The Central Idea
Two geometric figures may be compared in two important ways.
Congruent figures have the same shape and the same size.
Similar figures have the same shape, but their sizes may differ by a constant scale factor.
Symbolically,
\[ \triangle ABC\cong\triangle DEF \]means exact geometric sameness, while
\[ \triangle ABC\sim\triangle DEF \]means proportional sameness of shape.
Congruence Is Stronger Than Similarity
Every pair of congruent figures is also similar.
Congruence corresponds to similarity with scale factor
\[ 1. \]But similar figures need not be congruent because their scale factor may be different from \(1\).
Corresponding Parts
Congruence and similarity statements encode a vertex correspondence.
If
\[ \triangle ABC\cong\triangle DEF, \]then
\[ A\leftrightarrow D,\qquad B\leftrightarrow E,\qquad C\leftrightarrow F. \]This determines which sides and angles correspond.
Why Order Matters
The statement
\[ \triangle ABC\cong\triangle DEF \]tells us
\[ AB\leftrightarrow DE, \] \[ BC\leftrightarrow EF, \] \[ AC\leftrightarrow DF. \]Reordering the letters can change the stated correspondence.
Congruent Figures
If two figures are congruent, every corresponding length and angle measure is equal.
For congruent triangles,
\[ AB=DE,\qquad BC=EF,\qquad AC=DF \]and
\[ \angle A\cong\angle D,\qquad \angle B\cong\angle E,\qquad \angle C\cong\angle F. \]Rigid Motions Preserve Congruence
Translations, rotations, and reflections preserve distances and angle measures.
Therefore any image of a figure under a rigid motion is congruent to the original.
Congruence through Superposition
Conceptually, two figures are congruent if one can be moved by rigid motions so that it coincides exactly with the other.
This transformation view explains why congruence preserves both shape and size.
Why Triangle Criteria Matter
To prove two triangles congruent, we do not need to verify all six corresponding parts individually.
Certain smaller sets of information are sufficient to determine the whole triangle.
These are the triangle congruence criteria.
SSS Congruence
Side-Side-Side congruence states that if 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
\[ \boxed{\triangle ABC\cong\triangle DEF}. \]Why SSS Works
Three compatible side lengths determine a triangle up to rigid motion and reflection.
Once the three corresponding lengths agree, the angle measures are forced as well.
SSS and Construction
The standard three-side triangle construction illustrates SSS directly.
Once a base is fixed, the third vertex must lie at the intersection of two circles whose radii are the remaining side lengths.
The two possible intersections are mirror images and produce congruent triangles.
SAS Congruence
Side-Angle-Side congruence states that two sides and the included angle determine a triangle.
If
\[ AB=DE, \] \[ \angle B\cong\angle E, \]and
\[ BC=EF, \]then
\[ \boxed{\triangle ABC\cong\triangle DEF}. \]What Does Included Angle Mean?
The included angle is the angle formed by the two known sides.
In sides \(\overline{AB}\) and \(\overline{BC}\), the included angle is
\[ \angle ABC. \]SAS requires this angle, not an unrelated angle.
Why SAS Works
One side fixes a base length. The included angle fixes a direction from one endpoint, and the second side fixes the distance along that direction.
The third vertex is therefore determined.
ASA Congruence
Angle-Side-Angle congruence uses two angles and the included side.
If
\[ \angle A\cong\angle D, \] \[ AB=DE, \]and
\[ \angle B\cong\angle E, \]then
\[ \boxed{\triangle ABC\cong\triangle DEF}. \]AAS Congruence
Angle-Angle-Side congruence uses two angles and a nonincluded side.
Two known angles determine the third angle because triangle angles sum to
\[ 180^\circ. \]A known side then fixes the triangle's scale.
Why ASA and AAS Both Work
Once two angle measures are known, the shape of the triangle is fixed up to scale.
One corresponding side length then fixes that scale.
HL Congruence for Right Triangles
Hypotenuse-Leg congruence applies specifically to right triangles.
If two right triangles have congruent hypotenuses and one pair of congruent corresponding legs, then the triangles are congruent.
Why HL Is Special
In a right triangle, the right angle is already fixed.
Knowing the hypotenuse and one leg determines the remaining leg through the Pythagorean relationship:
\[ a^2+b^2=c^2. \]Thus the triangle is determined.
RHS Terminology
Some curricula call HL congruence RHS:
- Right angle;
- Hypotenuse;
- Side.
The underlying criterion is the same.
What Does Not Prove Congruence?
Some combinations of information are insufficient.
Important noncriteria include:
- AAA;
- SSA in general.
Why AAA Does Not Prove Congruence
Three equal corresponding angles determine shape but not size.
Two triangles can have identical angle measures while one is an enlarged version of the other.
Therefore AAA proves similarity, not congruence.
Example: AAA Failure for Congruence
Consider two equilateral triangles, one with side length \(3\) and the other with side length \(8\).
Every angle in both triangles measures
\[ 60^\circ. \]The triangles are similar but not congruent.
Why SSA Is Ambiguous
Side-Side-Angle information, when the angle is not included between the known sides, can sometimes produce more than one triangle.
This is the classical ambiguous case.
The Ambiguous Case
Suppose side \(a\), side \(b\), and angle \(A\) are given.
Depending on their values, there may be:
- no triangle;
- one triangle;
- two distinct triangles.
Therefore SSA is not a general congruence criterion.
Right-Triangle Exception
HL may look like an SSA pattern because the hypotenuse and a leg are known along with the right angle.
But the fixed right-angle structure removes the ordinary ambiguity.
Shared Sides in Congruence Proofs
Two triangles may share a side.
Then the reflexive property gives
\[ AB=AB. \]This shared side can supply one of the required congruence relationships.
Vertical Angles in Congruence Proofs
When two lines intersect, vertical angles are congruent.
This often supplies an angle relationship needed for ASA, AAS, or SAS.
Parallel Lines in Congruence Proofs
Parallel lines cut by a transversal can generate congruent corresponding or alternate interior angles.
These relationships frequently provide angle pairs for triangle congruence.
Midpoints in Congruence Proofs
If \(M\) is the midpoint of \(\overline{AB}\), then
\[ AM=MB. \]This equality often supplies a side pair in congruence arguments.
Angle Bisectors in Congruence Proofs
If ray \(\overrightarrow{BD}\) bisects \(\angle ABC\), then
\[ \angle ABD\cong\angle DBC. \]That angle equality often supports SAS, ASA, or AAS.
Perpendicular Lines in Congruence Proofs
Perpendicular lines create right angles.
All right angles are congruent, so perpendicularity often provides an angle pair automatically.
CPCTC
Once two triangles have been proved congruent, their corresponding parts are congruent.
This principle is commonly abbreviated CPCTC:
Corresponding Parts of Congruent Triangles are Congruent.
Congruence First, CPCTC Second
CPCTC cannot normally be used to prove the triangles congruent.
First establish triangle congruence with a valid criterion. Then use CPCTC to conclude additional side or angle relationships.
Typical Congruence Proof Pattern
A common structure is:
\[ \text{givens} \longrightarrow \text{three sufficient relationships} \longrightarrow \text{triangle congruence} \longrightarrow \text{CPCTC conclusion}. \]Example: SAS Congruence
Suppose
\[ AB=DE, \] \[ BC=EF, \]and
\[ \angle B\cong\angle E. \]Since the angle is included between the two known sides,
\[ \boxed{\triangle ABC\cong\triangle DEF} \]by SAS.
Example: ASA Congruence
Suppose
\[ \angle A\cong\angle D, \] \[ \angle B\cong\angle E, \]and
\[ AB=DE. \]Then
\[ \boxed{\triangle ABC\cong\triangle DEF} \]by ASA.
Example: AAS Congruence
Suppose
\[ \angle A\cong\angle D, \] \[ \angle C\cong\angle F, \]and
\[ AB=DE. \]The known side is not between the two known angles, so the criterion is AAS.
Example: HL Congruence
Suppose \(\triangle ABC\) and \(\triangle DEF\) are right triangles with
\[ AC=DF \]as congruent hypotenuses and
\[ AB=DE \]as congruent legs.
Then the triangles are congruent by HL.
Similarity
Similar figures have equal corresponding angle measures and proportional corresponding side lengths.
If
\[ \triangle ABC\sim\triangle DEF, \]then corresponding angles are congruent and
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}. \]Scale Factor
The common ratio between corresponding side lengths is the scale factor.
If
\[ \frac{DE}{AB}=k, \]then every length in the second figure is \(k\) times the corresponding length in the first.
Similarity through Dilations
A dilation multiplies all distances from its center by the same scale factor.
It preserves angle measure and creates proportional lengths.
Therefore dilations are the fundamental transformations of similarity.
Similarity Transformations
A similarity transformation can be built from:
- a dilation;
- possibly followed by translations, rotations, or reflections.
Such a transformation preserves shape.
AA Similarity
Angle-Angle similarity states that if two angles of one triangle are congruent to two angles of another, the triangles are similar.
The third angles are then automatically congruent because each triangle has angle sum
\[ 180^\circ. \]AAA and AA
AAA is enough to establish similarity.
But because two angles force the third, AA is the more efficient criterion.
SAS Similarity
Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent.
If
\[ \frac{AB}{DE} = \frac{BC}{EF} \]and
\[ \angle B\cong\angle E, \]then
\[ \boxed{\triangle ABC\sim\triangle DEF}. \]SSS Similarity
Two triangles are similar if all three pairs of corresponding side lengths are proportional.
If
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}, \]then
\[ \boxed{\triangle ABC\sim\triangle DEF}. \]Why SSS Similarity Works
If every side of one triangle is the same scale multiple of the corresponding side of another, one triangle can be dilated to match the other's side lengths.
SSS congruence then completes the argument.
Why SAS Similarity Works
A dilation can scale the two known sides of one triangle to match the corresponding sides of the other.
The included angle is preserved under dilation.
SAS congruence then forces the resulting triangles to match.
Similarity Ratios Must Match Correspondence
When forming proportions, corresponding sides must be paired consistently.
If
\[ A\leftrightarrow D,\quad B\leftrightarrow E,\quad C\leftrightarrow F, \]then valid ratios include
\[ \frac{AB}{DE},\qquad \frac{BC}{EF},\qquad \frac{AC}{DF}. \]Do Not Mix Ratio Directions
If one ratio is written first-triangle over second-triangle, all corresponding ratios in the same equality should use the same direction.
For example,
\[ \frac{AB}{DE} = \frac{BC}{EF} \]is consistent.
Mixing
\[ \frac{AB}{DE} = \frac{EF}{BC} \]generally reverses one ratio and breaks the proportion.
Finding Missing Lengths by Similarity
Suppose
\[ \triangle ABC\sim\triangle DEF \]with
\[ AB=6,\qquad DE=9,\qquad BC=10. \]The scale factor from the first triangle to the second is
\[ \frac{9}{6}=\frac32. \]Therefore,
\[ EF=10\left(\frac32\right)=15. \]Perimeter Scaling
Similar figures with scale factor \(k\) have perimeters related by the same factor:
\[ \boxed{P_2=kP_1}. \]Area Scaling
If corresponding lengths scale by \(k\), corresponding areas scale by
\[ \boxed{k^2}. \]Volume Scaling
For similar three-dimensional figures, corresponding volumes scale by
\[ \boxed{k^3}. \]Congruence as Scale Factor \(1\)
If two similar figures have scale factor
\[ k=1, \]corresponding lengths are equal.
The figures are congruent.
Parallel Lines Create Similar Triangles
When a line inside a triangle is parallel to one side, corresponding and alternate interior angle relationships often establish AA similarity.
This is one of the most common similarity configurations.
Example: Parallel Segment in a Triangle
In \(\triangle ABC\), let \(D\) lie on \(\overline{AB}\) and \(E\) lie on \(\overline{AC}\), with
\[ DE\parallel BC. \]Then
\[ \angle ADE\cong\angle ABC \]and
\[ \angle AED\cong\angle ACB. \]Therefore,
\[ \boxed{\triangle ADE\sim\triangle ABC} \]by AA.
Similarity and Proportional Division
From
\[ \triangle ADE\sim\triangle ABC, \]we obtain proportions such as
\[ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}. \]Similarity therefore converts angle information into length relationships.
Right-Triangle Similarity
Drawing the altitude from the right angle to the hypotenuse of a right triangle creates two smaller triangles.
Each smaller triangle is similar to the original and to the other smaller triangle.
This produces geometric-mean relationships among the sides.
Similarity and Indirect Measurement
Similar triangles allow unknown distances to be inferred from known proportional relationships.
This is the basis of many methods for estimating:
- heights;
- distances across inaccessible regions;
- map dimensions;
- scale-model measurements.
Shadow Measurement
If sunlight creates the same angle for two vertical objects at the same time, their height-shadow triangles are similar.
Thus
\[ \frac{h_1}{s_1} = \frac{h_2}{s_2}. \]Scale Drawings
A scale drawing represents a similar figure.
If a map scale is
\[ 1:50{,}000, \]then one unit on the map represents \(50{,}000\) of the same units in the actual region.
Congruence and Coordinates
Coordinate geometry can establish congruence through exact side lengths, slopes, midpoints, and angle relationships.
The distance formula can verify equal side lengths.
Coordinate SSS
Suppose two triangles are specified by coordinates.
Compute the three corresponding side lengths using
\[ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]If all three corresponding lengths match, SSS proves congruence.
Coordinate Similarity
Similarity can be established by showing proportional side lengths, equal slopes corresponding to parallel directions, or a coordinate transformation equivalent to a dilation and rigid motion.
Transformation Proof of Congruence
If one figure can be mapped onto another by a sequence of translations, rotations, and reflections, then the figures are congruent.
This is often more efficient than comparing many side and angle measures individually.
Transformation Proof of Similarity
If one figure can be mapped onto another using a dilation together with rigid motions, the figures are similar.
The dilation factor becomes the similarity scale factor.
Symmetry Creates Congruence
A reflection across a symmetry line maps one portion of a symmetric figure onto another.
Corresponding subfigures are therefore congruent.
Rotational Symmetry and Congruence
In a regular polygon, rotation by
\[ \frac{360^\circ}{n} \]maps each sector to the next.
The repeated component triangles are congruent.
Congruence Criteria Are Sufficient Conditions
SSS, SAS, ASA, AAS, and HL are sufficient conditions.
If the required relationships hold, congruence follows.
These are not arbitrary mnemonics; they summarize geometric determinacy.
Similarity Criteria Are Sufficient Conditions
AA, SAS similarity, and SSS similarity are sufficient to establish triangle similarity.
Once similarity is known, all corresponding angles and side ratios can be used.
Criteria and Proof Economy
A good proof uses the smallest sufficient collection of facts.
If AA already establishes similarity, there is no need to compute all three side ratios first.
If SAS proves congruence, proving extra corresponding parts beforehand may be unnecessary.
Common Error: Assuming Congruence from Appearance
Two triangles may look identical but differ slightly in side length or angle measure.
Use a valid congruence criterion rather than visual appearance.
Common Error: Assuming Similarity from Appearance
Two figures may appear to have the same shape but fail exact angle or proportional-side conditions.
Similarity must be established mathematically.
Common Error: Using AAA for Congruence
AAA determines shape but not scale.
Therefore AAA establishes similarity, not congruence.
Common Error: Using SSA as a General Congruence Criterion
SSA can produce an ambiguous case.
Unless a special structure such as HL applies, SSA is not a valid general triangle congruence criterion.
Common Error: Misidentifying the Included Angle
SAS uses the angle between the two known sides.
If the known angle is elsewhere, the configuration may be SSA rather than SAS.
Common Error: Misidentifying the Included Side
ASA uses the side between the two known angles.
If the known side is not between them, the criterion is AAS.
Common Error: Applying HL to Nonright Triangles
HL requires both triangles to be right triangles.
Without the right-angle condition, hypotenuse terminology does not apply and HL cannot be used.
Common Error: Mixing Correspondence
A congruence or similarity statement must preserve correct vertex order.
Incorrect correspondence can produce false side ratios or angle pairings even when the figures truly are congruent or similar.
Common Error: Mixing Ratio Directions
If one ratio is written
\[ \frac{\text{small}}{\text{large}}, \]all corresponding ratios in the same proportion should follow that direction.
Common Error: Using CPCTC Too Early
CPCTC applies after triangle congruence has been established.
It should not normally be used as one of the facts needed to prove the congruence itself.
Common Error: Assuming Similar Figures Have Equal Sides
Similar figures have proportional corresponding sides.
Equality occurs only when the scale factor is
\[ 1. \]Common Error: Scaling Angles
Under similarity, side lengths scale but angle measures do not.
If the scale factor is \(3\), a \(40^\circ\) angle remains
\[ 40^\circ, \]not \(120^\circ\).
Common Error: Scaling Area Linearly
If corresponding lengths scale by \(k\), areas scale by
\[ k^2, \]not \(k\).
Common Error: Scaling Volume by \(k^2\)
Similar three-dimensional figures scale volume by
\[ k^3. \]A Reliable Congruence Procedure
- Identify the two triangles.
- Write the intended vertex correspondence.
- Mark or list all known side and angle relationships.
- Derive useful relationships from midpoints, bisectors, parallel lines, perpendicular lines, vertical angles, or shared sides.
- Match the information to SSS, SAS, ASA, AAS, or HL.
- State the congruence in correct corresponding order.
- Use CPCTC only after congruence is established.
A Reliable Similarity Procedure
- Identify the two triangles.
- Establish correct vertex correspondence.
- Look first for two equal angle pairs.
- If angle information is incomplete, test proportional side relationships.
- Use AA, SAS similarity, or SSS similarity.
- Write the similarity statement in corresponding order.
- Use the resulting side proportions or angle congruences.
Choosing Congruence versus Similarity
Ask what the available information controls.
Equal corresponding lengths often point toward congruence.
Proportional lengths or angle-only information often point toward similarity.
Similarity may also be an intermediate step toward another proof.
Example: Shared-Side SAS
Suppose triangles \(\triangle ABC\) and \(\triangle ADC\) satisfy
\[ AB=AD, \] \[ \angle BAC\cong\angle CAD, \]and share
\[ AC=AC. \]Therefore,
\[ \boxed{\triangle ABC\cong\triangle ADC} \]by SAS.
Example: Parallel-Line AA Similarity
In \(\triangle ABC\), let \(D\in\overline{AB}\) and \(E\in\overline{AC}\), with
\[ DE\parallel BC. \]Then two pairs of corresponding angles are congruent, so
\[ \boxed{\triangle ADE\sim\triangle ABC} \]by AA.
Example: SSS Similarity
Suppose one triangle has side lengths
\[ 3,\quad4,\quad5 \]and another has side lengths
\[ 6,\quad8,\quad10. \]Then
\[ \frac36 = \frac48 = \frac5{10} = \frac12. \]Therefore the triangles are similar by SSS similarity.
Example: Similar but Not Congruent
The \(3\)-\(4\)-\(5\) triangle and the \(6\)-\(8\)-\(10\) triangle have scale factor
\[ 2. \]They are similar but not congruent because corresponding side lengths are not equal.
Example: Congruent and Similar
Two triangles with corresponding sides
\[ 5,\quad7,\quad9 \]and
\[ 5,\quad7,\quad9 \]are congruent by SSS.
They are also similar with scale factor
\[ 1. \]Example: Area Scale Factor
Suppose two similar triangles have length scale factor
\[ k=3. \]Their area scale factor is
\[ k^2=9. \]If the smaller area is \(12\), the larger area is
\[ \boxed{108}. \]Practice 1: Identify the Congruence Criterion
Two triangles have all three corresponding side lengths equal.
Which congruence criterion applies?
Practice 2: Included Angle
In a triangle, which angle is included between sides \(\overline{AB}\) and \(\overline{BC}\)?
Practice 3: AAA
Two triangles have three pairs of congruent corresponding angles.
What can be concluded?
Practice 4: Right-Triangle Congruence
Two right triangles have congruent hypotenuses and one pair of congruent corresponding legs.
Which congruence criterion applies?
Practice 5: AA Similarity
Two triangles have two pairs of congruent corresponding angles.
What can be concluded?
Practice 6: Find a Missing Similar Side
Two similar triangles have scale factor from the first to the second of
\[ \frac32. \]A side in the first triangle has length \(14\).
Find the corresponding side in the second triangle.
Practice 7: Area Scaling
Two similar figures have length scale factor
\[ 4. \]What is the area scale factor?
Practice 8: Volume Scaling
Two similar solids have length scale factor
\[ 2. \]What is the volume scale factor?
Practice 9: CPCTC
After proving
\[ \triangle ABC\cong\triangle DEF, \]what principle allows you to conclude
\[ AC=DF? \]Practice 10: Identify the Error
A student says, “The triangles have two corresponding sides equal and one corresponding angle equal, so they are congruent by SAS.”
What must still be checked?
Practice Answers
Practice 1
Three corresponding side pairs establish
\[ \boxed{\text{SSS congruence}}. \]Practice 2
The two sides meet at vertex \(B\), so the included angle is
\[ \boxed{\angle ABC}. \]Practice 3
AAA establishes the same shape but does not fix scale.
Therefore,
\[ \boxed{\text{the triangles are similar, not necessarily congruent}.} \]Practice 4
For right triangles, congruent hypotenuse and one corresponding leg give
\[ \boxed{\text{HL congruence}}. \]Practice 5
Two corresponding angle pairs establish
\[ \boxed{\text{AA similarity}}. \]Practice 6
Multiply by the scale factor:
\[ 14\left(\frac32\right)=21. \]Therefore the corresponding side is
\[ \boxed{21}. \]Practice 7
Area scales by the square of the length factor:
\[ 4^2=16. \]Therefore,
\[ \boxed{16}. \]Practice 8
Volume scales by the cube of the length factor:
\[ 2^3=8. \]Therefore,
\[ \boxed{8}. \]Practice 9
Once triangle congruence has been established, corresponding parts are congruent by
\[ \boxed{\text{CPCTC}}. \]Practice 10
SAS requires the known angle to be the angle included between the two known corresponding sides.
If it is not included, the information may instead form an SSA configuration, which is not a general congruence criterion.
What to Remember
Congruence and similarity criteria identify when limited geometric data are sufficient to force a complete relationship between figures.
- Congruent figures have the same shape and size.
- Similar figures have the same shape with proportional corresponding lengths.
- Triangle congruence criteria include SSS, SAS, ASA, AAS, and HL.
- AAA establishes similarity, not congruence.
- SSA is not a general congruence criterion.
- Triangle similarity criteria include AA, SAS similarity, and SSS similarity.
- Correspondence order controls which sides and angles are paired.
- CPCTC is used after congruence is established.
- Similarity scales lengths by \(k\), areas by \(k^2\), and volumes by \(k^3\).
- Rigid motions establish congruence; dilations combined with rigid motions establish similarity.
The essential habit is: identify the correspondence first, then use only a criterion whose required relationships have actually been established.
Where We Go Next
Congruence and similarity criteria provide the logical engines behind many geometric arguments.
The next lesson, Direct and Indirect Geometric Proof, develops how such facts, definitions, constructions, and theorems are assembled into complete arguments, including contradiction and other indirect methods.