Grade 12 Euclidean Geometry • Theorem 6

The Similarity Theorem

The Intuition (Plain English)

If two triangles have exactly the same angles (they are "equiangular"), then they are exactly the same shape. Because they are the same shape, one is just a "zoomed in" or "zoomed out" version of the other. This means their side lengths scale up or down by the exact same ratio.

The Formal Proof

This is the final examinable proof. It cleverly uses Congruency and the Proportion Theorem you just learned to prove Similarity.

📐 [Insert GeoGebra Diagram] Large Triangle ABC. Smaller Triangle DEF.

Given: ΔABC and ΔDEF with Â = D̂, B̂ = Ê, and Ĉ = F̂.

R.T.P: AB/DE = AC/DF

Construction: Mark off P on AB so that AP = DE, and mark off Q on AC so that AQ = DF. Join PQ.

Statement	Reason
1. In ΔAPQ and ΔDEF:
AP = DE	construction
Â = D̂	given
AQ = DF	construction
2. ∴ ΔAPQ ≡ ΔDEF	SAS
3. ∴ P̂₁ = Ê	from congruency
4. But B̂ = Ê	given
5. ∴ P̂₁ = B̂	both equal Ê
6. ∴ PQ ∥ BC	corresponding ∠s are equal
7. ∴ AB/AP = AC/AQ	prop theorem; PQ ∥ BC
8. ∴ AB/DE = AC/DF	since AP=DE and AQ=DF

How to apply this in an exam (The "Cross-Multiplication" Trick)

Once you prove two triangles are similar, you set up a 3-part ratio. The exam question will usually ask you to prove a statement like AB · DF = AC · DE.

Typical Exam Question:
Prove that ΔABC ||| ΔDEF. Hence, deduce that AB × DF = AC × DE.

Step-by-Step Solution:

First, explicitly state the three equal angles to earn your similarity mark.
Â = D̂
B̂ = Ê
Ĉ = F̂
∴ ΔABC ||| ΔDEF (||| ∆s OR equiangular ∆s)
Immediately set up the ratio using the letter order (1st&2nd / 1st&2nd, etc).
AB/DE = BC/EF = AC/DF
Look at what you are required to deduce. It uses AB, DF, AC, DE. Pick the two fractions that contain these letters:
AB/DE = AC/DF
Cross-multiply to get your final answer:
AB · DF = AC · DE

⚠️ Common Exam Traps

Writing the vertices in the wrong order: The statement ΔABC ||| ΔEDF means that B̂ = D̂ and Ĉ = F̂. If you mess up the letter ordering, your entire ratio step will be wrong, and you will lose all the following marks. ALWAYS map the equal angles perfectly: angle 1 to angle 1, angle 2 to angle 2.
Forgetting the third angle: When proving similarity, you only need to prove two pairs of angles are equal. The third pair is automatically equal due to the sum of angles in a triangle. Don't waste time trying to prove the third one with complex circle geometry—just use sum of ∠s in ∆!