Exercise 36 Page 180

a According to the Corresponding Angles Theorem, if two parallel lines are cut by a transversal, then the pair of corresponding angles are congruent. Let's illustrate this.

We want to show that

∠ 1 ≅ ∠ 2

using translations.

\overline{A B}

and

\overline{C D}

are parallel segments, we can view them as translations of each other. In other words, we can actually move one so that it covers the other

Because a translation is a rigid motion, and a rigid motion preserves length and angle measures, we know that the angle that

\overline{A B}

creates with the transversal, will be preserved when it's translated to

\overline{C D} .

Therefore, we can say that

∠ 1 ≅ ∠ 2

b According to the Corresponding Angles Converse, if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. Here we can use the same reasoning as in the previous question. Let's draw the diagram.

We want to show that

\overline{A B} ∥ \overline{C D}

using translations. If we translate

B

until it reaches the same intersection point with the transversal as

\overline{C D},

then the marked angle of the image will be congruent with it's preimage.

Since the two lines create the same angle with the transversal, they overlap and are parallel.

Subchapter links

Communicate Your Answer p.173

Monitoring Progress p.175-177

Exercises p.178-180