Drawing Perpendicular and Parallel Lines
Semester 4B · Chapter 16 · Lesson 3 · MATH-4B-C16-L03
Before you begin: prerequisite check
Recall: 16.01–16.02. Attempt these questions independently first.
Two drawn straight lines have perpendicular separation 8 cm at one location and 8 cm at another. Does this observation support the claim that their separation is constant?
Show answer and reasoning
Parallel lines must keep a constant perpendicular separation.
Two drawn straight lines have perpendicular separation 5 cm at one location and 7 cm at another. Does this observation support the claim that their separation is constant?
Show answer and reasoning
Parallel lines must keep a constant perpendicular separation.
Two drawn straight lines have perpendicular separation 3 cm at one location and 3 cm at another. Does this observation support the claim that their separation is constant?
Show answer and reasoning
Parallel lines must keep a constant perpendicular separation.
Key vocabulary
- perpendicular
- meeting at a right angle
- parallel
- same direction and no intersection in a plane
- quadrilateral
- four-sided polygon
- line of symmetry
- line reflecting one matching half onto the other
Understand the concept
To draw a perpendicular line, align a set square or a right-angle corner at the chosen point and trace its perpendicular edge. To draw a parallel line, keep the guiding edge in the same direction while sliding to a new location. A grid gives a useful reference but a correctly constructed line must remain parallel or perpendicular even when extended. Leave construction lines or right-angle marks visible so a reader can verify what was drawn.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
Using the vertical reference line on the grid, draw a perpendicular through grid row 2 and another distinct parallel vertical line at least 1 grid spaces away. Label both.
The horizontal makes 90° with the reference and the vertical has equal direction.
Check 1: try it yourself
Using the vertical reference line on the grid, draw a perpendicular through grid row 7 and another distinct parallel vertical line at least 1 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Worked example 2
Draw a line parallel to the red reference 2 grid spaces to its left, then a perpendicular at row 3.
Parallel means equal direction; perpendicular means a 90° intersection.
Check 2: try it yourself
Draw a line parallel to the red reference 3 grid spaces to its left, then a perpendicular at row 2.
Show answer and reasoning
Parallel means equal direction; perpendicular means a 90° intersection.
Worked example 3
Draw a horizontal line at row 2 and explain whether it is perpendicular to the vertical red reference.
Vertical and horizontal grid lines form 90° angles.
Check 3: try it yourself
Draw a horizontal line at row 7 and explain whether it is perpendicular to the vertical red reference.
Show answer and reasoning
Vertical and horizontal grid lines form 90° angles.
Guided practice
Using the vertical reference line on the grid, draw a perpendicular through grid row 5 and another distinct parallel vertical line at least 4 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Construct a new vertical line 4 grid spaces right of the red vertical line. What is their relationship?
Show answer and reasoning
Both lines have the same vertical direction and stay a fixed distance apart.
Construct a rectangle 6 grid spaces wide and 3 grid spaces high using parallel and perpendicular edges.
Show answer and reasoning
Opposite sides are parallel and adjacent sides meet at 90°.
Using the vertical reference line on the grid, draw a perpendicular through grid row 5 and another distinct parallel vertical line at least 2 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Construct a rectangle 3 grid spaces wide and 4 grid spaces high using parallel and perpendicular edges.
Show answer and reasoning
Opposite sides are parallel and adjacent sides meet at 90°.
Independent practice
Using the vertical reference line on the grid, draw a perpendicular through grid row 5 and another distinct parallel vertical line at least 1 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Draw a line parallel to the red reference 2 grid spaces to its left, then a perpendicular at row 6.
Show answer and reasoning
Parallel means equal direction; perpendicular means a 90° intersection.
Construct a rectangle 5 grid spaces wide and 6 grid spaces high using parallel and perpendicular edges.
Show answer and reasoning
Opposite sides are parallel and adjacent sides meet at 90°.
Construct a new vertical line 1 grid spaces right of the red vertical line. What is their relationship?
Show answer and reasoning
Both lines have the same vertical direction and stay a fixed distance apart.
Construct a rectangle 5 grid spaces wide and 4 grid spaces high using parallel and perpendicular edges.
Show answer and reasoning
Opposite sides are parallel and adjacent sides meet at 90°.
Using the vertical reference line on the grid, draw a perpendicular through grid row 7 and another distinct parallel vertical line at least 5 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Draw a line parallel to the red reference 4 grid spaces to its left, then a perpendicular at row 3.
Show answer and reasoning
Parallel means equal direction; perpendicular means a 90° intersection.
Draw a horizontal line at row 6 and explain whether it is perpendicular to the vertical red reference.
Show answer and reasoning
Vertical and horizontal grid lines form 90° angles.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
Using the vertical reference line on the grid, draw a perpendicular through grid row 6 and another distinct parallel vertical line at least 5 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
A learner claims: “An approximately right-angle sketch is sufficient proof of perpendicular lines.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: construct and verify a true 90° angle using a grid, set square, or suitable instrument. Explain the relevant quantities or properties, not merely the final answer.
Exit check
Using the vertical reference line on the grid, draw a perpendicular through grid row 4 and another distinct parallel vertical line at least 3 grid spaces away. Label both.
Show answer and reasoning
The horizontal makes 90° with the reference and the vertical has equal direction.
Draw a line parallel to the red reference 5 grid spaces to its left, then a perpendicular at row 3.
Show answer and reasoning
Parallel means equal direction; perpendicular means a 90° intersection.
Draw a horizontal line at row 5 and explain whether it is perpendicular to the vertical red reference.
Show answer and reasoning
Vertical and horizontal grid lines form 90° angles.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.