GSEB Class 6 Maths Solutions Chapter 5 Understanding Elementary Shapes Exercise 5.5

Question 1. Which of the following are models for perpendicular lines?
(a) The adjacent edges of a table top.
(b) The lines of a railway track.
(c) The line segments forming the letter 'L'.
(d) The letter V.
Answer:
(a) Yes, the nearby sides of a table surface create an example of perpendicular lines.
(b) No, because parallel lines never cross each other, train track lines do not form an example of perpendicular lines.
(c) Yes, the line parts that make up the letter 'L' are perpendicular lines.
(d) No, the line sections that form the letter 'V' do not show perpendicular lines.
In simple words: Perpendicular lines meet at a perfect right angle (90 degrees). A table's adjacent edges and the letter 'L' form such angles. Train tracks are parallel and never meet, and the letter 'V' forms an acute angle, so these are not perpendicular.

Exam Tip: To identify perpendicular lines, always look for a 90-degree angle at the point where two lines intersect.

Question 2. Let \( \overline{\mathrm{OP}} \) be the perpendicular to the line segment \( \overline{\mathrm{XY}} \). Let \( \overline{\mathrm{PQ}} \) and \( \overline{\mathrm{XY}} \) intersect in the point A. What is the measure of \( \angle PAY \)?
Answer: Since line segment \( \overline{\mathrm{OP}} \) is perpendicular to line segment \( \overline{\mathrm{XY}} \), it signifies that these two lines intersect at a 90-degree angle. The question indicates that \( \overline{\mathrm{PQ}} \) (which includes \( \overline{\mathrm{OP}} \) as part of the line that continues to Q) and \( \overline{\mathrm{XY}} \) meet at point A. Therefore, the angle \( \angle PAY \) will precisely measure 90 degrees, as perpendicular lines always create right angles where they cross.
In simple words: When two lines are perpendicular, they cross each other at a perfect square corner, which is a 90-degree angle. So, \( \angle PAY \) is 90 degrees.

Exam Tip: Remember that perpendicular lines always intersect at a right angle, which measures exactly 90 degrees.

Question 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?
Answer: The triangles formed by the corners of the two set-squares are shown below:The angle sizes of the first triangle are: 30°, 60° and 90°. The angle sizes of the second triangle are: 45°, 45° and 90°. Yes, they both share a 90° angle.
In simple words: One set-square has angles of 30°, 60°, and 90°. The other set-square has angles of 45°, 45°, and 90°. They both share a 90-degree angle.

Exam Tip: Know the standard angle measurements for the two types of set-squares: (30°, 60°, 90°) and (45°, 45°, 90°).

Question 4. Study the diagram. The line l is perpendicular to line m

(a) is CE = EG?
(b) Does PE bisect CG?
(c) Identify any two line segments for which PE is the perpendicular bisector.
(d) Are these true?
(i) AC > FG
(ii) CD = GH
(iii) BC < EH
Answer:
(a) E is at 4, C is at 2, G is at 6. So, the distance from C to E is 2 units, and the distance from E to G is also 2 units. This means CE and EG are equal in length.
(b) Yes, the line segment PE cuts the line segment CG exactly in half and is at a 90-degree angle to it, because E (at 4) is the midpoint of CG (from 2 to 6).
(c) Two line segments for which PE acts as the perpendicular bisector are CG and DF. (The midpoint of CG is \( \frac{(2+6)}{2} = 4 \); the midpoint of DF is \( \frac{(3+5)}{2} = 4 \), and PE passes through 4 and is perpendicular to line m, on which these segments lie.)
(d)
(i) Yes, AC is 2 units (from 0 to 2) and FG is 1 unit (from 5 to 6), so 2 > 1.
(ii) Yes, CD is 1 unit (from 2 to 3) and GH is 1 unit (from 6 to 7), so they are equal.
(iii) Yes, BC is 1 unit (from 1 to 2) and EH is 3 units (from 4 to 7), so 1 < 3.
In simple words: By looking at the number line, we confirmed that CE and EG have the same length. PE cuts CG into two equal pieces at a right angle. We also found that the lengths of AC, CD, and BC compare correctly to FG, GH, and EH.

Exam Tip: Carefully read coordinate points from diagrams to calculate lengths and midpoints accurately. Always double-check your calculations against the visual representation.