Triangles – Congruence Theorems Class 9: notes and important questions

Triangles – Congruence Theorems: revision notes

1. Triangles are rigid

Three rods pinned at their ends make a triangle that cannot change shape: the three side lengths fix the triangle. Four rods pinned into a quadrilateral can be pushed out of shape (a square becomes a rhombus). That is why bridges, roof trusses, cranes and towers are built from triangles, and why a gate or shelf is braced with a diagonal.

2. Congruent triangles

Two triangles are congruent if one can be placed exactly on the other: all three sides and all three angles match. The order of the letters shows the correspondence: \(\triangle ABC \cong \triangle PQR\) means \(A \leftrightarrow P\), \(B \leftrightarrow Q\), \(C \leftrightarrow R\), so \(AB = PQ\), \(BC = QR\), \(CA = RP\) and \(\angle A = \angle P\), and so on. Corresponding parts of congruent triangles are equal (CPCT).

3. The congruence conditions

• SAS (taken as an axiom): two sides and the angle between them.
• ASA: two angles and the side between them. AAS: two angles and a side not between them (the third angles are then equal too, by the angle sum).
• SSS: all three sides.
• RHS: in right triangles, the hypotenuse and one other side.

4. SSA is not a congruence condition

Two sides and an angle not between them can give two different triangles: with the angle and the first side fixed, the second side can swing to two positions. SSA does work when the given angle is a right angle (that is RHS), and more generally when the side opposite the given angle is the longer of the two given sides. AAA gives triangles of the same shape but not always the same size.

5. Isosceles triangles

• Angles opposite equal sides are equal; conversely, sides opposite equal angles are equal.
• In an isosceles triangle, the median to the base is also the altitude and the bisector of the vertex angle.
• A triangle is equilateral if and only if all its angles are \(60^\circ\).

6. Propositions, converses and counterexamples

A proposition is a statement that is either true or false. "If \(P\), then \(Q\)" has converse "If \(Q\), then \(P\)". A statement and its converse are different: one can be true and the other false. When both are true we write "\(P\) if and only if \(Q\)".

One counterexample, a case where \(P\) holds but \(Q\) fails, shows a general statement is false. Examples can never prove it true; that needs a proof.

7. Why diagrams must be accurate

A badly drawn figure can make unequal lengths look equal or hide a second possible triangle (as in SSA). Use the figure to plan, but prove every step from the given facts.

Worked example 1

\(\triangle ABC \cong \triangle LMN\) with \(\angle A = 40^\circ\) and \(\angle B = 75^\circ\). Find \(\angle N\).

\(N\) corresponds to \(C\): \(\angle N = \angle C = 180^\circ - 40^\circ - 75^\circ = 65^\circ\).

Worked example 2

In \(\triangle PQR\), \(PQ = PR\) and \(\angle P = 50^\circ\). Find \(\angle Q\).

\(\angle Q = \angle R = \dfrac{180^\circ - 50^\circ}{2} = 65^\circ\).

Worked example 3

Write the converse of "If two angles are vertically opposite, then they are equal" and decide whether it is true.

Converse: "If two angles are equal, then they are vertically opposite." False: the base angles of an isosceles triangle are equal but not vertically opposite.

Common errors

• Using SSA (or AAA) as a congruence condition.
• Writing the congruence in the wrong order, so CPCT pairs the wrong sides.
• Using SAS when the angle is not between the two sides.
• Assuming a converse is true because the original statement is true.

Exam tips

• List the three pairs of equal parts with a reason each, then name the rule, then use CPCT.
• For a converse question, write the converse in full before deciding whether it is true.