Teacher Guide: SAS Congruence

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

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Printable worksheet

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Worksheet Answer key

Class quiz

10 questions on Congruence & Similarity. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

State the SAS congruence criterion and identify the included angle of two sides
Explain with a counterexample why SSA does not prove congruence
Find hidden congruent parts in diagrams: vertical angles, shared sides, midpoints and bisectors
Use CPCTC after SAS to find unknown sides and angles
Choose the right criterion (SSS, SAS, ASA, AAS) from the given information

Prerequisites

• Introduction to Congruence: what congruent figures are and the symbol for congruence
• Congruent Triangles: corresponding parts and congruence statements
• ASA and AAS Congruence: included and non-included parts
• Angle sum of a triangle and vertical angles

Discussion Starters

1. If you know two side lengths of a triangle, what else do you need to know to draw exactly one triangle?
2. Why does it matter whether the angle is between the two sides or not?
3. Can you find vertical angles or a shared side in the room around you, for example in scissors or a window frame?
4. How could you measure the width of a river without crossing it?

Common Misconceptions

Any two sides and any angle prove congruence

Remediation: Build the SSA case with a compass: draw a $30°$ angle, mark $AB = 10$ , and swing a circle of radius $6$ from $B$ . Students see the circle cut the other ray twice and get two triangles.

SAS and SSA are the same, only written in a different order

Remediation: Stress that the letters describe positions, not a list. In SAS the A is between the two S; in SSA it is not. Have students color the two sides and check whether the angle sits where they meet.

Only parts given with numbers count

Remediation: Show diagrams with crossing segments and shared sides. Ask students to mark every equal pair before naming a criterion, including the ones nobody wrote down.

Differentiation Ideas

For Struggling Students:

• Give a checklist: find the two sides, find their shared point, is the angle at that point?
• Use colored highlighting for each pair of corresponding parts
• Start with triangles drawn in the same orientation before rotated or mirrored ones

For On-Level Students:

• Mix SSS, SAS, ASA, AAS and SSA cases and ask students to sort them
• Write short two-column proofs that use vertical angles or a shared side
• Use CPCTC after SAS to find missing sides and angles

For Advanced Students:

• Find when SSA gives zero, one or two triangles by comparing the opposite side with the height
• Prove that the base angles of an isosceles triangle are equal using SAS and the angle bisector
• Show with SAS that a quadrilateral whose diagonals bisect each other has equal opposite sides

Standards Alignment

HSG-CO.B.7 (CCSS.MATH.CONTENT.HSG.CO.B.7)
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and angles are congruent
HSG-CO.B.8 (CCSS.MATH.CONTENT.HSG.CO.B.8)
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions
HSG-SRT.B.5 (CCSS.MATH.CONTENT.HSG.SRT.B.5)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

Lesson Resources

visualSAS Triangle Pair
Two triangles with the matching sides and the included angle marked; the second one is rotated and mirrored
activityStraws and a Hinge
Students tape two straws at a fixed angle and compare the third side, then try SSA with a swinging straw
worksheetHidden Parts Hunt
Diagrams with vertical angles and shared sides; students mark every congruent pair and name the criterion

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

\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

AB

and

BC

the shared point is

B

, so the included angle is

\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.

Worked Examples

\triangle ABC

AB = 6

cm,

\angle B = 50°

BC = 8

cm. In

\triangle DEF

DE = 6

cm,

\angle E = 50°

EF = 8

cm. Are the triangles congruent? If

AC \approx 6.2

cm, how long is

DF

List the matching parts

AB = DE = 6

cm,

BC = EF = 8

cm,

\angle B = \angle E = 50°

→ Two pairs of sides and one pair of angles

Check that the angle is included

Sides

AB

and

BC

meet at

B

; sides

DE

and

EF

meet at

E

→

\angle B

and

\angle E

are the included angles

Apply SAS

Two sides and the included angle match →

\triangle ABC \cong \triangle DEF

Use corresponding parts

AC

corresponds to

DF

(first and third letters) →

DF = AC \approx 6.2

Answer: Yes, $\triangle ABC \cong \triangle DEF$ by SAS, so $DF \approx 6.2$ 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 $30°$ , $10$ , $6$ 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 $\triangle ABC \cong \triangle FED$ when $B$ matches $E$ and $A$ matches $D$ pairs the wrong parts and leads to wrong CPCTC conclusions.

Correct: Write the vertices in matching order. If $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ , write $\triangle ABC \cong \triangle DEF$ .

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 $\triangle ABC \cong \triangle DEF$ (for example by SAS), then use CPCTC for the remaining parts.

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 $25°$ 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 $30°$ .

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 $AB$ across a pond, pick a point $C$ on land. Extend $AC$ past $C$ to $D$ with $CD = AC$ , and $BC$ past $C$ to $E$ with $CE = BC$ . The angles at $C$ are vertical angles, so $\triangle ACB \cong \triangle DCE$ by SAS and $DE = AB$ .

2Try It Yourself

A surveyor measures $AC = 30$ m and $BC = 24$ m from point $C$ . She marks $D$ and $E$ as described, then measures $DE = 42$ m.

How wide is the pond, from $A$ to $B$ , in meters?

Step 1: Write the mathematical expression

Enter the length of $AB$ 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 $60°$ 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 $AB$ and $BC$ it is $\angle B$
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

Why is SSA not a congruence criterion?

Because the side opposite the given angle can swing like a compass and meet the third side at two points. With

\angle A = 30°

AB = 10

and

BC = 6

you get two triangles, one with

AC \approx 5.34

and one with

AC \approx 11.98

Does SSA ever work?

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.

Is SAS a postulate or a theorem?

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.

How do I choose between SSS, SAS, ASA and AAS?

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,\ \angle B = \angle E,\ BC = EF \Rightarrow \triangle ABC \cong \triangle DEF

The angle must be between the two sides

Criteria that work

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

Each one fixes a single triangle

Patterns that do not work

\text{SSA},\ \text{AAA}

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

CPCTC

\triangle ABC \cong \triangle DEF \Rightarrow AC = DF,\ \angle A = \angle D,\ \angle C = \angle F

Use it only after congruence is proven

Teacher Guide: SAS Congruence

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Folding A-frames

Measuring across a pond

Clock hands

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

SAS criterion

Criteria that work

Patterns that do not work

CPCTC

More in This Topic

Introduction to Congruence

Introduction to Similarity

Congruent Triangles

AA Similarity