Triangle Congruence: Step-by-Step Guide
Two triangles are congruent if they are exactly the same size and shape, so all their matching sides and angles are equal. You do not have to check all six pairs. There are shortcuts: SSS, SAS, ASA, AAS, and HL (for right triangles). Once you prove two triangles are congruent, all their matching parts are equal. This idea is called CPCTC, which stands for 'Corresponding Parts of Congruent Triangles are Congruent'.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- List what you know. Write down every pair of equal sides and equal angles you are given. Look for shared sides, vertical angles, and angles made by parallel lines.\[\text{Given: }\overline{AB} \cong \overline{DE},\;\angle A \cong \angle D,\;\dots\]
- Match a congruence shortcut. SSS: three pairs of equal sides. SAS: two pairs of equal sides with the equal angle between them. ASA: two pairs of equal angles with the equal side between them. AAS: two pairs of equal angles and a side that is not between them. HL: for right triangles, the hypotenuse and one leg.\[\text{SSS},\;\text{SAS},\;\text{ASA},\;\text{AAS},\;\text{HL}\]
- Write a two-column proof. Make a list of statements. Next to each one, write the reason, like 'Given', 'Vertical angles', or 'Shared side'.\[\text{Statement} \;|\; \text{Reason}\]
- Write the conclusion. Write 'Triangle ABC is congruent to triangle DEF by SAS' (or the one you used). List the vertices in matching order.\[\triangle ABC \cong \triangle DEF\]
- Use CPCTC if needed. After proving the triangles are congruent, any other matching sides or angles are also equal, and you can say CPCTC as your reason.\[\triangle ABC \cong \triangle DEF \Rightarrow \overline{AC} \cong \overline{DF}\]
Worked example
Given that AB = DE, BC = EF, and angle B = angle E, prove triangle ABC is congruent to triangle DEF.
Step 1: The givens are AB = DE (sides), angle B = angle E (angles), and BC = EF (sides).
Step 2: Angle B is between sides AB and BC. Angle E is between sides DE and EF. So we have two sides and the angle between them.
Step 3: That matches SAS (Side-Angle-Side).
Step 4: Two-column proof: 1) AB = DE, Given. 2) angle B = angle E, Given. 3) BC = EF, Given. 4) Triangle ABC is congruent to triangle DEF, by SAS.
Step 5: Conclusion: Triangle ABC is congruent to triangle DEF by SAS.
Step 6 (extra, using CPCTC): This means AC = DF, angle A = angle D, and angle C = angle F.
Tips and common mistakes
- Match the corners in order. Triangle ABC and triangle DEF means A goes with D, B goes with E, and C goes with F.
- In SAS, the angle must be between the two sides. In ASA, the side must be between the two angles.
- AAA does not prove congruence. Triangles with the same angles can be different sizes. AAA only proves they are similar.
- SSA (two sides and an angle that is not between them) does not work in general. The exception is HL for right triangles.
- Look for hidden information: a side that is shared by both triangles, vertical angles, or equal angles from parallel lines.
- HL only works when both triangles are right triangles.
- After you prove the triangles are congruent, use CPCTC to show other sides or angles are equal.
- Give a reason for every line of a proof. Even 'Given' counts as a reason.