Triangles — Study Notes

1. Introduction

The triangle is one of the most important and beautiful figures in geometry. It is the simplest polygon, having only three sides and three angles, yet it has a rich collection of properties. Triangles are the building blocks of all other polygons, because any polygon can be divided into triangles. In this chapter we will study triangles in detail: their congruence, their angle relationships, and the properties of isosceles triangles.

Two figures are said to be congruent when they have exactly the same shape and size, so that one can be placed exactly on top of the other. Congruence is the central idea of this chapter. We will learn the four standard criteria for congruence of triangles: SAS, ASA, SSS and RHS. Using these criteria we will prove many important properties, including the fact that angles opposite to equal sides of a triangle are equal, and the triangle inequality, which describes how the sides of a triangle must relate to each other.

2. Congruence of Triangles

Congruent Figures

Two plane figures are congruent if they have the same shape and the same size, and one can be superimposed exactly on the other. We write triangle ABC is congruent to triangle DEF, meaning every side and angle of one matches exactly with a corresponding side and angle of the other.

• Corresponding sides are equal: AB = DE, BC = EF, CA = FD.
• Corresponding angles are equal: angle A = angle D, angle B = angle E, angle C = angle F.
• The order of the letters in the congruence statement matters; it tells us which vertices correspond to which.

Congruence Criteria

We do not need to check all six pairs (three sides and three angles) to establish congruence. The following criteria are sufficient:

1. SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle, then the triangles are congruent.
2. ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
3. SSS (Side-Side-Side): If the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
4. RHS (Right angle-Hypotenuse-Side): If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the triangles are congruent.

AAS Criterion

If two angles and a side (not necessarily the included side) of one triangle are equal to the corresponding two angles and side of another triangle, the triangles are congruent. This is often called the AAS criterion; it follows from ASA because if two angles are equal, the third angle is also equal.

3. Some Important Theorems on Triangles

Theorem: Angles Opposite to Equal Sides are Equal

In an isosceles triangle, the angles opposite to equal sides are equal. In triangle ABC, if AB = AC, then angle B = angle C.

Proof: Draw the bisector of angle A meeting BC at D. Then in triangles ABD and ACD, AB = AC (given), angle BAD = angle CAD (D is the bisector), and AD = AD (common). Hence triangle ABD is congruent to triangle ACD by SAS. Therefore angle B = angle C (corresponding angles of congruent triangles).

Theorem: Sides Opposite to Equal Angles are Equal

The converse of the above theorem also holds: if two angles of a triangle are equal, then the sides opposite to them are equal. In triangle ABC, if angle B = angle C, then AB = AC.

The Angle Sum Property

We have already proved that the sum of the angles of a triangle is 180 degrees. In this chapter this property is used constantly, especially together with congruence theorems.

4. Inequalities in a Triangle

A triangle cannot be formed with arbitrary side lengths; the sides must satisfy certain inequalities.

Triangle Inequality

In any triangle, the sum of any two sides is greater than the third side. For a triangle ABC: - AB + BC is greater than AC. - BC + CA is greater than AB. - CA + AB is greater than BC.

Equivalently, the difference between any two sides is less than the third side. For example, sides of lengths 3, 4 and 8 cannot form a triangle because 3 + 4 = 7, which is less than 8.

Theorems on Sides and Angles

• If two sides of a triangle are unequal, then the angle opposite to the longer side is greater.
• In any triangle, the side opposite to the greater angle is longer.
• The sum of any two sides of a triangle is greater than the third side (triangle inequality).
• The perpendicular is the shortest distance between a point and a line; in a triangle, the hypotenuse is the longest side.

These results help us compare sides and angles of triangles and prove further theorems in geometry.

5. Applications in Quadrilaterals and Beyond

The congruence of triangles is the main tool used to prove properties of other figures. For example, the properties of quadrilaterals studied in the next chapter, such as the fact that the diagonals of a parallelogram bisect each other, are proved by dividing the quadrilateral into triangles and showing the triangles are congruent. Similarly, the midpoint theorem, the properties of the circle, and many mensuration results all depend on triangle congruence and the inequalities we studied here. A deep understanding of this chapter makes all later geometry much easier.

Quick Revision Tables

Table 1: Congruence Criteria

Table 2: Inequalities and Angle-Side Relationships

Mind Map

Important Diagrams (SVG)

Diagram 1: SAS Congruence Criterion

Diagram 2: Isosceles Triangle Property

Common Mistakes

1. Students often use SSA (Side-Side-Angle) as a congruence criterion. It is not valid; the angle must be included between the two sides (SAS).
2. A frequent error is writing the congruence statement in the wrong order. The order of vertices in "triangle ABC is congruent to triangle DEF" must match the corresponding vertices.
3. Students apply the RHS criterion to non-right triangles. RHS applies only to right triangles.
4. Many students believe that if two angles are equal in two triangles, the triangles are congruent (AAA). AAA does not prove congruence, only similarity.
5. When applying the triangle inequality, students sometimes write that the sum of two sides is equal to the third. It must be strictly greater.
6. Students forget that the angle opposite the longer side is greater, and mistakenly relate the length of a side to the adjacent angle.
7. In the isosceles triangle theorem, students confuse the equal sides with equal angles: equal sides AB and AC imply equal angles at C and B respectively (opposite those sides).
8. Students forget to check whether the triangles to be compared are in the same orientation before applying congruence criteria.

Exam Tips

1. When asked to prove two segments are equal, look for congruent triangles first: identify the triangles, then apply SAS, ASA, SSS or RHS, and finally write the equal parts from CPCT (corresponding parts of congruent triangles).
2. Always write "by CPCT" after concluding two corresponding parts are equal, as this standard phrase is expected in proofs.
3. In the triangle inequality, remember the quick test: add the two smaller sides; if their sum is greater than the largest side, a triangle can be formed.
4. For isosceles triangle questions, first mark the equal sides and then immediately note the equal opposite angles.
5. Practise the standard proof "prove that angles opposite to equal sides of a triangle are equal" since it is asked very frequently.
6. In inequality questions, compare the angles first to decide which side is longer; the longer side is opposite the greater angle.
7. Draw a clear figure for every proof question and mark the given equalities directly on the figure to make the solution easy to follow.

Conclusion

In this chapter we studied the congruence of triangles and the four main criteria: SAS, ASA, SSS and RHS, along with the AAS criterion. We proved important properties of isosceles triangles, showing that angles opposite equal sides are equal and its converse. We also studied inequalities in a triangle, establishing that the sum of any two sides is greater than the third side and that the greater angle lies opposite the longer side. The concept of CPCT (corresponding parts of congruent triangles) gives us a powerful tool for proving equalities in geometry. Since triangles are the building blocks of all polygons, these results will be used constantly in the study of quadrilaterals, circles and areas in the coming chapters.