What this chapter is about
This chapter introduces you to the idea of congruence in geometry. Two shapes are called congruent when they are exactly the same in size and shape. Think of them as geometric twins — if you cut one shape out and place it over the other, they match perfectly.
You will learn how to check whether two triangles are congruent. Triangles are special because once you know certain measurements (some sides and angles), you can tell if two triangles are twins without measuring everything. These shortcuts are called congruence rules or criteria.
After studying this chapter, you will be able to identify congruent figures, use different rules to test if two triangles are congruent, and understand why these rules work. This skill helps in construction, art, and solving many geometry problems.
Key ideas
- Congruent figures are figures that have the exact same shape and size. One can be placed exactly over the other.
- Two line segments are congruent if they have equal length. Two angles are congruent if they have equal measure in degrees.
- When two triangles are congruent, their corresponding parts (matching sides and matching angles) are equal.
- SSS rule: If all three sides of one triangle equal the three sides of another triangle, the triangles are congruent.
- SAS rule: If two sides and the angle between them in one triangle equal two sides and the included angle in another triangle, the triangles are congruent.
- ASA rule: If two angles and the side between them in one triangle equal two angles and the included side in another triangle, the triangles are congruent.
- RHS rule: For right-angled triangles, if the hypotenuse and one other side of one triangle equal the hypotenuse and one side of another, the triangles are congruent.
- The order of letters matters when writing congruence. If triangle PQR is congruent to triangle XYZ, then P matches X, Q matches Y, and R matches Z.
Formulas and facts to remember
- Congruent segments: AB = CD means segment AB is congruent to segment CD.
- Congruent angles: Angle P = Angle Q means the two angles have the same degree measure.
- SSS (Side-Side-Side): Three pairs of equal sides prove congruence.
- SAS (Side-Angle-Side): Two pairs of equal sides with the angle between them equal prove congruence.
- ASA (Angle-Side-Angle): Two pairs of equal angles with the side between them equal prove congruence.
- RHS (Right angle-Hypotenuse-Side): In right triangles, equal hypotenuse and one equal side prove congruence.
- If triangle ABC ≅ triangle DEF, then AB = DE, BC = EF, CA = FD, and angle A = angle D, angle B = angle E, angle C = angle F.
Worked examples
Example 1: Using SSS rule
In triangle LMN, LM = 5 cm, MN = 7 cm, NL = 6 cm. In triangle PQR, PQ = 5 cm, QR = 7 cm, RP = 6 cm. Are the triangles congruent?
Step 1: Compare the three sides. LM = PQ = 5 cm, MN = QR = 7 cm, NL = RP = 6 cm.
Step 2: All three pairs of sides are equal.
Step 3: By the SSS rule, triangle LMN ≅ triangle PQR.
Example 2: Using SAS rule
In triangle ABC, AB = 4 cm, angle B = 60°, BC = 6 cm. In triangle DEF, DE = 4 cm, angle E = 60°, EF = 6 cm. Check for congruence.
Step 1: AB = DE = 4 cm (one pair of sides equal).
Step 2: Angle B = Angle E = 60° (the angles between those sides are equal).
Step 3: BC = EF = 6 cm (second pair of sides equal).
Step 4: Two sides and the included angle match. By SAS rule, triangle ABC ≅ triangle DEF.
Example 3: Using ASA rule
In triangle GHI, angle G = 50°, GH = 8 cm, angle H = 70°. In triangle JKL, angle J = 50°, JK = 8 cm, angle K = 70°. Are they congruent?
Step 1: Angle G = Angle J = 50°.
Step 2: GH = JK = 8 cm (the side between these angles).
Step 3: Angle H = Angle K = 70°.
Step 4: Two angles and the included side are equal. By ASA rule, triangle GHI ≅ triangle JKL.
Common mistakes
- Comparing sides without checking if the angle is between them → For SAS, the angle must be the one formed by the two sides you are comparing.
- Writing vertices in wrong order, like saying triangle ABC ≅ triangle EDF when A should match D → Always match corresponding vertices carefully.
- Thinking any three equal measurements prove congruence → Three angles equal (AAA) does not prove congruence; the triangles could be different sizes.
- Applying RHS to triangles that are not right-angled → RHS works only when both triangles have a right angle.
- Forgetting that congruent means same size, not just same shape → Similar figures have the same shape but may differ in size; congruent figures must match exactly.
Quick revision
- Congruent figures are exact copies in shape and size.
- SSS: three sides equal → triangles congruent.
- SAS: two sides and included angle equal → triangles congruent.
- ASA: two angles and included side equal → triangles congruent.
- RHS: right angle, hypotenuse, and one side equal → right triangles congruent.
- Matching vertices must be written in correct order when stating congruence.