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Chapter 7 Congruence of Triangles (Concepts)
Building upon our previous exploration of triangles and their intrinsic properties, this chapter introduces the fundamental geometric concept of congruence, with a specific focus on its application to triangles. The term congruence essentially means identical in both shape and size. When we state that two geometric figures are congruent, we imply that one figure can be perfectly superimposed onto the other, matching exactly at every point. Think of it like having two identical copies of the same shape cut out from paper; if they are congruent, one will fit precisely over the other without any overlap or shortage.
While this concept of perfect duplication applies broadly – for instance, two line segments are congruent if they possess exactly the same length, and two angles are congruent if they have the same measure – our primary concern here is the congruence of triangles. Two triangles, say $\triangle ABC$ and $\triangle PQR$, are defined as congruent if all their corresponding parts match perfectly. This means two conditions must be simultaneously met:
- All three pairs of corresponding sides must be equal in length ($AB = PQ$, $BC = QR$, $AC = PR$).
- All three pairs of corresponding angles must be equal in measure ($\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$).
Establishing congruence by methodically verifying all six pairs of corresponding elements (three sides and three angles) can be rather tedious and often unnecessary. Fortunately, geometry provides us with specific sets of minimum conditions, known as the congruence criteria (or congruence rules/postulates), which are sufficient to guarantee that two triangles are congruent without checking every single part.
This chapter meticulously presents these powerful criteria:
- SSS (Side-Side-Side) Criterion: If the three sides of one triangle are respectively equal in length to the corresponding three sides of another triangle, then the two triangles are congruent.
- SAS (Side-Angle-Side) Criterion: If two sides and the included angle (meaning the angle formed *between* those two specific sides) of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.
- ASA (Angle-Side-Angle) Criterion: If two angles and the included side (meaning the side situated *between* those two specific angles) of one triangle are equal to the corresponding two angles and the included side of another triangle, then the triangles are congruent.
- RHS (Right angle-Hypotenuse-Side) Criterion: This criterion applies exclusively to right-angled triangles. If the hypotenuse and one leg (a side other than the hypotenuse) of one right-angled triangle are equal to the corresponding hypotenuse and one leg of another right-angled triangle, then the two triangles are congruent.
Students will learn to carefully analyze given information, whether presented in diagrams or textual statements, to identify pairs of equal corresponding parts (sides or angles). Based on the available information, the appropriate congruence criterion (SSS, SAS, ASA, or RHS) must be selected and correctly applied to formally prove that two triangles are congruent. Once congruence has been established, for example, $\triangle ABC \cong \triangle PQR$, we can invoke a profoundly useful consequence known as CPCT, which stands for "Corresponding Parts of Congruent Triangles are equal". This principle allows us to confidently deduce that any other pair of corresponding parts (those not initially used to prove congruence) must also be equal (e.g., if we used SSS, CPCT allows us to state that $\angle A = \angle P$). This makes congruence a cornerstone for geometric proofs and for understanding concepts like structural stability and rigidity. It is absolutely crucial to maintain the correct order of vertices when writing a congruence statement, as $\triangle ABC \cong \triangle PQR$ explicitly implies that vertex A corresponds to P, B corresponds to Q, and C corresponds to R.
Congruence of Plane Figures
In geometry, we classify shapes based on their properties like the number of sides, angle measures, etc. We also compare shapes. Sometimes, two shapes look exactly alike – they have the same shape and the same size. This concept of being identical in all respects is called Congruence. Understanding congruence is fundamental because it allows us to determine when two geometric figures are exact copies of each other.
What is Congruence?
Two geometric figures are said to be Congruent if they have the exact same shape and the exact same size. If you can pick up one figure and place it directly on top of the other figure such that they perfectly overlap and cover each other completely, then the two figures are congruent.
The symbol used to denote congruence is $\cong$. If figure F1 is congruent to figure F2, we write $F1 \cong F2$.
Think of two leaves from the same plant that are identical, two identical copies of the same photograph, or two coins of the same denomination and year. These are examples of congruent objects.
For two figures to be congruent, all their corresponding parts must be equal. For polygons, this means corresponding sides must have equal lengths, and corresponding angles must have equal measures.
Congruence of Line Segments
Two line segments are congruent if and only if they have the same length.
If line segment AB has a length of 5 cm, and line segment CD also has a length of 5 cm, then line segment AB is congruent to line segment CD.
If length(AB) = length(CD), then $AB \cong CD$.
Conversely, if two line segments are congruent, it means their lengths are equal.
If $AB \cong CD$, then length(AB) = length(CD).
Congruence of Angles
Two angles are congruent if and only if they have the same measure. The measure of an angle refers to the 'opening' between its two arms, typically measured in degrees or radians.
If the measure of $\angle ABC$ is $45^\circ$ and the measure of $\angle PQR$ is $45^\circ$, then $\angle ABC$ is congruent to $\angle PQR$.
If $m\angle ABC = m\angle PQR$, then $\angle ABC \cong \angle PQR$.
Conversely, if two angles are congruent, their measures are equal.
If $\angle ABC \cong \angle PQR$, then $m\angle ABC = m\angle PQR$.
Note: The lengths of the arms of the angle do not affect its measure or its congruence. The measure only depends on the amount of rotation between the two arms.
Congruence of Squares
A square is defined by having four equal sides and four right angles. Two squares are congruent if their corresponding side lengths are equal. Since all sides of a square are equal, checking if the side length of one square is equal to the side length of another square is sufficient to determine their congruence.
If square S1 has a side length of 4 cm and square S2 also has a side length of 4 cm, then $S1 \cong S2$. Because their side lengths are equal, all their corresponding parts will be equal (all sides 4 cm, all angles $90^\circ$), making them identical.
If the side lengths are different, the squares are similar (same shape) but not congruent (different size).
Congruence of Rectangles
A rectangle is defined by having opposite sides equal and parallel and four right angles. Two rectangles are congruent if their corresponding lengths are equal and their corresponding breadths are equal.
If rectangle R1 has length $l_1$ and breadth $b_1$, and rectangle R2 has length $l_2$ and breadth $b_2$, then R1 $\cong$ R2 if and only if $l_1 = l_2$ and $b_1 = b_2$. Note that the orientation might differ, but the dimensions must match. A rectangle with length 5 and breadth 3 is congruent to a rectangle with length 3 and breadth 5 if orientation is considered, but generally, we compare the set of dimensions.
Congruence of Circles
A circle is defined by its center and its radius (or diameter). Two circles are congruent if they have the exact same size, which is determined by their radius or diameter.
Two circles are congruent if and only if their radii are equal (or their diameters are equal, since diameter is twice the radius).
If circle C1 has radius $r_1$ and circle C2 has radius $r_2$, then C1 $\cong$ C2 if and only if $r_1 = r_2$.
The position of the center does not affect the congruence of circles; only their size matters.
Congruence of General Plane Figures
For any two plane figures, regardless of their complexity, they are congruent ($F1 \cong F2$) if one can be perfectly superimposed on the other. This is the fundamental test of congruence based on the idea of rigid motion (movements like sliding, rotating, or flipping a figure without changing its size or shape). If F1 can be moved to match F2 exactly, they are congruent.
This superposition principle means that all corresponding parts of two congruent figures are equal. For polygons, this implies that corresponding sides are equal in length, and corresponding angles are equal in measure. For curves, it means corresponding segments of the curves match perfectly.
Example 1. Fill in the blanks:
(a) Two line segments are congruent if __________.
(b) Among two congruent angles, one has a measure of $70^\circ$; the measure of the other angle is __________.
(c) When we write $\angle A = \angle B$, we actually mean __________.
Answer:
(a) Two line segments are congruent if they have the same length.
(b) Among two congruent angles, one has a measure of $70^\circ$; the measure of the other angle is $70^\circ$. (By the definition of congruent angles, their measures are equal).
(c) When we write $\angle A = \angle B$, we are often using a shorthand notation in geometry. What is strictly meant is that the measure of angle A is equal to the measure of angle B. So, we actually mean $m\angle A = m\angle B$. (This equality in measure implies that the angles are congruent, $\angle A \cong \angle B$).
Example 2. If $\triangle ABC \cong \triangle FED$ under the correspondence $ABC \leftrightarrow FED$, write all the corresponding congruent parts of the triangles.
Answer:
Given that $\triangle ABC \cong \triangle FED$. The order of the vertices in the congruence statement is very important as it tells us which vertex corresponds to which vertex, and thus which sides and angles correspond.
The correspondence $ABC \leftrightarrow FED$ means:
- Vertex A corresponds to Vertex F ($A \leftrightarrow F$).
- Vertex B corresponds to Vertex E ($B \leftrightarrow E$).
- Vertex C corresponds to Vertex D ($C \leftrightarrow D$).
When two triangles are congruent, all their corresponding parts are congruent (equal in measure for angles and equal in length for sides). This is often abbreviated as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Therefore, the corresponding congruent parts are:
Corresponding Angles: The angles at corresponding vertices are congruent.
- Angle at A corresponds to Angle at F. So, $\angle A \cong \angle F$ (or $m\angle A = m\angle F$).
- Angle at B corresponds to Angle at E. So, $\angle B \cong \angle E$ (or $m\angle B = m\angle E$).
- Angle at C corresponds to Angle at D. So, $\angle C \cong \angle D$ (or $m\angle C = m\angle D$).
Corresponding Sides: The sides connecting corresponding vertices are congruent. The side connecting the first two vertices of the first triangle corresponds to the side connecting the first two vertices of the second triangle, and so on.
- Side AB (connecting A and B) corresponds to Side FE (connecting F and E). So, $AB \cong FE$ (or length $AB = FE$).
- Side BC (connecting B and C) corresponds to Side ED (connecting E and D). So, $BC \cong ED$ (or length $BC = ED$).
- Side CA (connecting C and A) corresponds to Side DF (connecting D and F). So, $CA \cong DF$ (or length $CA = DF$). (Note the order CA matches DF based on vertex correspondence C$\leftrightarrow$D, A$\leftrightarrow$F).
In summary, because $\triangle ABC \cong \triangle FED$, the six corresponding parts are congruent:
- $\angle A \cong \angle F$
- $\angle B \cong \angle E$
- $\angle C \cong \angle D$
- $AB \cong FE$
- $BC \cong ED$
- $CA \cong DF$
Congruence of Triangles & its Criteria
We have established the general idea of congruence for plane figures – that they are identical in shape and size. Triangles are particularly important figures in geometry, and proving that two triangles are congruent is a common task. If two triangles are congruent, all their corresponding parts (sides and angles) are equal. However, checking all six pairs of corresponding parts can be tedious. Fortunately, there are minimum conditions that are sufficient to guarantee that two triangles are congruent. These conditions are known as congruence criteria or rules.
Congruence of Triangles
Two triangles are Congruent if and only if their corresponding sides and corresponding angles are equal. This means that if you could perfectly superimpose one triangle onto the other, they would match in all respects.
When we state that $\triangle ABC$ is congruent to $\triangle PQR$, written as $\triangle ABC \cong \triangle PQR$, the order of the vertices is extremely important. It specifies the correspondence between the vertices of the two triangles:
- The first vertex of the first triangle corresponds to the first vertex of the second triangle: Vertex A corresponds to Vertex P ($A \leftrightarrow P$).
- The second vertex of the first triangle corresponds to the second vertex of the second triangle: Vertex B corresponds to Vertex Q ($B \leftrightarrow Q$).
- The third vertex of the first triangle corresponds to the third vertex of the second triangle: Vertex C corresponds to Vertex R ($C \leftrightarrow R$).
This one-to-one correspondence between vertices implies the equality of corresponding parts:
Corresponding Sides are Equal in Length:
- Side AB (connecting A and B) corresponds to Side PQ (connecting P and Q). So, $AB = PQ$.
- Side BC (connecting B and C) corresponds to Side QR (connecting Q and R). So, $BC = QR$.
- Side CA (connecting C and A) corresponds to Side RP (connecting R and P). So, $CA = RP$.
Corresponding Angles are Equal in Measure:
- Angle at vertex A corresponds to Angle at vertex P. So, $m\angle A = m\angle P$.
- Angle at vertex B corresponds to Angle at vertex Q. So, $m\angle B = m\angle Q$.
- Angle at vertex C corresponds to Angle at vertex R. So, $m\angle C = m\angle R$.
The statement $\triangle ABC \cong \triangle PQR$ encapsulates all these six equalities of corresponding parts. The principle that corresponding parts of congruent triangles are congruent is so fundamental that it has its own acronym: CPCTC.
Criteria for Congruence of Triangles
Checking all six pairs of corresponding parts to prove congruence can be tedious. Fortunately, mathematicians have identified certain minimum conditions that, if met, are sufficient to guarantee that two triangles are congruent. These are the congruence criteria (sometimes called postulates or theorems).
1. SSS (Side-Side-Side) Congruence Criterion
Statement: If the three sides of one triangle are equal in length to the three corresponding sides of another triangle, then the two triangles are congruent.
If in $\triangle ABC$ and $\triangle PQR$, the following conditions are met:
- Side AB = Side PQ (First pair of corresponding sides)
- Side BC = Side QR (Second pair of corresponding sides)
- Side CA = Side RP (Third pair of corresponding sides)
Then, we can conclude that $\triangle ABC \cong \triangle PQR$ by the SSS Congruence Criterion.
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 equal to the corresponding two sides and the included angle of the other triangle, then the two triangles are congruent.
If in $\triangle ABC$ and $\triangle PQR$, the following conditions are met:
- Side AB = Side PQ (First pair of corresponding sides)
- The angle included between sides AB and BC, $\angle B$, is equal in measure to the angle included between sides PQ and QR, $\angle Q$. So, $m\angle B = m\angle Q$ (Included Angle).
- Side BC = Side QR (Second pair of corresponding sides)
Then, we can conclude that $\triangle ABC \cong \triangle PQR$ by the SAS (Side-Angle-Side) Congruence Criterion.
Important Note: The angle must be the included angle, i.e., the angle formed by the two sides whose lengths are being compared. If the angle is not the included angle (this situation is often called SSA or ASS), it is generally NOT a valid congruence criterion for all triangles. There are cases where SSA can lead to two different possible triangles.
3. ASA (Angle-Side-Angle) Congruence Criterion
Statement: If two angles and the included side (the side between these two angles) of one triangle are equal to the corresponding two angles and the included side of the other triangle, then the two triangles are congruent.
If in $\triangle ABC$ and $\triangle PQR$, the following conditions are met:
- Angle $\angle B$ is equal in measure to angle $\angle Q$. So, $m\angle B = m\angle Q$ (First Angle).
- The side included between angles $\angle B$ and $\angle C$, side BC, is equal in length to the side included between angles $\angle Q$ and $\angle R$, side QR. So, $BC = QR$ (Included Side).
- Angle $\angle C$ is equal in measure to angle $\angle R$. So, $m\angle C = m\angle R$ (Second Angle).
Then, we can conclude that $\triangle ABC \cong \triangle PQR$ by the ASA (Angle-Side-Angle) Congruence Criterion.
4. AAS (Angle-Angle-Side) Congruence Criterion
Statement: If two angles and a non-included side (a side that is not between the two angles) of one triangle are equal to the corresponding two angles and the non-included side of the other triangle, then the two triangles are congruent.
This criterion can be proven using the ASA criterion and the Angle Sum Property of a triangle. If two angles of a triangle are equal to two corresponding angles of another triangle, then the third angles must also be equal (because the sum of angles in a triangle is $180^\circ$). Once the third angles are known to be equal, the AAS case effectively becomes an ASA case.
If in $\triangle ABC$ and $\triangle PQR$, the following conditions are met:
- Angle $\angle B$ is equal in measure to angle $\angle Q$. So, $m\angle B = m\angle Q$ (First Angle).
- Angle $\angle C$ is equal in measure to angle $\angle R$. So, $m\angle C = m\angle R$ (Second Angle).
- A side that is NOT included between these two angles, say side AB, is equal in length to the corresponding side in the other triangle, PQ. So, $AB = PQ$ (Non-included Side corresponding to $\angle C$ and $\angle R$).
Then, we can conclude that $\triangle ABC \cong \triangle PQR$ by the AAS (Angle-Angle-Side) Congruence Criterion.
Note: The non-included side must correspond to the same non-included angle in the other triangle based on the vertex correspondence.
5. RHS (Right angle-Hypotenuse-Side) Congruence Criterion
This is a special criterion that applies only to right-angled triangles. It is the only case where SSA (Angle-Side-Side with the angle not included) works as a congruence criterion.
Statement: Two right-angled triangles are congruent if the hypotenuse and one side (leg) of one triangle are equal in length to the hypotenuse and the corresponding side (leg) of the other triangle.
If in right-angled $\triangle ABC$ (with $m\angle B = 90^\circ$) and right-angled $\triangle PQR$ (with $m\angle Q = 90^\circ$), the following conditions are met:
- Both triangles are right-angled ($m\angle B = m\angle Q = 90^\circ$). (The 'R' in RHS).
- The length of the hypotenuse of $\triangle ABC$, AC, is equal to the length of the hypotenuse of $\triangle PQR$, PR ($AC = PR$). (The 'H' in RHS).
- The length of one leg of $\triangle ABC$, say AB, is equal to the length of the corresponding leg of $\triangle PQR$, PQ ($AB = PQ$). (The 'S' in RHS). Or, the leg BC = QR.
Then, we can conclude that $\triangle ABC \cong \triangle PQR$ by the RHS (Right angle-Hypotenuse-Side) Congruence Criterion.
Example 1. In the figure, $AC = AD$ and $AB$ bisects $\angle CAD$.
(i) State three pairs of equal parts in $\triangle ABC$ and $\triangle ABD$.
(ii) Is $\triangle ABC \cong \triangle ABD$? Give reasons.
(iii) Is $BC = BD$? Give reasons.
Answer:
We are given a figure involving two triangles, $\triangle ABC$ and $\triangle ABD$. We are given that $AC = AD$ and that AB bisects $\angle CAD$.
(i) State three pairs of equal parts in $\triangle ABC$ and $\triangle ABD$.
Let's look for sides or angles that are given to be equal or are common to both triangles.
- We are given that $AC = AD$. This is a pair of corresponding sides.
$AC = AD$
(Given)
- We are given that AB bisects $\angle CAD$. This means AB divides $\angle CAD$ into two equal angles. The angle $\angle CAD$ is formed by rays AC and AD. Ray AB is between them. So, the angles formed are $\angle CAB$ and $\angle DAB$.
$m\angle CAB = m\angle DAB$
(Since AB bisects $\angle CAD$)
- Look at side AB. It is an arm of $\angle CAB$ in $\triangle ABC$ and also an arm of $\angle DAB$ in $\triangle ABD$. It is a side of $\triangle ABC$ and also a side of $\triangle ABD$. Thus, side AB is common to both triangles.
$AB = AB$
(Common side)
So, three pairs of equal parts are $AC = AD$, $m\angle CAB = m\angle DAB$, and $AB = AB$.
(ii) Is $\triangle ABC \cong \triangle ABD$? Give reasons.
To check for congruence, look at the equal parts we found in part (i):
- Side $AC$ in $\triangle ABC$ is equal to Side $AD$ in $\triangle ABD$.
- Angle $\angle CAB$ in $\triangle ABC$ is equal to Angle $\angle DAB$ in $\triangle ABD$. This angle is formed by sides AC and AB in $\triangle ABC$, and by sides AD and AB in $\triangle ABD$. Thus, it is the included angle between the two sides we are considering ($AC, AB$) and ($AD, AB$).
- Side $AB$ in $\triangle ABC$ is equal to Side $AB$ in $\triangle ABD$.
We have two sides and the included angle of $\triangle ABC$ equal to the corresponding two sides and the included angle of $\triangle ABD$. This matches the conditions of the SAS congruence criterion.
Yes, $\triangle ABC \cong \triangle ABD$.
Reason: By the SAS (Side-Angle-Side) congruence criterion, $\triangle ABC$ is congruent to $\triangle ABD$ because $AC = AD$ (Side), $m\angle CAB = m\angle DAB$ (Included Angle), and $AB = AB$ (Side).
(iii) Is $BC = BD$? Give reasons.
We have just proved that $\triangle ABC \cong \triangle ABD$ in part (ii). When two triangles are congruent, all their corresponding parts are equal (CPCTC).
We need to check if BC and BD are corresponding sides in the congruence $\triangle ABC \cong \triangle ABD$.
BC is the side opposite vertex A in $\triangle ABC$. BD is the side opposite vertex A in $\triangle ABD$. Based on the vertex correspondence $A \leftrightarrow A$, $B \leftrightarrow B$, $C \leftrightarrow D$, the side BC (connecting vertices B and C) corresponds to the side BD (connecting vertices B and D).
Yes, $BC = BD$.
Reason: Since $\triangle ABC \cong \triangle ABD$ (as proved in part ii), their corresponding parts are equal. $BC$ and $BD$ are corresponding sides of these congruent triangles. Therefore, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $BC = BD$.
Example 2. In $\triangle PQR$, $PQ = PR$. D is the mid-point of QR. Prove that $\triangle PQD \cong \triangle PRD$. What can you say about $\angle PDQ$?
Answer:
Given: In $\triangle PQR$, $PQ = PR$. D is the mid-point of QR.
To Prove: $\triangle PQD \cong \triangle PRD$.
Proof:
Consider the two triangles $\triangle PQD$ and $\triangle PRD$. We need to find three pairs of corresponding parts that are equal to apply a congruence criterion.
- We are given that $PQ = PR$. This is a pair of corresponding sides (PQ in $\triangle PQD$ and PR in $\triangle PRD$).
$PQ = PR$
(Given)
- We are given that D is the mid-point of QR. This means the line segment QR is divided into two equal parts at D. The segments QD and RD are sides of $\triangle PQD$ and $\triangle PRD$ respectively.
$QD = RD$
(Since D is the mid-point of QR)
- Look at the line segment PD. It is drawn from vertex P to point D on the opposite side. PD is a side of $\triangle PQD$ and also a side of $\triangle PRD$.
$PD = PD$
(Common side to both triangles)
We have found that all three sides of $\triangle PQD$ are equal to the corresponding three sides of $\triangle PRD$. This matches the SSS congruence criterion.
By the SSS (Side-Side-Side) congruence criterion, we have:
- PQ = PR (Side)
- QD = RD (Side)
- PD = PD (Side)
Therefore, $\triangle PQD \cong \triangle PRD$.
(By SSS congruence criterion)
This proves the congruence of the two triangles.
What can you say about $\angle PDQ$?
Since $\triangle PQD \cong \triangle PRD$ (as proved above), their corresponding parts are equal (CPCTC). We can look at the angles $\angle PDQ$ and $\angle PDR$. These angles are formed at point D on the line segment QR. $\angle PDQ$ is in $\triangle PQD$ and is opposite side PQ. $\angle PDR$ is in $\triangle PRD$ and is opposite side PR.
Since $PQ = PR$ (given), the angles opposite these sides in the congruent triangles should correspond. Based on the vertex correspondence $P \leftrightarrow P$, $Q \leftrightarrow R$, $D \leftrightarrow D$, the angle $\angle PDQ$ corresponds to angle $\angle PDR$.
$m\angle PDQ = m\angle PDR$
(CPCTC)
Also, $\angle PDQ$ and $\angle PDR$ are adjacent angles that form a straight line (QR is a line segment, and D is a point on it). Therefore, $\angle PDQ$ and $\angle PDR$ form a linear pair.
The sum of angles in a linear pair is $180^\circ$.
$m\angle PDQ + m\angle PDR = 180^\circ$
(Linear pair)
Since $m\angle PDQ = m\angle PDR$, let their common measure be $x$.
$x + x = 180^\circ$
$2x = 180^\circ$
$x = \frac{180^\circ}{2} = 90^\circ $.
So, $m\angle PDQ = 90^\circ$.
Therefore, $\angle PDQ$ is a right angle ($90^\circ$).
(This shows that in an isosceles triangle, the line segment from the vertex angle to the midpoint of the base is perpendicular to the base. This segment PD is also the median to the base QR in $\triangle PQR$).