Geometry · Step-by-step guide
SAS vs. SSS Triangle Congruence
Compare SAS and SSS congruence, identify included angles, and decide which theorem a diagram supports.
Start with the central idea
SSS uses three pairs of congruent sides. SAS uses two pairs of congruent sides plus the angle between them. If the known angle is not included, SAS does not apply.
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.
Count the side pairs
Three known corresponding side pairs point directly to SSS.
Find the included angle
Trace from one known side through their shared vertex to the other known side.
Do not infer from appearance
Only markings, givens, and proved facts count; a diagram need not be drawn to scale.
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.
Two triangles share a diagonal and have two other matching side pairs.
- Use the reflexive property for the shared diagonal.
- That creates three side pairs.
- Apply SSS.
Two side pairs meet at a pair of vertical angles.
- Vertical angles are congruent.
- The angle is included between the known sides.
- Apply SAS.
Two sides and a non-included angle are known.
- The angle is not between the sides.
- This is SSA.
- More information is required.
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Calling any two sides and one angle SAS.
Forgetting a shared side can supply a third S for SSS.
Assuming unmarked sides are congruent.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Congruent Triangles
Practice distinguishing the valid triangle congruence shortcuts.
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.
Do the diagram markings show SAS or SSS congruence?
Show hint
Count the marked pairs of sides.
Show worked solution
- Each triangle has three marked sides.
- The one-, two-, and three-tick sides correspond.
Answer: SSS.
Do the diagram markings show SAS or SSS congruence?
Show hint
Check whether the marked angle is included.
Show worked solution
- Two corresponding side pairs are marked.
- The marked angle lies between those two sides.
Answer: SAS.
Do these markings prove congruence by SAS?
Show hint
Locate the angle relative to the two marked sides.
Show worked solution
- Two corresponding sides are marked, but the angle is not between them.
- The information is SSA, which can describe more than one triangle.
Answer: No; SSA is not a general congruence theorem.
The corresponding side lengths are shown. Does SAS or SSS prove the triangles congruent?
Show hint
Count how many corresponding side lengths match.
Show worked solution
- The triangles each have side lengths 4, 5, and 6.
- All three corresponding side pairs match.
Answer: SSS.
The corresponding measurements are shown. Does SAS or SSS prove the triangles congruent?
Show hint
Identify the angle between the labeled sides.
Show worked solution
- Each triangle has corresponding sides of length 5 and 7.
- The 40° angle is included between those sides.
Answer: SAS.
Frequently asked questions
Quick answers before you move on.
What should I remember first?
SSS uses three pairs of congruent sides. SAS uses two pairs of congruent sides plus the angle between them. If the known angle is not included, SAS does not apply.
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.