Perpendicular Lines
Semester 4B · Chapter 16 · Lesson 1 · MATH-4B-C16-L01
Before you begin: prerequisite check
Recall: 15.01 right angles. Attempt these questions independently first.
What angle do a horizontal and vertical line make when they cross?
Show answer and reasoning
The two lines meet at a right angle.
How many sides does a quadrilateral have?
Show answer and reasoning
Quad means four; a quadrilateral is a four-sided polygon.
Does a square have four right angles?
Show answer and reasoning
Every interior angle of a square is 90°.
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
Two lines are perpendicular when they meet at an exact right angle, 90°. A pair of vertical and horizontal grid lines is an easy example, but a slanted pair can also be perpendicular. Line length and position do not determine the right-angle relationship. Mark the right angle with a small square at the intersection. Check using a set square or a ruler-based construction, and be careful not to call nearly-right angles perpendicular without evidence.
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
Two lines meet at a 75° angle. Are they perpendicular? Explain.
Only a 90° intersection is perpendicular.
Check 1: try it yourself
Two lines meet at a 160° angle. Are they perpendicular? Explain.
Show answer and reasoning
Only a 90° intersection is perpendicular.
Worked example 2
One angle at a crossing is 95°. Can the two entire straight lines be perpendicular?
Perpendicular straight lines form four right angles.
Check 2: try it yourself
One angle at a crossing is 50°. Can the two entire straight lines be perpendicular?
Show answer and reasoning
Perpendicular straight lines form four right angles.
Worked example 3
An angle beside another makes 180°. If the first angle is 120°, find the neighbor and decide if the lines are perpendicular.
Adjacent angles total 180°. Both are right angles only when the first is 90°.
Check 3: try it yourself
An angle beside another makes 180°. If the first angle is 30°, find the neighbor and decide if the lines are perpendicular.
Show answer and reasoning
Adjacent angles total 180°. Both are right angles only when the first is 90°.
Guided practice
Two lines meet at a 110° angle. Are they perpendicular? Explain.
Show answer and reasoning
Only a 90° intersection is perpendicular.
One angle at a crossing is 110°. Can the two entire straight lines be perpendicular?
Show answer and reasoning
Perpendicular straight lines form four right angles.
An angle beside another makes 180°. If the first angle is 80°, find the neighbor and decide if the lines are perpendicular.
Show answer and reasoning
Adjacent angles total 180°. Both are right angles only when the first is 90°.
If an intersection has one angle 25°, is that enough evidence of a right-angle meeting?
Show answer and reasoning
A right-angle meeting requires an angle of exactly 90°.
A carpenter checks two beams and measures 100° where they meet. How many degrees must that angle change to become perpendicular?
Show answer and reasoning
Compute the absolute difference between 100° and the target 90°.
Independent practice
An angle beside another makes 180°. If the first angle is 155°, find the neighbor and decide if the lines are perpendicular.
Show answer and reasoning
Adjacent angles total 180°. Both are right angles only when the first is 90°.
One angle at a crossing is 55°. Can the two entire straight lines be perpendicular?
Show answer and reasoning
Perpendicular straight lines form four right angles.
An angle beside another makes 180°. If the first angle is 160°, find the neighbor and decide if the lines are perpendicular.
Show answer and reasoning
Adjacent angles total 180°. Both are right angles only when the first is 90°.
If an intersection has one angle 140°, is that enough evidence of a right-angle meeting?
Show answer and reasoning
A right-angle meeting requires an angle of exactly 90°.
A carpenter checks two beams and measures 90° where they meet. How many degrees must that angle change to become perpendicular?
Show answer and reasoning
Compute the absolute difference between 90° and the target 90°.
Two lines meet at a 50° angle. Are they perpendicular? Explain.
Show answer and reasoning
Only a 90° intersection is perpendicular.
One angle at a crossing is 90°. Can the two entire straight lines be perpendicular?
Show answer and reasoning
Perpendicular straight lines form four right angles.
An angle beside another makes 180°. If the first angle is 135°, find the neighbor and decide if the lines are perpendicular.
Show answer and reasoning
Adjacent angles total 180°. Both are right angles only when the first is 90°.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
Two lines meet at a 155° angle. Are they perpendicular? Explain. Explain or sketch the mathematical relationship.
Show answer and reasoning
Only a 90° intersection is perpendicular. State which quantities or geometric properties justify the method.
A learner claims: “Only vertical and horizontal lines can be perpendicular.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: any intersecting lines that make a 90° angle are perpendicular, even when rotated. Explain the relevant quantities or properties, not merely the final answer.
Exit check
Two lines meet at a 70° angle. Are they perpendicular? Explain.
Show answer and reasoning
Only a 90° intersection is perpendicular.
If an intersection has one angle 150°, is that enough evidence of a right-angle meeting?
Show answer and reasoning
A right-angle meeting requires an angle of exactly 90°.
A carpenter checks two beams and measures 70° where they meet. How many degrees must that angle change to become perpendicular?
Show answer and reasoning
Compute the absolute difference between 70° and the target 90°.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.