CBSE Class 9 Maths: Geometry of Triangles - Exercise 7.1

Understanding Congruence: The Heart of Triangles

In mathematics, two figures are said to be congruent if they have exactly the same shape and the same size. Imagine placing one figure directly on top of the other; if they match perfectly, they are congruent. For triangles, this means all three corresponding sides are equal in length, and all three corresponding angles are equal in measure.

However, we don't always need to check all six conditions (three sides and three angles) to prove congruence. There are specific criteria, or postulates, that allow us to prove two triangles are congruent by checking only three corresponding parts. Exercise 7.1 primarily focuses on the Side-Angle-Side (SAS) and Angle-Side-Angle (ASA) congruence rules. The ability to correctly identify these corresponding parts and apply the rules is fundamental. Once two triangles are proven congruent, a very important consequence follows: their Corresponding Parts of Congruent Triangles are Congruent (CPCTC). This means any other corresponding sides or angles that weren't used to prove congruence can now be stated as equal. This concept is a cornerstone for solving many geometric proofs.

Key Definitions for Triangle Congruence

Congruent Figures

Two geometric figures are congruent if they have the same shape and the same size. One can be perfectly superimposed on the other.

Congruent Triangles

Two triangles are congruent if their corresponding sides and corresponding angles are equal. If $\triangle ABC \cong \triangle PQR$, it means $AB=PQ$, $BC=QR$, $CA=RP$, $\angle A=\angle P$, $\angle B=\angle Q$, and $\angle C=\angle R$.

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

This is a fundamental theorem stating that if two triangles are congruent, then every pair of their corresponding parts (sides and angles) are equal.

SAS Congruence Rule (Side-Angle-Side)

Two triangles are congruent if two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and the included angle of the other triangle.

ASA Congruence Rule (Angle-Side-Angle)

Two triangles are congruent if two angles and the included side (the side between those two angles) of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Worked Examples from NCERT Exercise 7.1

• Example 1: Proving Congruence using SAS Problem: In quadrilateral ABCD, AC = AD and AB bisects $\angle A$. Show that $\triangle ABC \cong \triangle ABD$. What can you say about BC and BD? Solution: Step 1: Identify the triangles we need to prove congruent. We need to prove $\triangle ABC \cong \triangle ABD$. Step 2: List the given information and conditions that make parts equal. Given: AC = AD (Side) Given: AB bisects $\angle A$. This means $\angle CAB = \angle DAB$ (Angle). Common side: AB = AB (Side) Step 3: Apply the appropriate congruence rule. We have two sides (AC, AB and AD, AB) and the included angle ($\angle CAB$ and $\angle DAB$) equal. Therefore, by SAS congruence rule, $\triangle ABC \cong \triangle ABD$. Step 4: Use CPCTC to deduce other equalities. Since $\triangle ABC \cong \triangle ABD$, their corresponding parts are congruent. Thus, BC = BD (by CPCTC). Final Answer: $\triangle ABC \cong \triangle ABD$ by SAS congruence rule, and BC = BD (by CPCTC).
• Example 2: Proving Congruence using ASA Problem: Line segment AB is parallel to another line segment CD. O is the midpoint of AD. Show that (i) $\triangle AOB \cong \triangle DOC$ and (ii) O is also the midpoint of BC. Solution: Step 1: Identify the triangles for congruence proof. We need to prove $\triangle AOB \cong \triangle DOC$. Step 2: List the given information and conditions. Given: AB $\| $ CD. Given: O is the midpoint of AD. This means AO = DO (Side). Angles: Since AB $\| $ CD and AD is a transversal, $\angle OAB = \angle ODC$ (Alternate interior angles). (Angle) Angles: $\angle AOB = \angle DOC$ (Vertically opposite angles). (Angle) Step 3: Apply the appropriate congruence rule. We have two angles ($\angle OAB$, $\angle AOB$ and $\angle ODC$, $\angle DOC$) and the included side (AO and DO) between the first angle and the second. Therefore, by ASA congruence rule, $\triangle AOB \cong \triangle DOC$. Step 4: Use CPCTC to deduce other equalities. Since $\triangle AOB \cong \triangle DOC$, their corresponding parts are congruent. From CPCTC, BO = CO. Step 5: Conclude based on the deduced equalities. Since BO = CO, it implies that O is the midpoint of BC. Final Answer: (i) $\triangle AOB \cong \triangle DOC$ by ASA congruence rule. (ii) O is the midpoint of BC (by CPCTC).

YoLearn's Exam Tip: Mastering Congruence Proofs

When attempting congruence problems in exams, precision is key!

1. Correct Order of Vertices: Always write the names of congruent triangles in corresponding order. For example, if $\triangle ABC \cong \triangle PQR$, it means vertex A corresponds to P, B to Q, and C to R. This ensures that sides (AB=PQ, BC=QR, AC=PR) and angles ($\angle A=\angle P$, $\angle B=\angle Q$, $\angle C=\angle R$) correspond correctly. A common mistake is writing $\triangle ABC \cong \triangle QPR$, which implies different correspondences.
2. Identify Included Angle/Side: For SAS, the angle MUST be between the two sides. For ASA, the side MUST be between the two angles. Not paying attention to "included" can lead to incorrect application of the rule.
3. State Reasons Clearly: Every step in your proof must be justified. Whether it's "Given", "Common side", "Vertically opposite angles", "Alternate interior angles", or the specific congruence rule (SAS, ASA), always provide a clear reason.
4. Don't Forget CPCTC: After proving triangles congruent, if the question asks to prove equality of other parts, remember to use CPCTC. It's a powerful tool to complete your proof.

Practice Questions with Solutions

• Q: In $\triangle ABC$ and $\triangle DEF$, AB = DE, BC = EF, and $\angle B = \angle E$. Are the triangles congruent? If yes, by which criterion? A: Step 1: List the given corresponding parts. Given: AB = DE (Side) Given: BC = EF (Side) Given: $\angle B = \angle E$ (Angle) Step 2: Check if the angle is included between the two sides. Yes, $\angle B$ is included between sides AB and BC, and $\angle E$ is included between sides DE and EF. Step 3: Apply the congruence rule. Since two sides and the included angle of $\triangle ABC$ are equal to the corresponding two sides and included angle of $\triangle DEF$, the triangles are congruent by the SAS (Side-Angle-Side) congruence rule. Final answer: Yes, $\triangle ABC \cong \triangle DEF$ by SAS congruence rule.
• Q: Given two triangles $\triangle PQR$ and $\triangle XYZ$ where $\angle Q = \angle Y$, $\angle R = \angle Z$, and QR = YZ. Prove that $\triangle PQR \cong \triangle XYZ$. A: Step 1: List the given corresponding parts. Given: $\angle Q = \angle Y$ (Angle) Given: $\angle R = \angle Z$ (Angle) Given: QR = YZ (Side) Step 2: Check if the side is included between the two angles. Yes, side QR is included between $\angle Q$ and $\angle R$, and side YZ is included between $\angle Y$ and $\angle Z$. Step 3: Apply the congruence rule. Since two angles and the included side of $\triangle PQR$ are equal to the corresponding two angles and included side of $\triangle XYZ$, the triangles are congruent by the ASA (Angle-Side-Angle) congruence rule. Final answer: $\triangle PQR \cong \triangle XYZ$ by ASA congruence rule.
• Q: In the given figure, AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB. A: Step 1: Identify the triangles for congruence. We need to prove $\triangle AOD \cong \triangle BOC$. Step 2: List the given information and conditions. Given: AD $\perp$ AB, BC $\perp$ AB. This means $\angle DAO = \angle CBO = 90^\circ$ (Angles). Given: AD = BC (Side). Angles: $\angle AOD = \angle BOC$ (Vertically opposite angles). (Angles) Step 3: Apply the appropriate congruence rule. We have two angles ($\angle DAO$, $\angle AOD$ and $\angle CBO$, $\angle BOC$) and a non-included side (AD and BC). This falls under AAS (Angle-Angle-Side) congruence. (Note: ASA requires the included side; AAS is a direct consequence of ASA and angle sum property of triangles). By AAS congruence rule, $\triangle AOD \cong \triangle BOC$. Step 4: Use CPCTC to deduce other equalities. Since $\triangle AOD \cong \triangle BOC$, their corresponding parts are congruent. Thus, AO = BO (by CPCTC). Step 5: Conclude based on the deduced equalities. Since AO = BO, it means CD bisects AB. Final answer: CD bisects AB, as $\triangle AOD \cong \triangle BOC$ by AAS congruence, leading to AO = BO by CPCTC.
• Q: In $\triangle ABC$, E is the midpoint of median AD. BE produced meets AC at F. Show that AF = (1/3)AC. A: Step 1: Construct a line. Draw a line DG parallel to BF, meeting AC at G. (Construction) Step 2: Apply Midpoint Theorem in $\triangle ADG$. In $\triangle ADG$, E is the midpoint of AD (given) and EF $\| $ DG (by construction, since F lies on BF which is parallel to DG). By converse of Midpoint Theorem, F is the midpoint of AG. So, AF = FG. (Equation 1) Step 3: Apply Midpoint Theorem in $\triangle BCF$. In $\triangle BCF$, D is the midpoint of BC (since AD is a median, D is midpoint of BC) and DG $\| $ BF (by construction). By converse of Midpoint Theorem, G is the midpoint of FC. So, FG = GC. (Equation 2) Step 4: Combine the results. From (1) and (2), we have AF = FG = GC. Since AC = AF + FG + GC, AC = AF + AF + AF (substituting FG and GC with AF) AC = 3AF. Step 5: Final conclusion. Therefore, AF = (1/3)AC. Final answer: AF = (1/3)AC.

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