Grade 11 Equiangular Triangles: Short Notes

11 ශ්‍රේණිය · 14 පරිච්ඡේදය · සමකෝණී ත්‍රිකෝණ

36%

of Grade 11 term tests include it (22 of 61)

224

marks in past papers (24 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Use the theorem: a line parallel to one side divides the other two sides proportionally
Use the converse to show that two lines are parallel
Show that two triangles are equiangular, giving reasons
Match corresponding sides and write equal ratios correctly
Find unknown lengths and prove product relations such as AB² = BD × BC

Measuring with shadows

On a sunny afternoon in Jaffna, a 1.5 m stick casts a 2 m shadow. At the same moment, the clock tower's shadow is 24 m long. Without climbing anything, you can find the tower's height. The secret: the two triangles have the same angles.

A line parallel to one side

Theorem: a straight line drawn parallel to one side of a triangle divides the other two sides proportionally.In △ABC, if DE ∥ BC, then AD/DB = AE/EC.Converse: if AD/DB = AE/EC, then DE ∥ BC.Use the theorem to find lengths; use the converse to prove lines are parallel.

Blue to amber, on both sides

DE ∥ BC. The blue parts are in the same ratio to the amber parts on both sides: AD : DB = AE : EC.

✏️ Finding a piece

In △ABC, DE ∥ BC with D on AB and E on AC. AD = 4 cm, DB = 6 cm and AE = 5 cm. Find EC.

1DE ∥ BC, so AD/DB = AE/EC.
24/6 = 5/EC
3EC = 5 × 6 ÷ 4 = 7.5 cm.

✅ EC = 7.5 cm

⚠️ The parallel line needs the whole side

If AD = 2, DB = 4 and BC = 9, many write DE/BC = AD/DB = 2/4 and get 4.5. Wrong! DE is a side of the small triangle ADE, so compare it with the whole side: DE/BC = AD/AB = 2/6.So DE = 9 × 2/6 = 3.Part : part works only for the pieces of the two sides, AD/DB = AE/EC.

Equiangular triangles

Two triangles are equiangular when the three angles of one equal the three angles of the other. Showing two pairs equal is enough: the third pair follows, since each triangle's angles add to 180°.Theorem: in equiangular triangles, the corresponding sides are proportional.If △ABC and △PQR are equiangular: AB/PQ = BC/QR = CA/RP.The converse is also true: proportional sides give equal angles.

Match by angles, not by position

△PQR is turned and flipped, but its angles match △ABC: A = P, B = Q, C = R. Sides opposite equal angles correspond: BC ↔ QR, CA ↔ RP, AB ↔ PQ.

✏️ The clock tower

Back to Jaffna: a 1.5 m stick casts a 2 m shadow while the clock tower casts a 24 m shadow. Find the height of the tower.

1Both stand vertically: right angles at the base.
2The sun's rays are parallel, so they meet the ground at the same angle. The triangles are equiangular.
3Height/shadow is the same: h/24 = 1.5/2.
4h = 24 × 1.5 ÷ 2 = 18 m.

✅ 18 m

⚠️ Pairing the wrong sides

The most common O/L error in this chapter: writing a ratio with sides that don't correspond. Before writing any ratio, list the equal angles, then pair the sides opposite them. Don't pair sides just because they look alike on the page.Also, equiangular is not congruent: don't write ≡. The triangles have the same shape but usually different sizes.

📌 Remember

• DE ∥ BC ⇔ AD/DB = AE/EC (theorem and converse).• For DE itself, use the whole side: DE/BC = AD/AB.• Two pairs of equal angles make triangles equiangular.• Equiangular triangles have proportional corresponding sides: pair the sides opposite equal angles.• Look for vertically opposite, alternate and common angles.

🧾 Quick revision sheet

DE ∥ BC in △ABC. Which ratios are equal?	AD/DB = AE/EC
Converse of the parallel line theorem	If AD/DB = AE/EC, then DE ∥ BC.
Equiangular triangles	Triangles whose three angles are equal pair by pair.
How many equal angle pairs must you show?	Two. The third follows from the angle sum of 180°.
Which sides correspond?	The sides opposite equal angles.
Equiangular vs congruent	Equiangular: same shape, sides in proportion. Congruent: same shape and same size.

Now test yourself: 5 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 12-minute lesson free ▶ Practise 5 past-paper questions