Congruence

Compare the three sides of both triangles:
• $AB = PQ = 5\text{ cm}$
• $BC = QR = 6\text{ cm}$
• $AC = PR = 7\text{ cm}$
Since all three corresponding sides are equal, the triangles are congruent by the SSS (Side-Side-Side) Criterion.
• Congruence Statement:

$$\mathbf{\triangle ABC \cong \triangle PQR}$$

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Examine the given parts:
• Side $DE = QR = 6\text{ cm}$
• Side $DF = PR = 8\text{ cm}$
• The angle $\angle D$ is formed between sides $DE$ and $DF$ (the included angle).
• The angle $\angle R$ is formed between sides $QR$ and $PR$ (the included angle).
• Given that $\angle D = \angle R = 50^\circ$.
Since two sides and the included angle of one triangle are equal to the corresponding parts of the other, the triangles are congruent by the SAS (Side-Angle-Side) Criterion.
• Vertex Correspondence: $D \leftrightarrow R$, $E \leftrightarrow Q$, $F \leftrightarrow P$.
• Congruence Statement: $\mathbf{\triangle DEF \cong \triangle RQP}$.

• Definition of Congruence: Requires both identical shape AND identical size.
• The AAA Counterexample: Consider two equilateral triangles:
1. $\triangle_1$ with side length $2\text{ cm}$: its angles are $60^\circ, 60^\circ, 60^\circ$.
2. $\triangle_2$ with side length $8\text{ cm}$: its angles are ALSO $60^\circ, 60^\circ, 60^\circ$.
All three corresponding angles are identical, but $\triangle_2$ is four times larger than $\triangle_1$! If placed over each other, they will not coincide.
• Conclusion: AAA guarantees only that figures have the same shape (Similarity), but tells us nothing about their size. Therefore, AAA cannot prove congruence.

• (a) Proof of $\triangle AOC \cong \triangle BOD$:
In $\triangle AOC$ and $\triangle BOD$:
1. $AO = BO$ (Given that $AB$ is bisected at $O$).
2. $\angle AOC = \angle BOD$ (Vertically Opposite Angles are equal).
3. $CO = DO$ (Given that $CD$ is bisected at $O$).
Therefore, by the SAS Congruence Criterion:

$$\mathbf{\triangle AOC \cong \triangle BOD}$$

• (b) Proof of $AC \parallel BD$:
Since $\triangle AOC \cong \triangle BOD$, by CPCTC (Corresponding Parts of Congruent Triangles):

$$\angle CAO = \angle DBO$$

Notice that $\angle CAO$ and $\angle DBO$ form a pair of Alternate Interior Angles with respect to line segments $AC$ and $BD$ cut by transversal $AB$.
Since alternate interior angles are equal, the lines must be parallel: $\mathbf{AC \parallel BD}$. (Proved!).

Examine the right-angled triangles $\triangle ABC$ and $\triangle PQR$:
1. $\angle B = \angle Q = 90^\circ$ (Right angles match).
2. Hypotenuse $AC = \text{Hypotenuse } PR = 13\text{ cm}$ (Hypotenuses match).
3. Leg $AB = \text{Leg } QR = 5\text{ cm}$ (One leg matches).
Therefore, by the RHS (Right angle-Hypotenuse-Side) Congruence Criterion:

$$\mathbf{\triangle ABC \cong \triangle PQR}$$

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By CPCTC, the remaining sides and angles ($BC = PQ = 12\text{ cm}$) are equal.

In $\triangle ABD$ and $\triangle ACD$:
1. $\angle BAD = \angle CAD$ (Since $AD$ bisects $\angle A$).
2. Side $AD = AD$ (Common side).
3. $\angle ADB = \angle ADC = 90^\circ$ (Since $AD \perp BC$).
Therefore, by the ASA (Angle-Side-Angle) Congruence Criterion:

$$\triangle ABD \cong \triangle ACD$$

By CPCTC:

$$AB = AC$$

Since two sides of $\triangle ABC$ are equal ($AB = AC$), $\triangle ABC$ is an Isosceles Triangle. (Proved!).

• Acronym: Corresponding Parts of Congruent Triangles are Congruent.
• Practical Importance: It is the primary engine of Euclidean geometric proofs.
• In any two triangles, there are six pairs of parts (3 sides and 3 angles). Once we establish congruence using a minimal 3-part criterion (such as SSS or SAS), we do not need to measure the remaining three parts: CPCTC guarantees with absolute mathematical certainty that all remaining corresponding sides and angles are identical.

• SSA is NOT a valid congruence criterion (it is an ambiguous case).
• If two sides and a non-included angle are given, it is often possible to construct two completely different triangles (one acute and one obtuse) with the exact same side lengths and non-included angle.
• Only when the non-included angle is a $90^\circ$ right angle (the RHS criterion) does SSA become uniquely valid.