Draw a diagram. Review the definition of an angle bisector.
See solution.
Practice makes perfect
Let's begin by drawing and marking a diagram that represents the situation. We have a transversal that intersects two parallel lines, and we are considering corresponding angles. Note that by the Corresponding Angles Theorem these angles are congruent.
We want to prove that the bisectors of ∠ 1 and ∠ 2 are parallel. Recall that an angle bisector is a ray that divides an angle into two congruent angles. Let's draw these rays and mark congruent angle pairs.
Notice that the blue rays represent the angle bisectors. We can extend them and get two lines a and b that are intersected by the transversal t. Now we can write a two-column proof to prove that a ∥ b.
Two-Column proof
Let's review the idea of a two-column proof. It lists each statement on the left and the justification is on the right. Each statement must follow logically from the steps before it.
In this case, we are given that a ∥ b, and this how we will begin our proof!
Statement 1)& a ∥ b Reason1)& Given
From our first diagram we can tell that ∠ 1 and ∠ 2 are corresponding angles, so by the Corresponding Angles Theorem they are congruent, ∠ 1 ≅ ∠ 2.
Statement2)& ∠ 1 ≅ ∠ 2 Reason 2)& Corresponding Angles Theorem
Next, we draw the angle bisectors and name the lines c and d as shown in the second diagram. An angle bisector divides the angle into two congruent angles. Since ∠ 1 ≅ ∠ 2, we can conclude that ∠ 3 ≅ ∠ 4.
Statement 3)& ∠ 3 ≅ ∠ 4 Reason3)& Definition of an angle bisector.
Notice that we have two lines, c and d, and a transversal t that form corresponding angles, ∠ 2 and ∠ 3, that are congruent. Thus, we can use the Converse of the Corresponding Angles Theorem and conclude that c ∥ d.
Statement 4)& c ∥ d Reason 4)& Converse of the
& Corresponding Angles Theorem
Finally, we can complete our two-column proof.
