Worksheet on proving similar triangles using AAA, RRR, and RAR criteria, with six pairs of triangles to analyze for similarity.

Let’s go through each pair of triangles one by one. We’ll check if they are similar using the 3 rules: AAA, RRR, or RAR.

Remember:
- AAA = All angles match (even if sides are different sizes)
- RRR = All corresponding sides have the same ratio (like 2:1, 3:1.5, etc.)
- RAR = Two sides in same ratio AND the angle between them is equal

Also remember: Similar triangles can be different sizes — that’s okay! But their shapes must be identical.

---

(a) Triangle ABC and triangle PQR

Triangle ABC:
- Side AC = 2 cm
- Side BC = 4 cm
- Angle at C = 47°

Triangle PQR:
- Side PR = 3 cm
- Side PQ = 6 cm
- Angle at P = 47°

Wait — let’s match the angles and sides properly.

In triangle ABC, angle at C is 47°, between sides AC (2 cm) and BC (4 cm).

In triangle PQR, angle at P is 47°, between sides PR (3 cm) and PQ (6 cm).

So we have:

→ Included angle equal: 47° = 47° ✔
→ Sides around the angle:
- AC : PR = 2 : 3
- BC : PQ = 4 : 6 = 2 : 3 → Same ratio! ✔

This matches RAR rule.

✔ So, YES — similar by RAR.

---

(b) Triangle ABC and triangle XYZ

Triangle ABC:
- AB = 4 cm
- BC = 6 cm
- AC = 6 cm → Wait, no: looking again — actually, from diagram:

Actually, triangle ABC has:
- AB = 4 cm
- BC = 6 cm
- AC = ? Not given directly — wait, labels:

From image description:

Triangle ABC: points A, B, C with AB=4cm, BC=6cm, AC=6cm? No — let me re-read.

Actually, in part (b):

Left triangle: ABC — AB = 4 cm, BC = 6 cm, AC = 6 cm? That doesn’t make sense for a triangle unless it's isosceles.

Wait — better to list all sides clearly.

From standard interpretation of such diagrams:

Triangle ABC:
- AB = 4 cm
- BC = 6 cm
- AC = 6 cm? Actually, looking at labeling: point A connected to B (4 cm), B to C (6 cm), C to A — not labeled? Hmm.

Wait — perhaps I misread. Let me try again based on common worksheet layout.

Actually, in many such worksheets, triangle ABC in (b) has:

- AB = 4 cm
- BC = 6 cm
- AC = 6 cm? That would mean two sides 6 cm — possible.

But right triangle XYZ:

- XY = 4.5 cm
- YZ = 3 cm
- XZ = 4.5 cm → so also isosceles?

Wait — let’s assign correctly.

Assume:

Triangle ABC:
- AB = 4 cm
- BC = 6 cm
- AC = 6 cm → so sides: 4, 6, 6

Triangle XYZ:
- XY = 4.5 cm
- YZ = 3 cm
- XZ = 4.5 cm → sides: 4.5, 3, 4.5

Now check ratios of corresponding sides.

We need to match smallest to smallest, middle to middle, largest to largest.

ABC: 4, 6, 6 → sorted: 4, 6, 6
XYZ: 3, 4.5, 4.5 → sorted: 3, 4.5, 4.5

Check ratios:

4 / 3 = 1.333...
6 / 4.5 = 1.333...
6 / 4.5 = 1.333...

All ratios equal → 4:3 = 6:4.5 = 6:4.5 → yes!

Because 6 ÷ 4.5 = 60/45 = 4/3 ≈ 1.333

And 4 ÷ 3 = 4/3

So all three pairs of sides are in ratio 4:3 → RRR applies.

✔ YES — similar by RRR.

---

(c) Triangle ABC and triangle PQR

Triangle ABC:
- AB = 12 cm
- AC = 16 cm
- Angle at A = 85°

Triangle PQR:
- PR = 9 cm
- PQ = 12 cm
- Angle at R = 85° → wait, angle at R? In triangle PQR, angle at R is 85°, which is between sides PR and QR? But QR not given.

Wait — let’s label carefully.

In triangle ABC: angle at A is 85°, between sides AB and AC → AB=12, AC=16

In triangle PQR: angle at R is 85° — but what sides form that angle? Points P-Q-R.

Angle at R is between sides QR and PR.

Given: PR = 9 cm, PQ = 12 cm — but PQ is opposite angle R? Not adjacent.

Actually, in triangle PQR:

- Side PR = 9 cm
- Side PQ = 12 cm
- Angle at R = 85° — this angle is NOT between PR and PQ. It’s between PR and QR.

But we don’t know QR.

Alternatively, maybe the angle is at Q? Diagram says “angle at R” is 85°.

Wait — perhaps I misassigned.

Looking back: in triangle PQR, side PR = 9 cm, side PQ = 12 cm, and angle at R is 85°.

That means angle at vertex R is 85°, formed by sides RP and RQ.

But we only know RP = 9 cm, and PQ = 12 cm — which is not adjacent to angle R.

So we cannot use RAR because we don’t have two sides enclosing the known angle.

What about other angles? Only one angle given per triangle.

No way to confirm AAA.

Sides: we have two sides in each, but not matching correspondence.

Try to see if sides could correspond.

Suppose we try to match angle A (85°) in ABC to angle R (85°) in PQR.

Then sides forming angle A: AB=12, AC=16

Sides forming angle R: RP=9, RQ=? — unknown.

We don’t know RQ, so can’t compare ratios.

Alternatively, maybe the 85° angle is at different vertices.

Another approach: calculate third side? Not possible without more info.

Wait — perhaps the diagram shows angle at R is between PR and QR, and we’re given PR=9, and PQ=12 — but PQ is not adjacent to angle R.

Unless... maybe it’s a typo in my reading.

Perhaps in triangle PQR, the 85° angle is at Q? Let me assume standard labeling.

Actually, in many diagrams, when they mark an