Explaining geometric reasoning | Year 9

01 / LEARN

Understand the idea.

Attach a reason to every step

A geometric explanation links given facts to a conclusion through justified steps. Record the relevant property beside each equation: for example, alternate angles in parallel lines, triangle angle sum, equal base angles or equal compass radii.

If the needed angle is not given directly, establish it first. You can then use it in the next equation. This prevents a correct numerical result from concealing an unsupported assumption about the diagram.

Draw only the justified conclusion

Draw only the conclusion supported by the evidence. Equal corresponding angles establish triangle similarity, which allows different sizes. Congruence additionally needs enough matching size information, such as the conditions of a valid congruence test.

For a construction, identify the equal distances created by the compass. For a transformation, name the properties it preserves. A convincing argument explains why the result follows, rather than assuming it or relying on how accurately the sketch was drawn.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Transfer the given angle to the triangle.
$C = 35^{\circ}$
The upper line is parallel to $B C$ , so the alternate interior angles made by transversal $A C$ are equal.
2
Write the triangle angle sum.
$A + 65^{\circ} + 35^{\circ} = 180^{\circ}$
The three interior angles of triangle $A B C$ total $180^{\circ}$ .
3
Calculate the angle at A.
$\begin{aligned} A & = 180^{\circ} - 65^{\circ} - 35^{\circ} \\ = 80^{\circ} \end{aligned}$
Subtract the two established angles from the triangle total.

Answer

80°, with both reasons stated.

EXAMPLE 02

Use equal sides

AB=AC and angle A=40°. Find B and C.

Worked solution

1
Use the equal sides.
$B = C$
Since $A B = A C$ , their opposite base angles $C$ and $B$ are equal.
2
Find the total of the base angles.
$B + C = 180^{\circ} - 40^{\circ} = 140^{\circ}$
Subtract the vertex angle from the triangle angle sum.
3
Share the total equally.
$B = C = \frac{140^{\circ}}{2} = 70^{\circ}$
The two equal base angles each occupy half the remaining angle sum.
4
Check all three angles.
$40^{\circ} + 70^{\circ} + 70^{\circ} = 180^{\circ}$
The result satisfies both the isosceles condition and the triangle sum.

Answer

B=C=70°. Equal base angles and triangle sum justify the result.

03 / YOUR TURN

Try it yourself.

Check your understanding

An isosceles triangle has two base angles of 72°. Find the vertex angle.

Show answer & reasoning

36°

The two base angles total $72^{\circ} + 72^{\circ} = 144^{\circ}$ . Subtracting from the triangle sum gives the vertex angle $180^{\circ} - 144^{\circ} = 36^{\circ}$ .

Keep practising

6 worksheet questions · 17 marks

Open worksheet Worked solutions

Read the questions online

An isosceles triangle has vertex angle $50^{\circ}$ . Find both base angles with reasons.
3 marks
Find the angle adjacent to $124^{\circ}$ 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
$A B$ 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 $50^{\circ}$ . Find both base angles with reasons.
3 marks
Equal sides give equal base angles.
The remaining sum is $180^{\circ} - 50^{\circ} = 130^{\circ}$ .
$130^{\circ} \div 2 = 65^{\circ}$ .
$65^{\circ}$ each.
Find the angle adjacent to $124^{\circ}$ on a straight line and name the fact used.
2 marks
$180^{\circ} - 124^{\circ} = 56^{\circ}$ .
$56^{\circ}$ ; adjacent angles on a line total $180^{\circ}$ .
Equilateral triangles have side lengths 2 cm and 5 cm. Are they similar, congruent or neither? Explain.
3 marks
All angles are $60^{\circ}$ , 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.
$U A$ and $U B$ are radii drawn with the same compass width.
$A B$ 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 $A M$ and $M B$ 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.

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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 ↗

In this lesson

Understand the idea.

Attach a reason to every step

Draw only the justified conclusion

See the method in action.

Transfer an angle first

Transfer the given angle to the triangle.

Write the triangle angle sum.

Calculate the angle at A.

Use equal sides

Use the equal sides.

Find the total of the base angles.

Share the total equally.

Check all three angles.

Try it yourself.

Check your understanding

Keep practising