Congruence of Triangles — Study Notes

1. Introduction

Two figures are called congruent if they have exactly the same shape and the same size. If you place one figure exactly on top of the other and they coincide completely, the figures are congruent. For example, two rupee coins of the same denomination, two identical stamps, or two sheets of the same size are congruent. In geometry, we use the symbol congruent with the sign drawn as an equals sign with a tilde above it. When two figures are congruent, one can be obtained from the other by a rigid motion such as sliding, turning or flipping, without changing its size.

The study of congruence of triangles is important because triangles are the basic building blocks of polygons, and knowing when two triangles are congruent helps in proving many geometric results. Instead of comparing all six parts (three sides and three angles), we use efficient rules that require only three conditions. These rules, called criteria for congruence, allow us to establish that two triangles are exactly alike with just a few measurements, which is how surveyors, engineers and architects verify shapes in the real world.

2. Congruence of Plane Figures

2.1 What Congruence Means

Two plane figures are congruent if one can be placed exactly over the other such that every point of one coincides with a corresponding point of the other. The order of the vertices in the congruence statement is important because it tells us which vertices correspond. For example, if triangle ABC is congruent to triangle PQR, written triangle ABC congruent to triangle PQR, then vertex A corresponds to P, B to Q and C to R.

2.2 Congruence of Line Segments

Two line segments are congruent if they have the same length. So congruence of segments is simply equality of lengths. If segment AB has length 5 cm and segment CD has length 5 cm, the segments are congruent.

2.3 Congruence of Angles

Two angles are congruent if they have the same measure. For example, two angles each of 45 degrees are congruent.

2.4 Congruence of Circles and Polygons

Two circles are congruent if they have equal radii. Two polygons are congruent if they have the same number of sides, equal corresponding sides and equal corresponding angles.

3. Congruence of Triangles

Two triangles are congruent if their corresponding sides are equal and their corresponding angles are equal. If triangle ABC is congruent to triangle PQR, then: - AB = PQ, BC = QR and CA = RP (corresponding sides) - angle A = angle P, angle B = angle Q and angle C = angle R (corresponding angles)

The congruence symbol with the triangle notation preserves the correspondence between vertices.

3.1 Correspondence of Vertices

When we write triangle ABC congruent to triangle PQR, the order matters. A corresponds to P, B to Q and C to R. Therefore AB corresponds to PQ, BC to QR and CA to RP. Writing the triangles in the correct order is essential for identifying equal parts.

4. Criteria for Congruence of Triangles

4.1 SSS (Side-Side-Side) Congruence

If the three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent. For example, if AB = PQ, BC = QR and CA = RP, then triangle ABC is congruent to triangle PQR by SSS. No angle measurement is needed.

4.2 SAS (Side-Angle-Side) Congruence

If two sides and the angle included between them in one triangle are equal to the two corresponding sides and the included angle of another triangle, the two triangles are congruent. The angle must be the angle between the two given sides. For example, if AB = PQ, angle B = angle Q and BC = QR, then the triangles are congruent by SAS.

4.3 ASA (Angle-Side-Angle) Congruence

If two angles and the side included between them in one triangle are equal to the two corresponding angles and the included side of another triangle, the triangles are congruent. For example, if angle A = angle P, AB = PQ and angle B = angle Q, then the triangles are congruent by ASA.

4.4 RHS (Right angle-Hypotenuse-Side) Congruence

If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle, the triangles are congruent. This rule applies only to right-angled triangles. For example, if angle B = angle Q = 90 degrees, AC = PR and BC = QR, then the triangles are congruent by RHS.

5. When Triangles are NOT Congruent

Some combinations of equal parts do not guarantee congruence: - AAA (Angle-Angle-Angle): equal angles guarantee similarity of shape but not equality of size, so AAA is not a valid congruence criterion. - SSA or ASS (Side-Side-Angle): two sides and a non-included angle do not always give a unique triangle, so this is not a valid criterion. - Only matching pairs of sides or angles must correspond; jumbled matches may not be congruent.

6. Applications of Congruence

Congruence of triangles is used to prove many geometric facts. For example, in an isosceles triangle, the angles opposite the equal sides are equal, and the theorem can be proved by drawing the median or altitude and showing the two resulting right triangles are congruent by RHS or SSS. Congruence also guarantees that corresponding parts of congruent triangles are equal, which is often abbreviated as CPCT. This is used to prove that certain lengths or angles in a figure are equal even when they are not directly measurable.

Quick Revision Tables

Table 1: Criteria for Congruence of Triangles

Table 2: Corresponding Parts of Congruent Triangles

Mind Map

Important Diagrams (SVG)

Diagram 1: Correspondence of Congruent Triangles

Diagram 2: SSS Congruence

Diagram 3: RHS Congruence

Common Mistakes

1. Using AAA as a congruence criterion. Equal angles give similar triangles, but the sizes can differ, so AAA never proves congruence.
2. Applying RHS to a triangle that is not right angled. RHS works only for right-angled triangles.
3. Using SAS with an angle that is not between the two given sides. The included angle must lie between the two sides.
4. Getting the correspondence wrong. Writing triangle ABC congruent to triangle PQR but comparing AB with QR instead of PQ is a common error.
5. Concluding congruence from only two equal parts. Congruence requires the specific three conditions of a valid criterion.
6. Believing SSA is a valid criterion. Two sides and a non-included angle can give two different triangles, so it is not reliable.
7. Forgetting that CPCT stands for corresponding parts of congruent triangles, and applying it to triangles that were not proved congruent.
8. Mixing up SAS and SSA. In SAS the angle is included between the sides; in SSA it is not, and SSA is invalid.

Exam Tips

1. Always state the criterion you are using (SSS, SAS, ASA or RHS) and list the equal parts before writing the congruence statement.
2. Write the congruence in the correct order so that equal parts line up; this makes CPCT statements obvious.
3. Check for common features in figures such as a shared side or a common angle, which are automatically equal and help prove congruence.
4. Remember that AAA gives similarity, not congruence, and is never a valid criterion.
5. For RHS problems, first confirm the triangles are right angled and identify the hypotenuse.
6. After proving triangles congruent, clearly write the equal corresponding parts using CPCT as the reason.
7. Practise proving simple results like base angles of an isosceles triangle being equal, since these use congruence cleverly.

Conclusion

Congruence of triangles tells us when two triangles are exactly alike in shape and size. By using the four valid criteria, SSS, SAS, ASA and RHS, we can establish congruence with only three conditions instead of comparing all six parts. The careful writing of vertex correspondence and the use of CPCT allow us to conclude that other corresponding parts are equal, which becomes a powerful tool in geometric proofs. Recognising invalid criteria like AAA and SSA protects us from false conclusions. Congruence is foundational for the study of quadrilaterals, similar triangles and advanced geometry in higher classes.

Extra Practice Problems

1. Under what conditions would triangle ABC be congruent to triangle PQR by SSS?
2. In triangles DEF and STU, DE = ST, angle E = angle T and EF = TU. Which criterion applies and write the congruence statement.
3. In two right triangles, the hypotenuse and one side are equal. Prove they are congruent.
4. Why is AAA not a criterion for congruence? Give an example.
5. In an isosceles triangle with AB = AC, show that angle B = angle C by drawing the altitude AD and using RHS.
6. If triangle XYZ congruent to triangle LMN, list all equal corresponding parts.
7. Two triangles have sides 5, 6, 7 each. Are they necessarily congruent? Explain.
8. In a figure, O is the midpoint of both AB and CD. Prove that triangle AOC is congruent to triangle BOD.