Congruence of Geometric Figures

Congruence of Geometric Figures - Congruence of Plane Figures: Definition and Examples

In geometry, we often need to compare figures to understand their relationships. One fundamental relationship is that of congruence. Two geometric figures are congruent if they are exact duplicates of each other – they have the same shape and the same size. If two figures are congruent, it means one can be transformed into the other by a sequence of rigid motions (translations, rotations, and reflections) without changing its size or shape.

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Definition of Congruence

Two geometric figures are defined as congruent if and only if one can be exactly superimposed upon the other. Superimposition means placing one figure on top of the other such that all points and lines of one figure perfectly match with the corresponding points and lines of the other figure. When figures are congruent, all corresponding lengths, angle measures, and areas are equal.

The concept of congruence captures the idea of being identical in every geometric aspect.

Superposition Principle:

The method of superposition is an intuitive way to understand congruence. If you can pick up a figure (mentally or physically) and move it, rotate it, or flip it so that it fits exactly onto another figure, then the two figures are congruent. This implies that rigid transformations (movements that preserve distances and angles) lead to congruent figures.

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Symbol for Congruence

The mathematical symbol used to denote congruence is $\cong$. This symbol is a combination of the equality sign ($=$) and the similarity sign ($\sim$), indicating that the figures are equal in size and similar in shape (which implies the same shape).

If figure F is congruent to figure G, we write it concisely as:

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Examples of Congruent Plane Figures

Real-world objects and geometric figures provide many examples of congruence:

• Manufactured Items:
  • Two coins of the same denomination and from the same minting batch (e.g., two brand new $\textsf{₹}10$ coins) are designed to be congruent.
  • Two identical postage stamps from the same sheet are congruent.
  • Nails, screws, or bolts of the exact same specification are congruent.
  • Printed photographs of the same size from the same digital image file are congruent.
• Geometric Shapes:
  • Two line segments of the exact same length are congruent.
  • Two angles with the exact same measure are congruent.
  • Two squares with sides of equal length are congruent. If square A has side length $s_1$ and square B has side length $s_2$, then square A $\cong$ square B if and only if $s_1 = s_2$.
  • Two circles with the exact same radius are congruent. If circle C1 has radius $r_1$ and circle C2 has radius $r_2$, then C1 $\cong$ C2 if and only if $r_1 = r_2$.
  • Two equilateral triangles with the same side length are congruent.

In the diagram, the two squares are congruent because they have the same side length and same shape. The two circles are congruent because they have the same radius.

The concept of congruence is fundamental when we discuss the relationships between geometric figures, particularly triangles, which we will explore in subsequent sections.

Congruence of Geometric Figures - Congruence of Line Segments and Angles

Having introduced the general concept of congruence for plane figures, we can now specifically apply this idea to the most basic geometric elements: line segments and angles. Understanding when these fundamental components are congruent is crucial for establishing congruence in more complex figures like triangles.

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Congruence of Line Segments

A line segment is a finite portion of a line defined by two endpoints. Its measurable attribute is its length. Congruence for line segments is defined directly in terms of their lengths.

Definition: Two line segments are congruent if and only if they have the same length.

If $\overline{\text{AB}}$ and $\overline{\text{CD}}$ are two line segments, then:

$\overline{\text{AB}} \cong \overline{\text{CD}}$ if and only if the length of $\overline{\text{AB}}$ is equal to the length of $\overline{\text{CD}}$.

Using notation for length:

Conversely, if $\overline{\text{AB}} \cong \overline{\text{CD}}$, then $\text{AB} = \text{CD}$.

For example, if line segment $\overline{\text{PQ}}$ measures 7 cm and line segment $\overline{\text{XY}}$ also measures 7 cm, then $\overline{\text{PQ}} \cong \overline{\text{XY}}$.

In the diagram, the single dash on $\overline{\text{AB}}$ and $\overline{\text{CD}}$ indicates that they have the same length, and thus are congruent.

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Congruence of Angles

An angle is formed by two rays sharing a common endpoint. Its measurable attribute is its measure (typically in degrees). Congruence for angles is defined based on their measures.

Definition: Two angles are congruent if and only if they have the same measure.

If $\angle \text{ABC}$ and $\angle \text{PQR}$ are two angles, then:

$\angle \text{ABC} \cong \angle \text{PQR}$ if and only if the measure of $\angle \text{ABC}$ is equal to the measure of $\angle \text{PQR}$.

Using notation for measure ($\text{m}\angle$) or simply equality for measure ($\angle$):

Conversely, if $\angle \text{ABC} \cong \angle \text{PQR}$, then $\text{m}\angle \text{ABC} = \text{m}\angle \text{PQR}$.

For example, if $\angle \text{X}$ measures $60^\circ$ and $\angle \text{Y}$ measures $60^\circ$, then $\angle \text{X} \cong \angle \text{Y}$.

In the diagram, the single arc on $\angle \text{ABC}$ and $\angle \text{PQR}$ indicates they have the same measure, and thus are congruent.

It is important to note that the lengths of the arms of the angle do not determine the angle's measure or congruence. The measure of an angle is determined only by the amount of rotation between the two rays from the vertex.

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Example

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Congruence of Geometric Figures - Congruence of Triangles: Definition

Triangles are fundamental polygons, and the concept of congruence is particularly important when applied to them. Two triangles are congruent if they are identical in shape and size. This means that one triangle can be moved (translated, rotated, or reflected) to perfectly coincide with the other. Congruence of triangles implies the equality of all their corresponding parts.

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Definition of Congruent Triangles

Two triangles are defined as congruent if and only if there is a correspondence between their vertices such that each pair of corresponding sides and each pair of corresponding angles are equal.

If $\triangle \text{ABC}$ is congruent to $\triangle \text{PQR}$, we denote this relationship using the congruence symbol:

This notation $\triangle \text{ABC} \cong \triangle \text{PQR}$ implies a specific correspondence between the vertices of the two triangles:

• Vertex A corresponds to Vertex P ($A \leftrightarrow P$)
• Vertex B corresponds to Vertex Q ($B \leftrightarrow Q$)
• Vertex C corresponds to Vertex R ($C \leftrightarrow R$)

This correspondence between vertices means that the parts connecting or located at these corresponding vertices are also corresponding parts. Specifically, if $\triangle \text{ABC} \cong \triangle \text{PQR}$ with the correspondence A $\leftrightarrow$ P, B $\leftrightarrow$ Q, C $\leftrightarrow$ R, then:

Corresponding Sides are Equal:

The sides connecting corresponding vertices are equal in length:

1. Side $\overline{\text{AB}}$ corresponds to side $\overline{\text{PQ}}$. Their lengths are equal: $\text{AB} = \text{PQ}$.
2. Side $\overline{\text{BC}}$ corresponds to side $\overline{\text{QR}}$. Their lengths are equal: $\text{BC} = \text{QR}$.
3. Side $\overline{\text{AC}}$ corresponds to side $\overline{\text{PR}}$. Their lengths are equal: $\text{AC} = \text{PR}$.

(Note: Using the congruence notation for segments, $\overline{\text{AB}} \cong \overline{\text{PQ}}$, $\overline{\text{BC}} \cong \overline{\text{QR}}$, $\overline{\text{AC}} \cong \overline{\text{PR}}$)

Corresponding Angles are Equal:

The angles at corresponding vertices are equal in measure:

1. Angle $\angle \text{A}$ (or $\angle \text{BAC}$) corresponds to angle $\angle \text{P}$ (or $\angle \text{QPR}$). Their measures are equal: $\text{m}\angle \text{A} = \text{m}\angle \text{P}$.
2. Angle $\angle \text{B}$ (or $\angle \text{ABC}$) corresponds to angle $\angle \text{Q}$ (or $\angle \text{PQR}$). Their measures are equal: $\text{m}\angle \text{B} = \text{m}\angle \text{Q}$.
3. Angle $\angle \text{C}$ (or $\angle \text{BCA}$) corresponds to angle $\angle \text{R}$ (or $\angle \text{PRQ}$). Their measures are equal: $\text{m}\angle \text{C} = \text{m}\angle \text{R}$.

(Note: Using the congruence notation for angles, $\angle \text{A} \cong \angle \text{P}$, $\angle \text{B} \cong \angle \text{Q}$, $\angle \text{C} \cong \angle \text{R}$)

So, by definition, if two triangles are congruent, all six pairs of corresponding parts are equal. Conversely, if all six pairs of corresponding parts of two triangles are equal, then the triangles are congruent.

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Importance of Order in Congruence Statement

When writing a congruence statement between two triangles, the order of the vertices is extremely important. The order establishes the vertex correspondence, which in turn dictates which sides and angles are corresponding.

For example, saying $\triangle \text{ABC} \cong \triangle \text{PQR}$ means:

This implies $\text{AB} = \text{PQ}$, $\text{BC} = \text{QR}$, $\text{AC} = \text{PR}$, and $\angle \text{A} = \angle \text{P}$, $\angle \text{B} = \angle \text{Q}$, $\angle \text{C} = \angle \text{R}$.

However, saying $\triangle \text{ABC} \cong \triangle \text{QRP}$ means:

This implies $\text{AB} = \text{QR}$, $\text{BC} = \text{RP}$, $\text{AC} = \text{QP}$, and $\angle \text{A} = \angle \text{Q}$, $\angle \text{B} = \angle \text{R}$, $\angle \text{C} = \angle \text{P}$. This is a completely different set of equalities unless the triangles have further specific properties (like being isosceles or equilateral).

Therefore, when stating that two triangles are congruent, always ensure that the vertices are listed in the correct corresponding order.

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The definition of congruence requires the equality of all six parts. However, proving the equality of all six parts every time would be tedious. Fortunately, there are specific minimum sets of conditions (involving only three pairs of parts) that are sufficient to guarantee the congruence of two triangles. These are known as the congruence criteria or postulates/theorems, which we will discuss next.

Congruence of Geometric Figures - Criteria for Congruence of Triangles (SSS, SAS, ASA, AAS, RHS)

As defined earlier, two triangles are congruent if and only if all six pairs of their corresponding parts (three sides and three angles) are equal. However, checking all six conditions every time can be time-consuming and is often unnecessary. Mathematicians have identified minimum sets of conditions that are sufficient to prove the congruence of two triangles. These are known as the Congruence Criteria or postulates/theorems.

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Minimum Conditions for Triangle Congruence

If any one of the following criteria is met between two triangles, then the triangles are congruent, and consequently, all their remaining corresponding parts are also equal (by CPCTC, which we will discuss next).

1. SSS (Side-Side-Side) Congruence Criterion

Statement: If the three sides of one triangle are respectively equal in length to the three corresponding sides of another triangle, then the two triangles are congruent.

Condition: In $\triangle \text{ABC}$ and $\triangle \text{PQR}$, if:

• $\text{AB} = \text{PQ}$ (First pair of corresponding sides)
• $\text{BC} = \text{QR}$ (Second pair of corresponding sides)
• $\text{AC} = \text{PR}$ (Third pair of corresponding sides)

Conclusion: $\triangle \text{ABC} \cong \triangle \text{PQR}$.

This is often taken as a postulate.

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2. SAS (Side-Angle-Side) Congruence Criterion

Statement: If two sides and the included angle (the angle formed by these two sides) of one triangle are respectively equal to two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

Condition: In $\triangle \text{ABC}$ and $\triangle \text{PQR}$, if:

• $\text{AB} = \text{PQ}$ (First pair of corresponding sides)
• $\angle \text{BAC} = \angle \text{QPR}$ (or $\angle \text{A} = \angle \text{P}$) (The angle included between sides AB and AC, corresponding to the angle included between sides PQ and PR)
• $\text{AC} = \text{PR}$ (Second pair of corresponding sides)

Conclusion: $\triangle \text{ABC} \cong \triangle \text{PQR}$.

Important Note: The position of the angle is critical. The angle must be the one formed by the two sides that are given as equal. SSA (Side-Side-Angle) or ASS (Angle-Side-Side), where the angle is not included between the two given sides, is generally not a valid criterion for congruence because in some cases, two different triangles can be formed with the same SSA information (this is known as the ambiguous case in trigonometry).

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3. ASA (Angle-Side-Angle) Congruence Criterion

Statement: If two angles and the included side (the side common to both angles) of one triangle are respectively equal to two corresponding angles and the included side of another triangle, then the two triangles are congruent.

Condition: In $\triangle \text{ABC}$ and $\triangle \text{PQR}$, if:

• $\angle \text{ABC} = \angle \text{PQR}$ (or $\angle \text{B} = \angle \text{Q}$) (First pair of corresponding angles)
• $\text{BC} = \text{QR}$ (The side included between angles B and C, corresponding to the side included between angles Q and R)
• $\angle \text{BCA} = \angle \text{QRP}$ (or $\angle \text{C} = \angle \text{R}$) (Second pair of corresponding angles)

Conclusion: $\triangle \text{ABC} \cong \triangle \text{PQR}$.

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4. AAS (Angle-Angle-Side) Congruence Criterion

Statement: If two angles and a non-included side of one triangle are respectively equal to two corresponding angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

Condition: In $\triangle \text{ABC}$ and $\triangle \text{PQR}$, if:

• $\angle \text{BAC} = \angle \text{QPR}$ (or $\angle \text{A} = \angle \text{P}$) (First pair of corresponding angles)
• $\angle \text{ABC} = \angle \text{PQR}$ (or $\angle \text{B} = \angle \text{Q}$) (Second pair of corresponding angles)
• $\text{BC} = \text{QR}$ (A non-included side in $\triangle \text{ABC}$, corresponding to a non-included side in $\triangle \text{PQR}$)

Conclusion: $\triangle \text{ABC} \cong \triangle \text{PQR}$.

Justification: The AAS criterion can be proven using the ASA criterion and the Angle Sum Property. If two angles of one triangle are equal to two corresponding angles of another triangle ($\angle \text{A} = \angle \text{P}$ and $\angle \text{B} = \angle \text{Q}$), then the third pair of angles must also be equal because the sum of angles in any triangle is $180^\circ$. That is, $\text{m}\angle \text{C} = 180^\circ - (\text{m}\angle \text{A} + \text{m}\angle \text{B})$ and $\text{m}\angle \text{R} = 180^\circ - (\text{m}\angle \text{P} + \text{m}\angle \text{Q})$. Since $\text{m}\angle \text{A} = \text{m}\angle \text{P}$ and $\text{m}\angle \text{B} = \text{m}\angle \text{Q}$, it follows that $\text{m}\angle \text{C} = \text{m}\angle \text{R}$. Now, if we are given that a non-included side, say BC, is equal to the corresponding non-included side QR ($\text{BC} = \text{QR}$), we can re-examine the situation. We have $\angle \text{B} = \angle \text{Q}$ (given), $\text{BC} = \text{QR}$ (given), and $\angle \text{C} = \angle \text{R}$ (just proven). This matches the conditions for the ASA congruence criterion. Therefore, $\triangle \text{ABC} \cong \triangle \text{PQR}$.

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5. RHS (Right angle-Hypotenuse-Side) Congruence Criterion

Statement: If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side of the other triangle, then the two triangles are congruent.

Condition: In two triangles $\triangle \text{ABC}$ and $\triangle \text{PQR}$:

• One angle is a right angle: $\angle \text{B} = 90^\circ$ and $\angle \text{Q} = 90^\circ$ (R - Right angle)
• The hypotenuses are equal: $\text{AC} = \text{PR}$ (H - Hypotenuse)
• One pair of corresponding sides (legs) are equal: $\text{AB} = \text{PQ}$ (S - Side) OR $\text{BC} = \text{QR}$ (S - Side)

Conclusion: $\triangle \text{ABC} \cong \triangle \text{PQR}$.

Note: This criterion is a special case applicable only to right-angled triangles. It is not the same as SSA, as the properties of right triangles allow us to prove congruence in this specific scenario (e.g., using the Pythagorean theorem to show the third sides are also equal, leading to SSS congruence).

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Summary of Congruence Criteria

To prove triangle congruence, check for one of these minimum sets of equal corresponding parts:

• SSS: 3 sides equal
• SAS: 2 sides and the included angle equal
• ASA: 2 angles and the included side equal
• AAS: 2 angles and a non-included side equal
• RHS: (For right triangles) Right angle, Hypotenuse, and one Leg equal

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Congruence of Geometric Figures - CPCTC (Corresponding Parts of Congruent Triangles)

After proving that two triangles are congruent using one of the congruence criteria (SSS, SAS, ASA, AAS, RHS), we gain a powerful tool for deducing the equality of the remaining corresponding parts. This principle is so frequently used in geometric proofs that it is referred to by the acronym CPCT.

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Statement of the CPCT Principle

CPCT stands for Corresponding Parts of Congruent Triangles are Congruent (or equal in measure/length).

Principle: If two triangles are proven to be congruent, then every pair of corresponding sides is equal in length, and every pair of corresponding angles is equal in measure.

This principle is a direct consequence of the definition of congruence. Once you establish congruence based on a minimum set of conditions (like SSS), it guarantees that the triangles are identical in every respect, including the parts you didn't initially use for the congruence proof.

If you have proven that $\triangle \text{ABC} \cong \triangle \text{PQR}$ (with the specific vertex correspondence A $\leftrightarrow$ P, B $\leftrightarrow$ Q, C $\leftrightarrow$ R), then you can confidently state, using CPCT as the reason, that:

• Corresponding Sides: $\text{AB} = \text{PQ}$, $\text{BC} = \text{QR}$, $\text{AC} = \text{PR}$
• Corresponding Angles: $\text{m}\angle \text{A} = \text{m}\angle \text{P}$, $\text{m}\angle \text{B} = \text{m}\angle \text{Q}$, $\text{m}\angle \text{C} = \text{m}\angle \text{R}$

Even if, for example, you used SSS to prove the triangles congruent (using $\text{AB}=\text{PQ}$, $\text{BC}=\text{QR}$, $\text{AC}=\text{PR}$), CPCT allows you to conclude that $\angle \text{A}=\angle \text{P}$, $\angle \text{B}=\angle \text{Q}$, and $\angle \text{C}=\angle \text{R}$.

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Usage of CPCT in Proofs

CPCT is an essential step in many geometric proofs. Often, the goal of a proof is not merely to show that two triangles are congruent, but to use that congruence to prove that certain sides or angles within those triangles are equal. CPCT provides the justification for this step.

The sequence is typically:

1. Identify the triangles you want to prove congruent.
2. Find three pairs of corresponding parts that satisfy one of the congruence criteria (SSS, SAS, ASA, AAS, RHS).
3. State that the two triangles are congruent, citing the specific criterion used.
4. Use CPCT to conclude that any other pair of corresponding sides or angles is equal.

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