38. The Four Right Angles · What makes a rectangle, a rectangle?

Try to break it

Drag corner A and watch the angles. Bend it away from square and the ✓ RECTANGLE badge vanishes — it is no longer a rectangle. That is exactly the definition: four right angles. Break even one and you have broken the rectangle.

How you build it

Construct a rectangle.

• Place point A — the first corner of the rectangle.
• Place point B a good distance from A. AB will be one side of the rectangle.
• Draw side AB (Segment tool: click A, then B).
• With the Perp tool, click point B, then click a little to one side — it raises a line exactly square to AB at B. This fixes a right angle at B.
• Place point C on the perpendicular line. The angle ABC is now exactly 90°.
• Draw side BC (Segment tool: click B, then C).
• With the Parallel tool, click point C, then click on side AB — it draws a line through C parallel to AB.
• With the Parallel tool, click point A, then click on side BC — it draws a line through A parallel to BC.
• Mark point D where the two parallel lines cross — your click snaps onto the exact crossing. That completes rectangle ABCD: with the right angle at B and both pairs of opposite sides parallel, all four angles are 90°.

The proof, step by step

Prove that all four angles of the rectangle are right angles.

• A rectangle is defined as a quadrilateral with four right angles.
• Each angle measures exactly 90°.
• The sum of the four angles is 360°.
• Opposite sides are equal and parallel.

Worked example

In quadrilateral PQRS, ∠P = ∠Q = ∠R = ∠S = 90°. Which of the following best describes PQRS?

By definition, a quadrilateral with four 90° angles is a rectangle. While a square also has four 90° angles, the definition given applies to all rectangles, not just squares.

• A square only
• A rectangle — correct
• A parallelogram with unequal adjacent sides
• A rhombus