The idea, step by step
State the angle reason
A diagram can suggest equal angles without proving them. Use a stated relationship: corresponding or alternate angles with parallel lines, vertically opposite angles, or the angle sum in a triangle.
Write each conclusion beside its reason. To use the converse of an angle fact, state that the relevant equal angles imply the two lines are parallel. Do not assume parallelism from appearance.
Build similarity from angles
Two equal corresponding angles are enough for AA similarity because the third pair also sums to the same total. Write vertices in matching order before forming side ratios.
Similarity gives equal angles and proportional lengths, not necessarily equal lengths. Establish similarity first, then apply the common scale factor.
A common trap
Treating an equal side ratio as sufficient proof of similarity.
Instead: One pair of proportional sides alone proves nothing. Use AA, three proportional sides, or two proportional sides with their included angle equal.
See the thinking
Parallel lines
Two parallel lines are cut by a transversal. One acute angle is 58°. Find the corresponding acute angle and the adjacent obtuse angle.
- Corresponding angles on parallel lines are equal, so the acute angle is 58°.
- Angles on a straight line sum to 180°, so the obtuse angle is 122°.
Similar triangles
Triangles ABC and DEF have ∠A=∠D=90°, AB=6, AC=4, DE=9, DF=6. Explain similarity and find the larger hypotenuse.
- and , with equal included angles.
- SAS similarity gives correspondence A↔D, B↔E, C↔F.
- The larger hypotenuse is .
Check your understanding
Two triangle angles are 47° and 68°. Find the third angle.
Show answer
65°
180−47−68=65.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- Two parallel lines are cut by a transversal. An alternate angle is 72°. Find its matching alternate angle and state the reason.2 marks
- A triangle has angles 36° and 91°. Find the third.2 marks
- Triangles have angle sets 50°, 60°, 70° and 70°, 50°, 60°. Are they similar? Explain.2 marks
- Similar triangles ABC and DEF have A↔D, B↔E, C↔F. AB=8, DE=12, BC=10. Find EF.2 marks
- A transversal makes equal corresponding angles with two lines. What can be concluded about the lines?2 marks
- A student says two triangles are similar because both contain a 40° angle. Give a counterexample using angle sets.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Two parallel lines are cut by a transversal. An alternate angle is 72°. Find its matching alternate angle and state the reason.2 marks
Alternate angles between parallel lines are equal.
72° - A triangle has angles 36° and 91°. Find the third.2 marks
180−36−91=53.
53° - Triangles have angle sets 50°, 60°, 70° and 70°, 50°, 60°. Are they similar? Explain.2 marks
Match equal angles; AA similarity applies.
Yes. - Similar triangles ABC and DEF have A↔D, B↔E, C↔F. AB=8, DE=12, BC=10. Find EF.2 marks
Scale factor 12/8=1.5.
EF=10×1.5=15.
15 - A transversal makes equal corresponding angles with two lines. What can be concluded about the lines?2 marks
Use the converse of the corresponding-angle theorem.
They are parallel. - A student says two triangles are similar because both contain a 40° angle. Give a counterexample using angle sets.3 marks
Both contain 40° but the other angles differ.
One equal angle is insufficient.
For example 40°, 60°, 80° and 40°, 70°, 70°.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗