Mathematics • Year 10 • Unit 3 • Lesson 12
Similar Triangles — Tests and Proofs Skill Drill
Build fluency with the three similarity tests from Lesson 12: AAA (two equal angles is enough), SSS ratio (all three sides in the same ratio), and SAS ratio + included angle. Practise writing short formal proofs and using similarity to find unknown sides.
1. I do — fully worked example
Read every step. Each line gives the statement and the reason — that is what a formal proof looks like in Year 10.
Problem. In triangle ABC and triangle DEF, ∠A = ∠D = 50° and ∠B = ∠E = 60°. Prove that the two triangles are similar.
Step 1 — State the given equal angles.
∠A = ∠D = 50° (given), ∠B = ∠E = 60° (given)
Reason: copying down the data shows the marker exactly what you are using.
Step 2 — Find the third pair of angles.
∠C = 180° − 50° − 60° = 70°
∠F = 180° − 50° − 60° = 70°
Reason: angles in a triangle sum to 180°. So ∠C = ∠F.
Step 3 — State the similarity and the test used.
△ABC ~ △DEF (AAA)
Reason: all three pairs of corresponding angles are equal.
Answer: △ABC ~ △DEF (AAA).
2. We do — fill in the missing steps (SSS ratio proof)
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. △PQR has sides PQ = 3 cm, QR = 4 cm, PR = 5 cm. △XYZ has sides XY = 6 cm, YZ = 8 cm, XZ = 10 cm. Prove they are similar and state the scale factor.
Step 1 — Match shortest to shortest, middle to middle, longest to longest:
PQ = 3 ↔ XY = ______ ; QR = 4 ↔ YZ = ______ ; PR = 5 ↔ XZ = ______
Step 2 — Calculate the three ratios:
XY / PQ = ______ / 3 = ______
YZ / QR = ______ / 4 = ______
XZ / PR = ______ / 5 = ______
Step 3 — Check all three ratios are equal: ✓ / ✗ ? ______
Step 4 — Write the conclusion:
△PQR ~ △________ (________), scale factor k = ______
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — pick the correct test
3.1 △ABC has angles 40°, 60°, 80°. △DEF has angles 40°, 60°, 80°. Which similarity test applies? 1 mark
3.2 △ABC has sides 6, 8, 10. △DEF has sides 9, 12, 15. Which test applies? Calculate the common ratio. 2 marks
3.3 △ABC has AB = 4 cm, AC = 6 cm, ∠A = 50°. △DEF has DE = 8 cm, DF = 12 cm, ∠D = 50°. Which test applies? 1 mark
3.4 Why does AAA need only two pairs of equal angles, not three? 1 mark
Standard — find unknown lengths from a similarity
3.5 △ABC ~ △DEF (AAA). AB = 5 cm, BC = 8 cm, DE = 15 cm. Find the scale factor k and the length EF. 2 marks
3.6 A 1.5 m stick casts a 2 m shadow on a sunny day at Bondi. At the same moment, a palm tree casts an 18 m shadow. Find the tree's height. State which similarity test you are using. 3 marks
Extension — write a short proof
3.7 In △ABC, point D lies on AB and point E lies on AC such that DE ∥ BC. Prove that △ADE ~ △ABC. (Hint: parallel lines create equal corresponding angles. Plus, ∠A is common.) 3 marks
3.8 A surveyor walks 3 m from a creek and sights a tree directly across. She walks a further 9 m along the creek bank and the tree is no longer in line. From a 5 m × 12 m × 13 m setup of similar right triangles, she calculates the creek is 12 m wide. Briefly explain how similarity makes this kind of indirect measurement possible. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (3-4-5 ↔ 6-8-10)
Step 1: PQ ↔ XY = 6; QR ↔ YZ = 8; PR ↔ XZ = 10.
Step 2: 6/3 = 2, 8/4 = 2, 10/5 = 2.
Step 3: ✓ all equal 2.
Step 4: △PQR ~ △XYZ (SSS), k = 2.
3.1 — All angles equal
AAA. All three corresponding angles equal — the triangles must be similar.
3.2 — Sides 6,8,10 and 9,12,15
SSS ratio: 9/6 = 12/8 = 15/10 = 3/2. All three ratios equal so the triangles are similar.
3.3 — Two sides and an included angle
SAS. The ratio of the two given sides is 8/4 = 12/6 = 2, and the included angle ∠A = ∠D = 50°.
3.4 — Why two angles is enough
Because the three angles in any triangle must sum to 180°. If two pairs are equal, the third pair is forced by 180° − (first) − (second), which is the same on both sides. So we only need to check two pairs.
3.5 — △ABC ~ △DEF
k = DE / AB = 15 / 5 = 3.
EF = BC × k = 8 × 3 = 24 cm.
3.6 — Stick and palm tree at Bondi
Similar triangles by AAA: both have a right angle at the base and the sun's rays make the same angle of elevation (parallel rays).
Set up the proportion: h / 1.5 = 18 / 2 = 9.
h = 1.5 × 9 = 13.5 m tall.
3.7 — △ADE ~ △ABC (proof)
∠A is common to both triangles.
∠ADE = ∠ABC (corresponding angles, DE ∥ BC).
Therefore △ADE ~ △ABC (AAA).
Two equal angles is enough — the third pair (∠AED = ∠ACB) is automatic.
3.8 — Why similarity allows indirect measurement
Similar triangles have proportional sides. So if two triangles can be set up with the same angles (using parallel lines, the sun's rays, or a sighting line), then once we know any one length in each triangle, every other length follows by multiplication by the scale factor. We can therefore measure something inaccessible (the width of the creek) using something accessible (steps along the bank).