SAS Congruence

Prove triangles congruent with two sides and the included angle, spot hidden equal parts, and see why SSA is not enough.

Intermediate25 min18 problems
Practice Problems

Definition

The SAS (Side-Angle-Side) criterion says: if two sides of one triangle and the angle between them are congruent to two sides and the angle between them in another triangle, then the two triangles are congruent.
AB‾≅DE‾,∠B≅∠E,BC‾≅EF‾⟹△ABC≅△DEF\overline{AB} \cong \overline{DE}, \quad \angle B \cong \angle E, \quad \overline{BC} \cong \overline{EF} \quad \Longrightarrow \quad \triangle ABC \cong \triangle DEF
The angle must be the included angle: the angle formed by the two given sides. Its vertex is the point the two sides share. For sides ABAB and BCBC the shared point is BB, so the included angle is ∠B\angle B.
Why it works: think of two sticks joined by a hinge. Once the lengths of the sticks and the opening of the hinge are fixed, the distance between the free ends is fixed too. The third side has only one possible length, so there is only one possible triangle.
Careful: two sides and an angle that is not between them (SSA) do not prove congruence. The side opposite the angle can sometimes swing into two different positions, giving two different triangles.

Try it now

In △ABC\triangle ABC, which angle is included between the sides ABAB and BCBC?

Worked Examples

In △ABC\triangle ABC: AB=6AB = 6 cm, ∠B=50°\angle B = 50°, BC=8BC = 8 cm. In △DEF\triangle DEF: DE=6DE = 6 cm, ∠E=50°\angle E = 50°, EF=8EF = 8 cm. Are the triangles congruent? If AC≈6.2AC \approx 6.2 cm, how long is DFDF?
1

List the matching parts

AB=DE=6AB = DE = 6 cm, BC=EF=8BC = EF = 8 cm, ∠B=∠E=50°\angle B = \angle E = 50° → Two pairs of sides and one pair of angles
2

Check that the angle is included

Sides ABAB and BCBC meet at BB; sides DEDE and EFEF meet at EE → ∠B\angle B and ∠E\angle E are the included angles
3

Apply SAS

Two sides and the included angle match → △ABC≅△DEF\triangle ABC \cong \triangle DEF
4

Use corresponding parts

ACAC corresponds to DFDF (first and third letters) → DF=AC≈6.2DF = AC \approx 6.2 cm

Common Mistakes

Using SAS when the angle is not between the two sides

Why it's wrong: Two sides and a non-included angle (SSA) can describe two different triangles, as the 30°30°, 1010, 66 example shows.

Correct: Find the point the two sides share. The angle at that vertex is the only one SAS may use.

Matching the vertices in the wrong order

Why it's wrong: Writing △ABC≅△FED\triangle ABC \cong \triangle FED when BB matches EE and AA matches DD pairs the wrong parts and leads to wrong CPCTC conclusions.

Correct: Write the vertices in matching order. If A↔DA \leftrightarrow D, B↔EB \leftrightarrow E, C↔FC \leftrightarrow F, write △ABC≅△DEF\triangle ABC \cong \triangle DEF.

Thinking there is not enough information because a part is not written in the problem

Why it's wrong: Vertical angles and shared sides are equal even when no one says so. Students miss them and give up on a proof that works.

Correct: Before you decide, look for crossing segments (vertical angles) and a side that belongs to both triangles (shared side).

Using CPCTC before the triangles are proven congruent

Why it's wrong: Corresponding parts are equal only because the triangles are congruent. Without that, you are assuming what you want to prove.

Correct: First prove △ABC≅△DEF\triangle ABC \cong \triangle DEF (for example by SAS), then use CPCTC for the remaining parts.

Interactive Visual

Congruent triangles

68ABC68DEF△ABC ≅ △DEF

SASTwo pairs of sides and the angle between them are equal.

  • AB = DE = 6
  • BC = EF = 8
  • ∠B = ∠E = 50°

Tap a criterion to see which matching parts it needs. Parts with the same marks are equal.

Function graph

0246810120246xyA10B6C₁≈ 5.34C₂6
All points 6 units from B

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
In △ABC\triangle ABC, which angle is included between the sides ABAB and BCBC?

Why It Matters

SAS is one of the most used tools in geometry proofs, because the information it needs often comes for free:
  • Vertical angles: when two segments cross, the opposite angles are equal, so you already have the angle
  • Shared sides: when two triangles share a side, that side is congruent to itself
  • Midpoints and bisectors: a midpoint gives two equal segments, an angle bisector gives two equal angles
Once SAS proves two triangles congruent, every other pair of corresponding parts is equal too (CPCTC). That is how surveyors measure across a pond they cannot walk through, and how builders know two hinged frames opened to the same angle are identical.

Real World Applications

Folding A-frames

Sawhorses, ladders and easels are two legs joined by a hinge. The leg lengths and the opening angle decide the whole shape.

Example:

Two easels each have legs of 150 cm opened to 25°25° at the hinge. By SAS the triangles they form are congruent, so their feet are the same distance apart.

1Try It Yourself

A carpenter builds two sawhorse frames. Each has two legs of 90 cm, and both are locked so the angle at the hinge is 30°30°.

Which congruence criterion guarantees the two frames are identical?

Step 1: Choose the right answer

Choose the criterion:

Measuring across a pond

Surveyors use SAS to find a distance they cannot walk, by building a congruent copy of the triangle on dry land.

Example:

To measure ABAB across a pond, pick a point CC on land. Extend ACAC past CC to DD with CD=ACCD = AC, and BCBC past CC to EE with CE=BCCE = BC. The angles at CC are vertical angles, so △ACB≅△DCE\triangle ACB \cong \triangle DCE by SAS and DE=ABDE = AB.

2Try It Yourself

A surveyor measures AC=30AC = 30 m and BC=24BC = 24 m from point CC. She marks DD and EE as described, then measures DE=42DE = 42 m.

How wide is the pond, from AA to BB, in meters?

Step 1: Write the mathematical expression

Enter the length of ABAB in meters:

Clock hands

The two hands of a clock are two sides with a fixed length, and the time sets the angle between them.

Example:

At 2:00 and at 10:00 the angle between the hands is 60°60° both times, so by SAS the tips are the same distance apart at both times.

3Try It Yourself

A clock has an hour hand 3 cm long and a minute hand 4 cm long. At 3:00 the hands form a right angle and their tips are 5 cm apart.

How far apart, in cm, are the tips at 9:00?

Step 1: Write the mathematical expression

Enter the distance in cm:

Key Takeaways

  • 1SAS: two sides and the included angle of one triangle congruent to those of another means the triangles are congruent
  • 2The included angle is at the vertex the two sides share: for ABAB and BCBC it is ∠B\angle B
  • 3SSA is not a criterion: two sides and a non-included angle can give two different triangles
  • 4Look for hidden congruent parts: vertical angles where segments cross, a shared side, midpoints and angle bisectors
  • 5After proving congruence, use CPCTC to get the remaining sides and angles
  • 6Criteria that work: SSS, SAS, ASA, AAS. Patterns that do not: SSA and AAA

Frequently Asked Questions

Because the side opposite the given angle can swing like a compass and meet the third side at two points. With ∠A=30°\angle A = 30°, AB=10AB = 10 and BC=6BC = 6 you get two triangles, one with AC≈5.34AC \approx 5.34 and one with AC≈11.98AC \approx 11.98.
Because the side opposite the given angle can swing like a compass and meet the third side at two points. With ∠A=30°\angle A = 30°, AB=10AB = 10 and BC=6BC = 6 you get two triangles, one with AC≈5.34AC \approx 5.34 and one with AC≈11.98AC \approx 11.98.
Yes, in one special case: when the given angle is opposite the longer of the two given sides, only one triangle is possible. The best known example is a right angle opposite the hypotenuse (the HL criterion). In general, though, do not use SSA.
Many textbooks accept SAS as a postulate (a basic rule without proof) and use it to prove ASA and SSS. Euclid proved it as his Proposition 4 by placing one triangle on top of the other. Either way, you may use it in proofs.
Count what you know. Three sides: SSS. Two sides: check whether the known angle is between them (SAS) or not (SSA, no conclusion). Two angles: check whether the known side is between them (ASA) or not (AAS).

Glossary

SAS congruence
Two triangles are congruent if two sides and the included angle of one are congruent to two sides and the included angle of the other
Included angle
The angle formed by two given sides; its vertex is their shared endpoint
SSA
Two sides and a non-included angle; not a congruence criterion, because it can describe two different triangles
Vertical angles
The opposite angles formed by two crossing lines; they are always equal
Shared side
A side that belongs to two triangles at once; it is congruent to itself (reflexive property)
CPCTC
Corresponding parts of congruent triangles are congruent: once triangles are proven congruent, all matching sides and angles are equal

Formula Card

SAS criterion

AB=DE, ∠B=∠E, BC=EF⇒△ABC≅△DEFAB = DE,\ \angle B = \angle E,\ BC = EF \Rightarrow \triangle ABC \cong \triangle DEF

The angle must be between the two sides

Criteria that work

SSS, SAS, ASA, AAS\text{SSS},\ \text{SAS},\ \text{ASA},\ \text{AAS}

Each one fixes a single triangle

Patterns that do not work

SSA, AAA\text{SSA},\ \text{AAA}

SSA can give two triangles; AAA fixes the shape but not the size

CPCTC

△ABC≅△DEF⇒AC=DF, ∠A=∠D, ∠C=∠F\triangle ABC \cong \triangle DEF \Rightarrow AC = DF,\ \angle A = \angle D,\ \angle C = \angle F

Use it only after congruence is proven

Today's challengeOne problem, three tries

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