Angle and similarity reasoning | Year 10

01 / LEARN

Understand the idea.

State the angle reason

An angle calculation needs a reason linking the given information to the unknown. Parallel lines give equal corresponding and alternate angles; vertically opposite angles are equal; angles on a straight line total $180^{\circ}$ .

Check the hypotheses before using a theorem. Equal corresponding angles can prove lines parallel by the converse theorem, but lines that merely look parallel cannot justify an angle equality.

Build similarity from angles

To prove similarity, establish AA, SAS similarity or SSS similarity. AA needs two matching angle pairs; SAS needs two proportional side pairs and equality of the angle between them.

Write vertices in corresponding order before forming ratios. All corresponding lengths use one scale factor; angles stay equal. A scale factor different from $1$ changes size, so similar triangles need not be congruent.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Match the corresponding acute angles.
$θ = 58^{\circ}$
Corresponding angles are equal because the two lines are parallel.
2
Subtract from a straight angle.
$ϕ = 180^{\circ} - 58^{\circ} = 122^{\circ}$
The adjacent acute and obtuse angles form a straight line.

Answer

58° and 122°

EXAMPLE 02

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.

Worked solution

1
Compare the two pairs of legs.
$\begin{aligned} \frac{A B}{D E} & = \frac{6}{9} = \frac{2}{3} \\ \frac{A C}{D F} & = \frac{4}{6} = \frac{2}{3} \end{aligned}$
Both corresponding leg pairs have the same ratio.
2
Check the angle between those legs.
$∠ B A C = ∠ E D F = 90^{\circ}$
The included angles are the given right angles.
3
State the similarity in matching order.
$△ A B C \sim △ D E F$
SAS similarity matches A with D, B with E and C with F.
4
Find the larger hypotenuse.
$\begin{aligned} E F^{2} & = 9^{2} + 6^{2} = 117 \\ E F & = \sqrt{117} = 3 \sqrt{13} \end{aligned}$
Pythagoras applies in the larger right triangle; a length takes the positive root.

Answer

$3 \sqrt{13}$ ; triangles are similar.

03 / YOUR TURN

Try it yourself.

Check your understanding

Two triangle angles are 47° and 68°. Find the third angle.

Show answer & reasoning

65°

The three interior angles total $180^{\circ}$ , so the missing angle is $180^{\circ} - 47^{\circ} - 68^{\circ} = 65^{\circ}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Two parallel lines are cut by a transversal. An alternate angle is $72^{\circ}$ . Find its matching alternate angle and state the reason.
2 marks
A triangle has angles $36^{\circ}$ and $91^{\circ}$ . Find the third.
2 marks
Triangles have angle sets $50^{\circ}, 60^{\circ}, 70^{\circ}$ and $70^{\circ}, 50^{\circ}, 60^{\circ}$ . Are they similar? Explain.
2 marks
Similar triangles ABC and DEF have $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$ . $A B = 8, D E = 12, B C = 10$ . 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 $40^{\circ}$ 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 $72^{\circ}$ . Find its matching alternate angle and state the reason.
2 marks
Alternate angles between parallel lines are equal.
$72^{\circ}$
A triangle has angles $36^{\circ}$ and $91^{\circ}$ . Find the third.
2 marks
$180 - 36 - 91 = 53$ .
$53^{\circ}$
Triangles have angle sets $50^{\circ}, 60^{\circ}, 70^{\circ}$ and $70^{\circ}, 50^{\circ}, 60^{\circ}$ . Are they similar? Explain.
2 marks
Match equal angles; AA similarity applies.
Yes.
Similar triangles ABC and DEF have $A \leftrightarrow D, B \leftrightarrow E, C \leftrightarrow F$ . $A B = 8, D E = 12, B C = 10$ . Find EF.
2 marks
Scale factor $\frac{12}{8} = 1.5$ .
$E F = 10 \times 1.5 = 15$ .
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 $40^{\circ}$ angle. Give a counterexample using angle sets.
3 marks
Both contain $40^{\circ}$ but the other angles differ.
One equal angle is insufficient.
For example $40^{\circ}, 60^{\circ}, 80^{\circ}$ and $40^{\circ}, 70^{\circ}, 70^{\circ}$ .

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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 ↗

In this lesson

Understand the idea.

State the angle reason

Build similarity from angles

See the method in action.

Parallel lines

Match the corresponding acute angles.

Subtract from a straight angle.

Similar triangles

Compare the two pairs of legs.

Check the angle between those legs.

State the similarity in matching order.

Find the larger hypotenuse.

Try it yourself.

Check your understanding

Keep practising