Triangles — Class 9 IMO

Congruence of Triangles

What is Congruence of Triangles?

In simple English, congruence means identical in every single way. If two shapes have the exact same shape and the exact same size, they are said to be congruent.

Imagine two identical brand-new one-rupee coins placed directly on top of each other. They cover each other completely because their boundaries match perfectly. In geometry, when two triangles are congruent, it means that if you cut one out and place it over the other, it will cover it exactly.

When two triangles, Triangle ABC and Triangle PQR, are congruent, we write it using the special symbol "≅":

Triangle ABC ≅ Triangle PQR

This symbol combines "~" (same shape) and "=" (same size).

When two triangles are congruent, their corresponding parts match up perfectly. This gives us a very important rule called CPCT, which stands for Corresponding Parts of Congruent Triangles. This means:

Corresponding sides are equal.
Corresponding angles are equal.

Instead of measuring all three sides and all three angles every time, mathematicians discovered four main shortcut rules or criteria for congruence:

SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. (Note: The angle must be trapped tightly between the two sides).
ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.
SSS (Side-Side-Side): Two triangles are congruent if all three sides of one triangle are equal to the corresponding three sides of the other triangle.
RHS (Right angle-Hypotenuse-Side): Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.

Let us look at a quick comparison table of these rules:

Congruence Rule	Full Name	What Must Be Equal?	Position Rule
ASA	Angle-Side-Angle	2 Angles and 1 Side	The side must be between the two angles.
SSS	Side-Side-Side	3 Sides	All three matching pairs of sides must be equal.
RHS	Right angle-Hypotenuse-Side	1 Right Angle, Hypotenuse, 1 Side	Applies only to right-angled triangles.

Example 1: Triangles share two sides and the angle between them equal. Which rule?

SAS — side-angle-side.

Example 2: Right triangles with equal hypotenuse and one equal leg — congruent?

Yes, by the RHS rule.

Quick recap

Congruent triangles have all corresponding parts equal.
Rules: SSS, SAS, ASA, AAS, RHS.

✓ Quick check

In a right-angled triangle ABC, where ∠B = 90°, if AD is the median drawn to the hypotenuse AC, then which of the following is always true?

In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. Hence, AD = CD = BD = ½ AC.

Which of the following is NOT a sufficient condition for congruence of two triangles?

Three angles (AAA) establishes similarity but does not guarantee that the sides are of the same length, hence it is not a sufficient criterion for congruence.

Isosceles Triangles

What are the Properties of a Triangle's Angles and Sides?

Triangles have beautiful internal mathematical relationships connecting their sides and angles. When you change the length of a side, the angle opposite to it changes too.

Here are the fundamental rules governing these properties:

Isosceles Triangle Property: An isosceles triangle is a triangle that has two equal sides. The rule states that the angles opposite to the equal sides of an isosceles triangle are also equal. For example, if side AB = side AC in Triangle ABC, then Angle B (opposite side AC) must equal Angle C (opposite side AB).
Converse of Isosceles Property: If two angles of a triangle are equal, then the sides opposite to those equal angles must also be equal in length.
Equilateral Triangle Property: An equilateral triangle has all three sides equal. Because all sides are equal, all three internal angles must also be equal to one another. Since the total sum of angles in a triangle is always 180°, each angle of an equilateral triangle measures exactly 60° (180° divided by 3).

Let us check how these properties look:

Triangle Type	Side Condition	Angle Condition	Special Value
Equilateral	All three sides are equal	All three angles are equal	Each angle is always 60°
Scalene	No sides are equal	No angles are equal	All angles are different

Example 1: An isosceles triangle has a vertex angle of 40°. Each base angle?

(180° − 40°) ÷ 2 = 70°.

Example 2: Two base angles of a triangle are equal. What follows?

The sides opposite them are equal — the triangle is isosceles.

Quick recap

Equal sides ⇒ equal opposite (base) angles, and conversely.
The apex-to-base-midpoint line is the perpendicular and angle bisector.

✓ Quick check

In an isosceles triangle ABC with AB = AC, if AD is a perpendicular dropped on BC, then which of the following criteria proves ΔABD ≅ ΔACD?

In right ΔABD and right ΔACD: Hypotenuse AB = Hypotenuse AC, and side AD is common. So, they are congruent by RHS. Alternatively, AB = AC, AD = AD, and BD = CD (perpendicular from vertex to base bisects it in an isosceles triangle), so SAS is also applicable via included angle ∠BAD = ∠CAD.

In ΔABC, if ∠A = 60°, ∠B = 40°, and ∠C = 80°, arrange the sides AB, BC, and AC in ascending order of their lengths.

The side opposite to the smallest angle is the shortest. ∠B = 40° is smallest → opposite side AC is shortest. ∠A = 60° → opposite side BC is next. ∠C = 80° is largest → opposite side AB is longest. Order: BC < AC < AB.

Inequalities and the Mid-point Theorem

What are Triangle Inequalities?

In the previous topic, we looked at what happens when sides and angles are perfectly equal. But what happens when they are not equal? Inequalities deal with relationships of "greater than" (>) or "less than" (<).

There are two major laws that govern inequalities in triangles:

The Side-Angle Inequality Theorem: If two sides of a triangle are unequal in length, the angle opposite to the longer side is always larger than the angle opposite to the shorter side. Think of it like a crane or a laptop lid: the wider you open the angle, the longer the distance between the tips becomes! Conversely, the side opposite to the larger angle is always the longer side.
The Triangle Inequality Theorem: This is a fundamental rule of nature. It states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. If this rule is broken, the two sides will lie flat or won't reach each other, and you cannot form a closed triangle at all!

Let us see how the Triangle Inequality Theorem works with an example. Suppose a triangle has sides of length \(a\), \(b\), and \(c\). For this triangle to exist, all three of these conditions must be true:

\(a + b > c\)
\(b + c > a\)
\(a + c > b\)

A helpful shortcut rule derived from this is: The difference between any two sides of a triangle is always less than the third side (\(|a - b| < c\)).

Example 1: Which is longest: the side opposite a 100° angle or a 40° angle?

The side opposite the 100° angle — greater angle, greater side.

Example 2: Midpoints of two sides are joined; the third side is 10 cm. Segment length?

Half of 10 = 5 cm, and it is parallel to that side.

Quick recap

Greater angle ⇒ greater opposite side; any two sides sum to more than the third.
Mid-point segment is parallel to and half the third side.

✓ Quick check

In an equilateral triangle ABC, the measure of each interior angle is:

An equilateral triangle has all three sides equal, hence all three interior angles are equal. Let each angle be x. x + x + x = 180° → 3x = 180° → x = 60°.

In ΔABC, D, E and F are the mid-points of sides AB, BC and CA respectively. If the perimeter of ΔABC is 24 cm, then the perimeter of ΔDEF is:

By the mid-point theorem, DE = AC/2, EF = AB/2, and FD = BC/2. Perimeter of ΔDEF = DE + EF + FD = (AC + AB + BC)/2 = (Perimeter of ΔABC)/2 = 24/2 = 12 cm.