The idea, step by step
Attach a reason to every step
Start with the given facts, use named properties and reach the requested conclusion. Reasons may include alternate angles in parallel lines, the triangle angle sum, equal base angles, or equal radii.
Establish an intermediate fact before using it. A correct angle with no explanation does not show whether you identified the relevant relationships.
Draw only the justified conclusion
Two matching angles show similarity, not necessarily congruence. Matching size information is also needed for congruence. Enlargement preserves angle relationships while changing size.
For a construction proof, identify the distances created by the compass. For a transformation, state the preserved properties. Avoid “it looks right” and arguments that assume the result they are meant to prove.
A common trap
Equal angles make triangles the same size.
Instead: They establish similarity. Congruence also needs corresponding size information.
See the thinking
Transfer an angle first
The line through A is parallel to BC. The angle between AC and the upper line is 35°; B is 65°. Find A.
- C=35° by alternate angles in parallel lines.
- A+B+C=180° by the triangle angle sum.
- A=180°−65°−35°=80°.
Use equal sides
AB=AC and angle A=40°. Find B and C.
- Equal sides give equal opposite base angles.
- B+C=180°−40°=140°.
- Each base angle is 140°÷2=70°.
Check your understanding
An isosceles triangle has two base angles of 72°. Find the vertex angle.
Show answer
36°
The base angles total 144°, leaving 36°.
Your turn
6 questions · 17 marks · Work at your own pace.
Read the questions online
- An isosceles triangle has vertex angle 50°. Find both base angles with reasons.3 marks
- Find the angle adjacent to 124° on a straight line and name the fact used.2 marks
- Equilateral triangles have side lengths 2 cm and 5 cm. Are they similar, congruent or neither? Explain.3 marks
- Why is a crossing U in the equal-radius bisector construction equally far from A and B?2 marks
- AB has midpoint M. Explain why a positive enlargement maps M to the midpoint of the image segment.3 marks
- A student says “two corresponding angles are equal, so the triangles are congruent”. Correct this and give sufficient extra information.4 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- An isosceles triangle has vertex angle 50°. Find both base angles with reasons.3 marks
Equal sides give equal base angles.
The remaining sum is 180°−50°=130°.
130°÷2=65°.
65° each. - Find the angle adjacent to 124° on a straight line and name the fact used.2 marks
180°−124°=56°.
56°; adjacent angles on a line total 180°. - Equilateral triangles have side lengths 2 cm and 5 cm. Are they similar, congruent or neither? Explain.3 marks
All angles are 60°, so they are similar.
Their corresponding lengths differ.
Similar, not congruent. - Why is a crossing U in the equal-radius bisector construction equally far from A and B?2 marks
The unchanged compass radius makes both distances equal.
UA and UB are radii drawn with the same compass width. - AB has midpoint M. Explain why a positive enlargement maps M to the midpoint of the image segment.3 marks
Enlargement preserves collinearity.
Equal lengths multiplied by the same factor remain equal.
The equal lengths AM and MB scale by the same factor and the three image points remain collinear. - A student says “two corresponding angles are equal, so the triangles are congruent”. Correct this and give sufficient extra information.4 marks
The third angles match by the triangle sum.
Matching angles allow different sizes.
One equal corresponding side fixes the similar triangles at the same size.
They are similar. An equal corresponding side would then fix the scale at 1 and establish congruence.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Curriculum
Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.
Official curriculum source ↗