The idea, step by step
The right angle fixes the hypotenuse
The hypotenuse is the side opposite the 90° angle. It is always the longest side of a right triangle, even when the drawing is rotated or is not to scale.
The other two sides meet at the right angle. The two remaining angles are acute and add to 90°. Mark this information before using a formula.
Opposite and adjacent depend on the angle
Choose the acute reference angle. The opposite side does not touch it. The adjacent leg touches it, but is not the hypotenuse.
Changing the reference angle exchanges the opposite and adjacent legs; it does not change the hypotenuse. A useful sketch with named vertices prevents side labels drifting away from their meaning.
A common trap
Calling the bottom side adjacent without first choosing an angle.
Instead: Adjacent describes a relationship to the reference angle, not a position on the page.
See the thinking
Worked example 1
Triangle ABC has B=90° and A=34°. Name the hypotenuse and find C.
- The side opposite B is AC, so AC is the hypotenuse.
- The two acute angles sum to 90°.
- C=90°−34°=56°.
Worked example 2
In the same triangle, identify opposite and adjacent legs first for angle A, then for angle C.
- AC remains the hypotenuse for both choices.
- BC does not touch A; AB is the leg touching A.
- After choosing C, the roles of AB and BC exchange.
Check your understanding
A triangle has angles 90° and 27°. Find its third angle.
Show answer
63°
180°−90°−27°=63°.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Triangle PQR has Q=90°. Which side is the hypotenuse?2 marks
- In right triangle ABC, B=90° and A=41°. Find C.2 marks
- Triangle XYZ has Z=90°. Relative to angle X, name the opposite leg and adjacent leg.2 marks
- In the previous triangle, choose Y instead. Which leg is now opposite?2 marks
- Could a right triangle have another angle of 94°? Explain.3 marks
- A right triangle is drawn upside down. A student calls its shortest side the hypotenuse. Explain the correction.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Triangle PQR has Q=90°. Which side is the hypotenuse?2 marks
The hypotenuse is opposite the right angle Q.
PR - In right triangle ABC, B=90° and A=41°. Find C.2 marks
C=180°−90°−41°=49°.
49° - Triangle XYZ has Z=90°. Relative to angle X, name the opposite leg and adjacent leg.2 marks
XY is the hypotenuse.
YZ does not touch X; XZ is the remaining leg touching X.
Opposite YZ; adjacent XZ. - In the previous triangle, choose Y instead. Which leg is now opposite?2 marks
The side not touching Y is XZ; the hypotenuse stays XY.
XZ - Could a right triangle have another angle of 94°? Explain.3 marks
90°+94° already exceeds 180°.
Both non-right angles in a right triangle must be acute.
No. - A right triangle is drawn upside down. A student calls its shortest side the hypotenuse. Explain the correction.3 marks
Orientation does not change side relationships.
Find the marked right angle, then the opposite side.
The hypotenuse is opposite the 90° angle and must be the longest side.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗