29. The Right-Angled Triangle · Meet the hypotenuse & the 90° corner

Try to break it

Try dragging P or R so the triangle gets very skinny or very tall. Does the corner at Q stay exactly 90°? Watch the angle measures at P and R add up to 90° every time!

How you build it

Make a right-angled triangle.

• Place point Q — the corner that will hold the right angle.
• Place point P a little way from Q. Segment QP will be the first leg of the triangle.
• Draw segment QP (Segment tool: click Q, then P) — the first leg of the triangle.
• Switch to the Perp tool. Click on Q, then click a little above the segment — it draws a line exactly square to QP, with a right-angle mark.
• Place point R on the perpendicular line. QR becomes the second leg, and the angle at Q is a true 90°.
• Draw segment QR (Segment tool: click Q, then R) — the second leg.
• Draw the hypotenuse PR (Segment tool: click P, then R) — the longest side, opposite the right angle at Q.

The proof, step by step

Prove that the triangle is right-angled at Q.

• The three angles of any triangle always add up to 180°.
• In the figure, the two acute angles ∠P and ∠R add up to 90°.
• So ∠Q = 180° − (∠P + ∠R) = 180° − 90° = 90°.
• An angle of exactly 90° is a right angle — so the triangle is right-angled at Q, and side PR opposite it is the hypotenuse.

Worked example

In right-angled triangle XYZ, ∠Y = 90°. If ∠X = 35°, what is the measure of ∠Z?

In a right-angled triangle, the sum of the two acute angles is always 90°. Therefore, ∠Z = 90° - 35° = 55°.

• 35°
• 45°
• 55° — correct
• 90°