Geometry · Step-by-step guide
Triangle Congruence Rules
Learn SSS, SAS, ASA, AAS, and HL triangle congruence rules—and why SSA and AAA do not prove congruence.
Start with the central idea
Triangles are congruent when corresponding sides and angles force exactly one size and shape. Valid shortcuts are SSS, SAS, ASA, AAS, and HL for right triangles.
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Match corresponding parts
Name congruent triangles in corresponding order so each vertex lines up with its partner.
Use only enough information
A valid congruence theorem avoids proving all six pairs of corresponding parts separately.
Watch the included part
In SAS the angle lies between the two known sides; in ASA the side lies between the two known angles.
See the structure
A picture makes the relationships easier to remember.
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
Three sides of one triangle match three sides of another.
- Mark the three pairs of corresponding sides.
- No angle information is needed.
- State congruence in matching vertex order.
Two sides and their included angle match.
- Locate the angle between the known sides.
- Verify it is the corresponding angle.
- Apply SAS.
Right triangles have equal hypotenuses and one equal leg.
- Confirm both triangles are right.
- Match hypotenuses and a pair of legs.
- Use the right-triangle-only HL theorem.
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Using SSA, which can produce two different triangles.
Using AAA, which proves similarity rather than congruence.
Writing triangle names out of corresponding order.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Congruent Triangles
Identify valid congruence reasons and match corresponding parts.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Which congruence theorem is shown by the markings in the diagram?
Show hint
Count the marked pairs of corresponding sides.
Show worked solution
- One-tick sides correspond, two-tick sides correspond, and three-tick sides correspond.
- All three pairs of corresponding sides are congruent; no angle fact is needed.
Answer: SSS.
Which congruence theorem is shown by the markings in the diagram?
Show hint
Check whether the marked angle lies between the two marked sides.
Show worked solution
- Two pairs of corresponding sides are marked congruent.
- The marked angle is included between those sides.
Answer: SAS.
Which congruence theorem is shown by the markings in the diagram?
Show hint
Decide whether the marked side is between the marked angles.
Show worked solution
- Two pairs of corresponding angles are marked congruent.
- The marked side joins those two angles, so it is included.
Answer: ASA.
Which congruence theorem is shown by the markings in the diagram?
Show hint
Notice where the marked side sits relative to the two marked angles.
Show worked solution
- Two pairs of corresponding angles are marked congruent.
- The marked side is not between those angles.
Answer: AAS.
Which right-triangle congruence theorem is shown by the markings?
Show hint
Identify the hypotenuse and the marked leg.
Show worked solution
- The square markings establish that both figures are right triangles.
- Their hypotenuses and one pair of legs are marked congruent.
Answer: HL.
Frequently asked questions
Quick answers before you move on.
What should I remember first?
Triangles are congruent when corresponding sides and angles force exactly one size and shape. Valid shortcuts are SSS, SAS, ASA, AAS, and HL for right triangles.
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.