What does congruent mean?
Congruent means exactly the same shape and the same size. If you cut one figure out, it covers the other one perfectly. We write ΔABC ≅ ΔPQR.
The order of letters matters. ΔABC ≅ ΔPQR means A sits on P, B sits on Q and C sits on R. So AB = PQ, BC = QR, CA = RP, ∠A = ∠P, ∠B = ∠Q and ∠C = ∠R.
Once two triangles are shown congruent, all their matching parts are equal. We call this CPCT: corresponding parts of congruent triangles.
Why triangles are rigid
Join three sticks with loose pins. Push a corner. Nothing moves. Now do the same with four sticks. The square leans over and becomes a diamond shape.
Why? Three lengths allow only one triangle. Four lengths allow many 4-gons. So a triangle is rigid: its shape is fixed by its sides. This single idea is behind the SSS rule.
The five congruence rules
A triangle has six parts: 3 sides and 3 angles. These rules say which three are enough.
- SAS (side-angle-side): two sides and the included angle (the angle between them) are equal. We take this one as the basic axiom.
- ASA (angle-side-angle): two angles and the side between them.
- AAS (angle-angle-side): two angles and any one other side. It follows from ASA because the third angle is 180° minus the other two.
- SSS (side-side-side): all three sides.
- RHS (right angle-hypotenuse-side): both triangles have a right angle, the hypotenuse (longest side, facing the right angle) is equal, and one more side is equal.
AAA is not a rule. A small and a big equilateral triangle have the same angles but different sizes.
Why SSA does not work
In SSA the angle is not between the two sides. Fix side AB = 5 and ∠A = 35°. Now side BC = 3.5 must reach the base line. Swing it round B like a door. It can hit the base at two different points, C₁ and C₂. Both triangles have AB = 5, ∠A = 35° and BC = 3.5, but they are not the same.
There is one safe case: when the given angle is 90°. Then the swinging side meets the base at only one point on each side. That special case is exactly the RHS rule.
Isosceles triangles
An isosceles triangle has two equal sides. Say AB = AC.
Theorem: the angles opposite the equal sides are equal, ∠B = ∠C.
Why: let M be the midpoint of BC and join AM. Then AB = AC, BM = CM and AM is shared. By SSS, ΔABM ≅ ΔACM. By CPCT, ∠B = ∠C. As a bonus, ∠AMB = ∠AMC and they add to 180°, so AM ⟂ BC.
Converse: if two angles of a triangle are equal, the sides opposite them are equal (proof uses AAS).
An equilateral triangle is isosceles in every direction, so each angle is 60°.
Propositions and converses
A proposition is a statement of the form “if P, then Q”. Its converse swaps the two parts: “if Q, then P”.
- “If two sides are equal, the opposite angles are equal.” Converse: “If two angles are equal, the opposite sides are equal.” Both are true.
- “If two triangles are congruent, their angles are equal.” Converse: “If the angles are equal, the triangles are congruent.” This is false (AAA fails).
So a converse must be proved on its own. It is never true just because the original is true.
Try it at home
Take 7 ice-cream sticks or straws and some pins or thread. Make a triangle with 3 and a square with 4. Push each one gently. Which one keeps its shape?
Next, cut two paper triangles using a ruler and protractor: sides 5 cm and 4 cm with 60° between them. Place one on the other. They match every time, whoever cuts them. That is SAS in your hands. In the 3D above, open free play, pick SSA and slide BC to see when two different triangles appear.
Key formulas and definitions
- ΔABC ≅ ΔPQR ⇒ AB = PQ, BC = QR, CA = RP and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R (CPCT)
- SAS, ASA, AAS, SSS, RHS → congruent
- SSA and AAA → not enough
- AB = AC ⇔ ∠B = ∠C (isosceles triangle)
- Angle sum: ∠A + ∠B + ∠C = 180°
Worked examples
1. In ΔABC and ΔPQR, AB = PQ = 6 cm, ∠B = ∠Q = 50° and BC = QR = 8 cm. Are they congruent? Which rule?
Step 1: two sides are equal (AB = PQ, BC = QR). Step 2: the angle between them is equal (∠B = ∠Q). Step 3: so by SAS, ΔABC ≅ ΔPQR.
2. In ΔABC, AB = AC and ∠A = 40°. Find ∠B and ∠C.
Step 1: AB = AC, so ∠B = ∠C (isosceles). Step 2: ∠B + ∠C = 180° − 40° = 140°. Step 3: each is 140° ÷ 2 = 70°.
3. Two triangles have ∠A = ∠P = 45°, ∠B = ∠Q = 75° and AC = PR = 7 cm. Are they congruent?
Step 1: two angles are equal. Step 2: AC is not between ∠A and ∠B, but it matches the same side PR in the other triangle. Step 3: by AAS, ΔABC ≅ ΔPQR.
4. A ladder 5 m long leans on a wall with its foot 3 m away. A second 5 m ladder also has its foot 3 m away from the same wall. Show the two triangles are congruent.
Step 1: the wall meets the ground at 90° in both. Step 2: the hypotenuse (ladder) is 5 m in both. Step 3: one side (ground distance) is 3 m in both. Step 4: by RHS the triangles are congruent, so both ladders reach the same height (4 m).
5. Line l bisects ∠A. From a point P on l, PB and PC are drawn perpendicular to the two arms of the angle. Prove PB = PC.
Step 1: in ΔAPB and ΔAPC, ∠PAB = ∠PAC (l bisects ∠A). Step 2: ∠ABP = ∠ACP = 90°. Step 3: AP is common. Step 4: by AAS, ΔAPB ≅ ΔAPC. Step 5: by CPCT, PB = PC.
6. In ΔABC, ∠B = ∠C = 65° and AB = 9 cm. Find AC and ∠A.
Step 1: ∠B = ∠C, so the sides opposite them are equal (converse of isosceles theorem): AC = AB = 9 cm. Step 2: ∠A = 180° − 65° − 65° = 50°.
Common mistakes
- Writing the letters in the wrong order. If A matches Q, you cannot write ΔABC ≅ ΔPQR.
- Using SSA as a rule. The angle must be between the two sides (SAS), unless it is a right angle (RHS).
- Thinking AAA proves congruence. Equal angles only prove the same shape, not the same size.
- Using CPCT before proving congruence. First prove ≅ with a rule, then use CPCT.