The Central Idea
If one triangle can be moved by rigid motions so that it coincides exactly with another triangle, the two triangles are congruent.
Rigid motions preserve distances and angle measures. They include:
- translations;
- rotations;
- reflections;
- and combinations of these motions.
Congruent triangles may occupy different positions or orientations, but their corresponding geometric measurements are identical.
Congruence Notation
If triangle \(ABC\) is congruent to triangle \(DEF\), we write
\[ \boxed{\triangle ABC\cong\triangle DEF}. \]The order of the letters communicates correspondence:
\[ A\leftrightarrow D,\qquad B\leftrightarrow E,\qquad C\leftrightarrow F. \]Corresponding Parts
From
\[ \triangle ABC\cong\triangle DEF, \]we may identify corresponding sides:
\[ AB=DE,\qquad BC=EF,\qquad AC=DF, \]and corresponding angles:
\[ \angle A\cong\angle D,\qquad \angle B\cong\angle E,\qquad \angle C\cong\angle F. \]Correct correspondence is essential. A congruence statement written in the wrong order can pair the wrong parts.
Congruence Versus Equality
Numbers and measures are equal:
\[ AB=DE \]and
\[ m\angle A=m\angle D. \]Geometric figures and geometric parts are congruent:
\[ \overline{AB}\cong\overline{DE} \]and
\[ \angle A\cong\angle D. \]Why We Need Congruence Criteria
A triangle has six basic measurements: three side lengths and three angle measures. Comparing all six would prove congruence, but it is unnecessary.
Certain smaller collections of information determine a triangle uniquely up to rigid motion. These collections are the triangle congruence criteria.
SSS: Side-Side-Side
If all three sides of one triangle are congruent to the three corresponding sides of another triangle, then the triangles are congruent.
\[ AB=DE,\qquad BC=EF,\qquad AC=DF \]implies
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by SSS}}. \]Why SSS Works
Three compatible side lengths determine a triangle up to rigid motion and reflection. If one side is fixed, the third vertex must lie at a prescribed distance from each endpoint, so it lies at an intersection of two circles.
The two possible intersection positions are mirror images, and reflection is a rigid motion. Therefore, both constructions represent congruent triangles.
SAS: Side-Angle-Side
If two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
If
\[ AB=DE,\qquad AC=DF, \]and
\[ \angle A\cong\angle D, \]then
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by SAS}}. \]What Does Included Angle Mean?
The included angle is the angle formed by the two given sides. If the known sides are \(\overline{AB}\) and \(\overline{AC}\), their included angle is
\[ \angle BAC. \]SAS requires the angle between the two known sides, not an unrelated angle.
ASA: Angle-Side-Angle
If two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent.
If
\[ \angle A\cong\angle D,\qquad AB=DE,\qquad \angle B\cong\angle E, \]then
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by ASA}}. \]AAS: Angle-Angle-Side
If two angles and a corresponding nonincluded side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
For example, if
\[ \angle A\cong\angle D,\qquad \angle B\cong\angle E,\qquad AC=DF, \]then
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by AAS}}. \]Why AAS Works
If two pairs of corresponding angles are congruent, the third pair must also be congruent because every triangle has angle sum \(180^\circ\).
AAS information can therefore be converted into ASA information by finding the third angle.
HL: Hypotenuse-Leg
HL is a special congruence criterion for right triangles. If two right triangles have congruent hypotenuses and one pair of corresponding congruent legs, then the triangles are congruent.
\[ \boxed{\text{right triangles}+\text{hypotenuse}+\text{leg}\Longrightarrow\text{HL congruence}}. \]Why HL Is Special
In a right triangle, the \(90^\circ\) angle is already known. If the hypotenuse and one leg are fixed, the remaining leg is determined by the Pythagorean relationship.
Thus the triangle's size and shape are fixed, making HL sufficient for congruence.
AAA Does Not Prove Congruence
Three equal corresponding angles determine shape but not size. Triangles with the same three angle measures can be built at many different scales.
\[ \boxed{\text{AAA proves similarity, not congruence}}. \]SSA Does Not Generally Prove Congruence
Knowing two sides and a nonincluded angle is called SSA. In general, SSA can produce no triangle, one triangle, or two noncongruent triangles.
This is the classical ambiguous case.
\[ \boxed{\text{SSA is not a general triangle congruence criterion}}. \]Do Not Confuse SSA with HL
HL may look superficially like side-side-angle information, but the right-angle condition adds essential structure.
HL requires two right triangles, congruent hypotenuses, and one corresponding pair of congruent legs. Without the right-triangle condition, ordinary SSA remains insufficient.
A Congruence Criteria Reference
| Criterion | Given Information | Valid for Congruence? |
|---|---|---|
| SSS | Three corresponding sides | Yes |
| SAS | Two sides and included angle | Yes |
| ASA | Two angles and included side | Yes |
| AAS | Two angles and nonincluded side | Yes |
| HL | Right triangles, hypotenuse, and one leg | Yes |
| AAA | Three corresponding angles | No; similarity only |
| SSA | Two sides and nonincluded angle | Not in general |
Shared Sides and the Reflexive Property
Two triangles in the same diagram may share a side. A segment is congruent to itself:
\[ \boxed{\overline{AB}\cong\overline{AB}}. \]Shared sides frequently provide one of the pieces needed for SSS or SAS.
Vertical Angles as Congruence Information
When two lines intersect, vertical angles are congruent. Thus, if two triangles meet at an intersection, the vertical-angle theorem may supply an angle pair needed for ASA, AAS, or SAS.
Parallel Lines as Congruence Information
If parallel lines are cut by a transversal, corresponding and alternate interior angles can provide congruent angle pairs. These relationships can then be combined with side information to establish ASA, AAS, or SAS.
Midpoints as Congruence Information
If \(M\) is the midpoint of \(\overline{AB}\), then
\[ AM=MB. \]The definition of midpoint therefore supplies a pair of congruent segments.
Angle Bisectors as Congruence Information
If ray \(\overrightarrow{BD}\) bisects \(\angle ABC\), then
\[ \angle ABD\cong\angle DBC. \]Combined with a shared side or known side equality, this may lead directly to SAS or ASA.
Perpendicular Lines as Congruence Information
Perpendicular lines form right angles. Since all right angles are congruent, perpendicular relationships can provide congruent angle pairs.
In right triangles, perpendicularity may also help establish the right-angle condition required for HL.
Example: SSS with a Shared Side
Suppose triangles \(ABC\) and \(ADC\) satisfy
\[ AB=AD, \] \[ BC=DC, \]and they share \(\overline{AC}\). By the reflexive property,
\[ AC=AC. \]Therefore,
\[ \boxed{\triangle ABC\cong\triangle ADC\text{ by SSS}}. \]Example: SAS with an Angle Bisector
Suppose \(\overrightarrow{AD}\) bisects \(\angle BAC\), and
\[ AB=AC. \]Compare triangles \(BAD\) and \(CAD\). The angle bisector gives
\[ \angle BAD\cong\angle DAC. \]The triangles share
\[ AD=AD. \]Therefore,
\[ \boxed{\triangle BAD\cong\triangle CAD\text{ by SAS}}. \]Example: ASA from Parallel Lines
Suppose two triangles contain one pair of corresponding angles created by parallel lines and another pair of vertical angles. If the side between those two angle pairs is congruent, then the triangles are congruent by ASA.
The important step is to identify exactly which side lies between the two established angle pairs.
Example: HL
Suppose \(\triangle ABC\) and \(\triangle DEF\) are right triangles with right angles at \(C\) and \(F\). If
\[ AB=DE \]and
\[ AC=DF, \]then the hypotenuses and one corresponding pair of legs are congruent. Therefore,
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by HL}}. \]CPCTC
Once triangle congruence has been established, every pair of corresponding parts is congruent. This principle is commonly abbreviated CPCTC:
Corresponding Parts of Congruent Triangles are Congruent.
CPCTC is used after congruence has been proved.
Using CPCTC
Suppose
\[ \triangle ABC\cong\triangle DEF. \]Then CPCTC allows conclusions such as
\[ AC=DF \]or
\[ \angle B\cong\angle E. \]Do Not Use CPCTC Too Early
CPCTC cannot normally be used to establish the triangle congruence that it depends on.
- Establish enough corresponding parts to use a valid congruence criterion.
- Conclude that the triangles are congruent.
- Use CPCTC to conclude additional corresponding parts are congruent.
Proof Structure: Given, Reason, Conclusion
A triangle congruence proof usually combines three kinds of information:
- given facts from the problem;
- derived facts from definitions and earlier theorems;
- a congruence criterion that supports the conclusion.
The proof should make clear why every claimed correspondence is valid.
Example: A Short Congruence Proof
Given \(AB=AC\) and \(AD\) bisects \(\angle BAC\), prove \(\triangle BAD\cong\triangle CAD\).
We know
\[ AB=AC \]by the given information,
\[ \angle BAD\cong\angle DAC \]by the definition of angle bisector, and
\[ AD=AD \]by the reflexive property. Therefore,
\[ \boxed{\triangle BAD\cong\triangle CAD\text{ by SAS}}. \]Congruence Can Prove Other Theorems
Triangle congruence is not merely a classification technique. It is a proof engine. Once two triangles are shown congruent, CPCTC can establish equal sides, equal angles, bisected segments, bisected angles, and properties of larger polygons.
Congruence and Isosceles Triangles
Congruence provides a natural route to the isosceles triangle theorem. If two sides of a triangle are congruent, an appropriate auxiliary segment can create two congruent triangles. CPCTC then shows that the angles opposite those equal sides are congruent.
Congruence and Perpendicular Bisectors
Suppose point \(P\) lies on the perpendicular bisector of \(\overline{AB}\), meeting it at midpoint \(M\). Then
\[ AM=MB, \] \[ PM=PM, \]and
\[ \angle AMP\cong\angle BMP \]because both are right angles. Therefore,
\[ \triangle AMP\cong\triangle BMP \]by SAS, and CPCTC gives
\[ \boxed{PA=PB}. \]Coordinate Verification of Congruence
Coordinate geometry can verify triangle congruence by computing distances, slopes, or angle relationships. If all three corresponding side lengths are equal by the distance formula, then SSS establishes congruence.
Example: Coordinate SSS
Consider
\[ A(0,0),\quad B(4,0),\quad C(0,3) \]and
\[ D(2,1),\quad E(6,1),\quad F(2,4). \]The first triangle has side lengths
\[ AB=4,\qquad AC=3,\qquad BC=5. \]The second has
\[ DE=4,\qquad DF=3,\qquad EF=5. \]Therefore,
\[ \boxed{\triangle ABC\cong\triangle DEF\text{ by SSS}}. \]Congruence and Rigid Transformations
The classical congruence criteria and the transformation definition describe the same underlying idea. If SSS, SAS, ASA, AAS, or HL proves that two triangles are congruent, then a sequence of rigid motions can map one onto the other.
Orientation Does Not Affect Congruence
One triangle may be a mirror image of another. Reflection reverses orientation but preserves all distances and angle measures. Therefore, mirror-image triangles can still be congruent.
Congruence Versus Similarity
Congruent figures have the same shape and the same size. Similar figures have the same shape but may have different sizes.
Congruence can be viewed as the special case of similarity with scale factor
\[ \boxed{k=1}. \]Common Error: Matching the Wrong Vertices
If \(\triangle ABC\cong\triangle DEF\), then \(A\) corresponds to \(D\), not automatically to whichever vertex looks closest in a diagram. Always use the congruence order and geometric evidence to establish correspondence.
Common Error: Using SAS with a Nonincluded Angle
SAS requires the angle between the two known sides. Two sides plus a different angle form SSA, which is not generally sufficient.
Common Error: Using AAA for Congruence
Equal corresponding angles establish shape but not scale. AAA therefore establishes similarity, not congruence.
Common Error: Assuming SSA Works
SSA can create an ambiguous case in which two different triangles satisfy the same measurements. Unless the special right-triangle HL conditions apply, do not use SSA as a triangle congruence criterion.
Common Error: Using CPCTC Before Congruence
CPCTC is a consequence of established triangle congruence. First prove the triangles congruent by a valid criterion; then use CPCTC for additional corresponding parts.
Common Error: Trusting the Picture
A diagram may make two sides or angles look equal without marking or proving them equal. Congruence proofs must use given information, definitions, established theorems, or valid calculations—not visual appearance alone.
A Reliable Congruence Procedure
- Identify the two triangles being compared.
- Determine the intended correspondence between their vertices.
- Mark all given congruent sides and angles.
- Look for shared sides, vertical angles, midpoint facts, bisectors, parallel lines, or right angles.
- Decide whether the evidence matches SSS, SAS, ASA, AAS, or HL.
- Check included versus nonincluded parts carefully.
- Write the triangle congruence statement in matching order.
- Use CPCTC only after congruence has been established.
- Verify that no unsupported assumption came from the drawing.
Practice 1: SSS
Suppose \(AB=DE\), \(BC=EF\), and \(AC=DF\). Which congruence criterion proves \(\triangle ABC\cong\triangle DEF\)?
Practice 2: SAS
Suppose \(AB=DE\), \(AC=DF\), and \(\angle A\cong\angle D\). Which congruence criterion applies?
Practice 3: ASA or AAS?
Suppose \(\angle A\cong\angle D\), \(\angle B\cong\angle E\), and \(AB=DE\). Which congruence criterion most directly applies?
Practice 4: AAS
Suppose \(\angle A\cong\angle D\), \(\angle B\cong\angle E\), and \(BC=EF\). Which congruence criterion applies?
Practice 5: HL
Two right triangles have congruent hypotenuses and one pair of corresponding congruent legs. Which criterion proves them congruent?
Practice 6: Invalid Criterion
A student knows that all three pairs of corresponding angles in two triangles are congruent. Can the student conclude that the triangles are congruent?
Practice 7: Shared Side
Triangles \(ABC\) and \(ADC\) share side \(\overline{AC}\). If \(AB=AD\) and \(BC=DC\), prove the triangles congruent.
Practice 8: Angle Bisector
In triangle \(ABC\), \(AD\) bisects \(\angle BAC\), and \(AB=AC\). Which criterion proves \(\triangle BAD\cong\triangle CAD\)?
Practice 9: CPCTC
Given \(\triangle ABC\cong\triangle DEF\), identify the side corresponding to \(\overline{BC}\).
Practice 10: Correspondence
If \(\triangle PQR\cong\triangle XYZ\), which angle corresponds to \(\angle Q\)?
Practice 11: Side-Length Evidence
One triangle has side lengths \(5\), \(7\), and \(9\). Another triangle also has side lengths \(5\), \(7\), and \(9\). What conclusion can be made?
Practice 12: Identify the Error
A student says, “Two sides of one triangle equal two sides of another, and one angle is equal, so the triangles are congruent by SAS.” Explain what additional check is required.
Practice Answers
Practice 1
All three corresponding side pairs are congruent.
\[ \boxed{\text{SSS}} \]Practice 2
The given angle is included between the two given sides.
\[ \boxed{\text{SAS}} \]Practice 3
Side \(\overline{AB}\) lies between \(\angle A\) and \(\angle B\).
\[ \boxed{\text{ASA}} \]Practice 4
The given side is not the side included between the two given angles.
\[ \boxed{\text{AAS}} \]Practice 5
\[ \boxed{\text{HL}} \]Practice 6
No. AAA fixes the triangle's shape but not its size.
\[ \boxed{\text{AAA establishes similarity, not congruence.}} \]Practice 7
We have \(AB=AD\), \(BC=DC\), and by the reflexive property, \(AC=AC\).
\[ \boxed{\triangle ABC\cong\triangle ADC\text{ by SSS}}. \]Practice 8
We have \(AB=AC\), \(\angle BAD\cong\angle DAC\), and \(AD=AD\).
\[ \boxed{\triangle BAD\cong\triangle CAD\text{ by SAS}}. \]Practice 9
From the congruence order, \(B\leftrightarrow E\) and \(C\leftrightarrow F\).
\[ \boxed{\overline{BC}\leftrightarrow\overline{EF}}. \]Practice 10
The correspondence is \(P\leftrightarrow X\), \(Q\leftrightarrow Y\), and \(R\leftrightarrow Z\).
\[ \boxed{\angle Q\leftrightarrow\angle Y}. \]Practice 11
The three corresponding side lengths match, so the triangles are congruent by SSS.
\[ \boxed{\text{The triangles are congruent by SSS.}} \]Practice 12
The angle must be the angle included between the two known sides. If it is not included, the information is SSA rather than SAS, and SSA is not generally a valid congruence criterion.
What to Remember
- congruent triangles have the same size and shape;
- the order of a congruence statement determines correspondence;
- SSS uses three corresponding side pairs;
- SAS requires the angle included between two known sides;
- ASA uses two angles and their included side;
- AAS uses two angles and a nonincluded side;
- HL is a special criterion for right triangles;
- AAA establishes similarity, not congruence;
- SSA is not generally sufficient;
- CPCTC is used only after triangle congruence has been established.
The most important strategic question is: What information fixes this triangle uniquely up to rigid motion?
Where We Go Next
Congruence fixes both shape and size. Geometry also needs a way to describe figures that preserve shape while allowing size to change.
The next lesson, Similarity and Scale, develops proportional correspondence, dilation, triangle similarity criteria, and the way linear scale affects geometric measurements.