Exercise 5.5

1. Which of the following are models for perpendicular lines?

(a) The adjacent edges of a tabletop

(b) The lines of a railway track

(c) The line segments forming the letter ‘L’

(d) The letter V

(a) Yes, adjacent edges of a tabletop are perpendicular.
(b) No, the lines of a railway track are parallel.
(c) Yes, the vertical and horizontal bars of 'L' form a 90° angle.
(d) No, the two lines in 'V' meet at an acute angle.
Correct models are (a) and (c).
2. Let \(\overline{PQ}\) be the perpendicular to the line segment \(\overline{XY}\). Let \(\overline{PQ}\) and \(\overline{XY}\) intersect in the point A. What is the measure of \(\angle PAY\)?
Perpendicular lines PQ and XY
Since \(\overline{PQ} \perp \overline{XY}\), the angle between them is a right angle.
The point of intersection is A.
∴ \(\angle PAY = 90^\circ\)
3. There are two set-squares in your box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common?
Two set-squares
The measures of the angles in the two set-squares are:
(a) 30°, 60° and 90°
(b) 45°, 45° and 90°
Yes, they have a common angle measure of 90°.
4. Study the diagram. The line \(l\) is perpendicular to line \(m\).
Perpendicular lines l and m with points
(a) Is CE = EG?
Yes. Since \(CE = 2\) units (from 3 to 5) and \(EG = 2\) units (from 5 to 7).

(b) Does PE bisect CG?
Yes, because E is the midpoint of CG (\(CE = EG\)).

(c) Identify any two line segments for which PE is the perpendicular bisector.
The line segments are \(\overline{BG}\) (since \(BE = EG = 3\) units)

and

\(\overline{DF}\) (since \(DE = EF = 1\) unit).

(d) Are these true?
(i) AC > FG   →   True (\(2 > 1\))
(ii) CD = GH   →   True (\(1 = 1\))
(iii) BC < EH   →   True (\(1 < 3\))