Exercise 16 Page 146

Practice makes perfect

a We are asked to identify two pairs of parallel lines formed on a tennis court. Recall that parallel lines do not intersect and are coplanar.

We can identify three pairs of parallel lines in the diagram. Two of these pairs are in different planes. The first plane consists of the whole court and the second plane includes the net.

The example pairs are k and m as well as p and q. Keep in mind that there are more possible pairs that meet the conditions.

b We are asked to find two pairs of perpendicular lines. Recall that perpendicular lines form a 90^(∘)-angle — also known as a right angle — at their point of intersection.

In the case of the given diagram, lines m and n form a right angle. Similarly, lines k and n also form a right angle. Therefore, these are two pairs of perpendicular lines.

c We want to identify two pairs of skew lines in the given diagram. Recall that lines are called skew when they do not intersect and are not coplanar. Let's take a look at the given diagram.

Note that line l is skew to all other lines in the diagram, because it is located in a different plane and does not intersect with any other line. Therefore, we can pair l with two arbitrary lines, for example q and m.

d To prove that ∠ 1 ≅ ∠ 2, let's first look at the tennis court from above and focus on the points of intersection of line n with lines k and m.

From the diagram, we note that lines m and n form a right angle. Therefore, by the definition of perpendicular lines, m and n are perpendicular lines. Therefore, again by the definition of perpendicular lines, all four angles formed around their point of intersection are right angles.

By following the same reasoning, all four angles formed around the point of intersection of lines k and n are also right angles.

Since all right angles are congruent, we conclude that ∠ 1 and ∠ 2 are congruent angles.

Completed Proof

Considering the given information, we can summarize all the steps in a paragraph proof. Given& Linesn andk form a right angle, & Linesn andm form a right angle Prove & ∠ 1 ≅ ∠ 2 Proof. By the definition of perpendicular lines, since n and k form a right angle, they are perpendicular lines and all four angles around their point of intersection are right angles. Following the same reasoning, all four angles around the point of intersection of n and m are right angles. Since all right angles are congruent ∠ 1 anf ∠ 2 are congruent.

Subchapter links

Exercises p.146

Exercise 16 Page 146

Hints

Check the answer

Practice exercises

Asses your skill

Completed Proof