Fundamental Concepts
7. The Perfect Cross · meeting at exactly 90 degrees
Lines AB and CD form a right angle at O.
Two lines are perpendicular when they meet at a right angle (exactly 90°). The small green square at O marks the right angle. Drag C — only when ∠BOC = 90° do the lines become perpendicular.
What this lesson covers
Try to break it
Grab point C and drag it around the circle. Watch the angle change! When it stops being 90 degrees, the lines are just intersecting, not perpendicular anymore.
How you build it
Construct a perpendicular using ruler and compass.
- Place point A on the canvas.
- Place point B somewhere else.
- Draw the line through A and B.
- With centre A and a radius wider than half of AB, draw an arc that crosses above and below the line.
- With centre B and the SAME radius you used at A, draw another arc. The two arcs intersect at two points — one above the line, one below.
- Draw the line through the two points where the arcs cross. By construction, this line is perpendicular to AB.
The proof, step by step
Prove that lines AB and CD are perpendicular.
- By definition, two lines are perpendicular if they intersect at an angle of exactly 90 degrees.
- The right-angle marker at point O visually confirms that the angle between the lines is 90 degrees.
- Therefore, the lines satisfy the condition and are perpendicular to each other.
Worked example
Line l and line m intersect at point P. If the angle between them measures exactly 90°, how are lines l and m related?
By definition, two lines that intersect at a right angle (90°) are called perpendicular lines. The 90° measurement directly satisfies this condition.
- They are parallel
- They are perpendicular — correct
- They are coincident
- They are skew