Supplementary and congruent angles are right angles.
See solution.
Practice makes perfect
Let's begin by analyzing the construction of a line perpendicular to line l. We begin with the line l and with a compass, we mark the points A and B using the same compass setting.
Then, we make the compass opening greater than AP, put the compass point on A, and draw an arc. With the same compass setting, we draw an arc from point B and name the point of intersection C.
Now, let's answer the questions from the exercise.
How Can You Use Congruent Triangles?
Notice that if we draw the segments between the points A, B, and C, we will get two congruent triangles.
Considering the congruent angles can help us justify the construction of a perpendicular line.
Which Lengths or Distances Are Equal?
We used the same compass setting to identify the points A and B. Therefore, we can conclude that AP ≅ BP. We also used the same compass setting to identify the point C. Thus, AC ≅ BC. Finally, from the fact that the triangles share the side PC and from the Reflexive Property of Congruence, PC ≅ PC.
AP ≅ BP, AC ≅ BC, and PC ≅ PC
By the Side-Side-Side (SSS) Theorem, we can conclude that △ APC ≅ △ BPC. Corresponding parts of congruent triangles are congruent. Therefore, ∠ APC ≅ ∠ BPC. Notice that these angles form a linear pair, so they are supplementary angles. If two angles are supplementary and congruent, then they are right angles.
Therefore, by the definition of perpendicular lines, CP ⊥ l.
