.

Explanation:

Step1: Recall the property of parallel lines and alternate interior angles

When two parallel lines are cut by a transversal, alternate interior angles are congruent.
Lines \(j\) and \(k\) are parallel. For angle \(1\) and angle \(5\), the line that intersects \(j\) and \(k\) (let's call it the trans - versal) creates alternate interior angles. So, \(\angle1\cong\angle5\) by the Alternate Interior Angles Theorem.
For angle \(3\) and angle \(6\), the line that intersects \(j\) and \(k\) (the trans - versal) creates alternate interior angles. So, \(\angle3\cong\angle6\) by the Alternate Interior Angles Theorem.

Answer:

Angle \(1\) is congruent to angle \(5\) because of the Alternate Interior Angles Theorem (when two parallel lines \(j\) and \(k\) are cut by a transversal, alternate interior angles are congruent). Angle \(3\) is congruent to angle \(6\) because of the Alternate Interior Angles Theorem.