Maharashtra Board Class 8 Maths Chapter 13 Congruence of Triangles Set 13.2 Solutions

Question 1. In each pair of triangles given below, parts shown by identical marks are congruent. State the test and the one-to-one correspondence of vertices by which triangles in each pair are congruent. Also state the remaining congruent parts.
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो समकोण त्रिभुज MST और TBM दिखाए गए हैं। इसमें कोण S और कोण B समकोण हैं, भुजा MS भुजा TB के बराबर है, और MT दोनों त्रिभुजों का उभयनिष्ठ कर्ण है।
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो त्रिभुज PRQ और TRS को दर्शाता है। इसमें भुजा PR भुजा TR के बराबर है और भुजा RQ भुजा RS के बराबर है। बिंदु R पर बनने वाले कोण PRQ और TRS ऊर्ध्वाधर विपरीत कोण हैं।
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो त्रिभुज DCH और DCF दिखाए गए हैं। इसमें कोण CDH और कोण CDF बराबर हैं, कोण DHC और कोण DFC बराबर हैं, और भुजा DC दोनों त्रिभुजों की उभयनिष्ठ भुजा है।
Answer: i. In \( \triangle \)MST and \( \triangle \)TBM,
\( \therefore \) side MS \( \cong \) side TB ... [Given]
m\( \angle \)MST = m\( \angle \)TBM = 90° ... [Given]
hypotenuse MT = hypotenuse MT ...[Common side]
\( \therefore \triangle \)MST \( \cong \triangle \)TBM ...[by hypotenuse-side test]
\( \therefore \) side ST \( \cong \) side BM ...[Corresponding sides of congruent triangles]
\( \angle \)SMT \( \cong \angle \)BTM ...[Corresponding sides of congruent triangles]
\( \angle \)STM \( \cong \angle \)BMT ...[Corresponding sides of congruent triangles]
ii. In \( \triangle \)PRQ and \( \triangle \)TRS,
side PR = side TR ... [Given]
\( \angle \)PRQ = \( \angle \)TRS ...[Vertically opposite angles]
side RQ = side RS ... [Given]
\( \therefore \triangle \)PRQ \( \cong \triangle \)TRS ...[by SAS test]
\( \therefore \) side PQ \( \cong \) side TS ... [Corresponding sides of congruent triangles]
\( \angle \)RPQ \( \cong \angle \)RTS ...[Corresponding sides of congruent triangles]
\( \angle \)PQR \( \cong \angle \)TSR ... [Corresponding sides of congruent triangles]
iii. In \( \triangle \)DCH and \( \triangle \)DCF,
\( \angle \)DCH \( \cong \angle \)DCF ...[Given]
\( \angle \)DHC \( \cong \angle \)DFC ...[Given]
side DC = side DC ...[Common side]
\( \therefore \triangle \)DCH \( \cong \triangle \)DCF ...[by AAS test]
\( \therefore \) side HC \( \cong \) side FC ...[Corresponding sides of congruent triangles]
side DH \( \cong \) side DF...[Corresponding sides of congruent triangles]
\( \angle \)HDC \( \cong \angle \)FDC ....[Corresponding sides of congruent triangles] In simple words: For two triangles to be congruent, specific criteria like Hypotenuse-Side, SAS (Side-Angle-Side), or AAS (Angle-Angle-Side) tests must be met, ensuring that corresponding parts (sides and angles) are equal. Once congruence is established, the remaining corresponding parts are also congruent.

🎯 Exam Tip: Clearly identify the given congruent parts and the type of congruence test applicable (e.g., SSS, SAS, ASA, AAS, Hypotenuse-Side) to score full marks. State the one-to-one correspondence of vertices accurately.

Question 2. In the given figure, seg AD = seg EC. Which additional information is needed to show that \( \triangle \)ABD and \( \triangle \)EBC will be congruent by AAS test?
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो त्रिभुज ABD और EBC दिखाए गए हैं। इसमें भुजा AD भुजा EC के बराबर है। बिंदु B पर कोण ABD और CBE ऊर्ध्वाधर विपरीत कोण हैं।
Answer: In \( \triangle \)ABD and \( \triangle \)CBE,
\( \therefore \) seg AD = seg CE ...[Given]
\( \angle \)ABD = \( \angle \)CBE ...[Vertically opposite angles]
\( \therefore \) The necessary condition for the two triangles to be congruent by AAS test is
\( \angle \)ADB = \( \angle \)CEB, or
\( \angle \)DAB = \( \angle \)ECB In simple words: To prove congruence by AAS test for two triangles, given one side and one pair of angles are equal, an additional pair of corresponding angles must also be equal. This means either the angle adjacent to the given side or the third angle must be congruent.

🎯 Exam Tip: For AAS congruence, ensure that the two angles and the non-included side are corresponding parts. Identify which additional angle pair will satisfy the AAS criterion based on the given information.

Maharashtra Board Class 8 Maths Chapter 13 Congruence Of Triangles Practice Set 13.2 Intext Questions And Activities

Question 1. Draw \( \triangle \)ABC and \( \triangle \)LMN such that two pairs of their sides and the angles included by them are congruent. Draw \( \triangle \)ABC and \( \triangle \)LMN, I(AB) = I(LM), I(BC) = I(MN), m\( \angle \)ABC = m\( \angle \)LMN. Copy \( \triangle \)ABC on a tracing paper. Place the paper on \( \triangle \)LMN in such a way that point A coincides with point L, side AB overlaps side LM. What do you notice?(Textbook pg. no. 83)
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो त्रिभुज ABC और LMN दिखाए गए हैं। इनमें भुजा AB, भुजा LM के बराबर है; भुजा BC, भुजा MN के बराबर है; और इनके बीच के कोण ABC और LMN बराबर हैं, जो SAS कसौटी को दर्शाता है।
Answer: We notice that \( \triangle \)ABC \( \cong \triangle \)LMN. In simple words: When two triangles have two pairs of corresponding sides and the included angle congruent (SAS test), they perfectly overlap when placed one over the other, indicating they are congruent.

🎯 Exam Tip: When drawing and comparing triangles for congruence, ensure accurate measurements for sides and angles. The act of using tracing paper helps visually confirm congruence, reinforcing the geometric concept.

Question 2. Draw \( \triangle \)PQR and \( \triangle \)XYZ such that I(PQ) = I(XY), I(Q R) = I(YZ), I(RP) = I(ZX). Copy \( \triangle \)PQR on a tracing paper. Place it on \( \triangle \)XYZ observing the correspondence P \( \leftrightarrow \) X, Q \( \leftrightarrow \) Y, R \( \leftrightarrow \) Z. What do you notice? (Textbook pg. no. 84)
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो त्रिभुज PQR और XYZ दिखाए गए हैं। इनमें सभी संगत भुजाएँ बराबर हैं: PQ=XY, QR=YZ, और RP=ZX, जो SSS कसौटी को दर्शाता है।
Answer: We notice that \( \triangle \)PQR \( \cong \triangle \)XYZ. In simple words: When all three corresponding sides of two triangles are equal (SSS test), the triangles are congruent, meaning they can perfectly overlap each other.

🎯 Exam Tip: The SSS (Side-Side-Side) congruence test is fundamental. Ensure that all three pairs of corresponding sides are congruent for a valid SSS congruence claim. Visual inspection with tracing paper helps confirm congruence.

Question 3. Draw \( \triangle \)XYZ and \( \triangle \)DEF such that, I(XZ) = I(DF), \( \angle \)X = \( \angle \)D and \( \angle \)Z = \( \angle \)F. Copy \( \triangle \)XYZ on a tracing paper and place it over \( \triangle \)DEF. What do you notice?(Textbook pg. no. 84)
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो त्रिभुज XYZ और DEF दिखाए गए हैं। इनमें भुजा XZ, भुजा DF के बराबर है; कोण X, कोण D के बराबर है; और कोण Z, कोण F के बराबर है, जो ASA कसौटी को दर्शाता है।
Answer: We notice that \( \triangle \)XYZ \( \cong \triangle \)DEF in the correspondence X \( \leftrightarrow \) D, Y \( \leftrightarrow \) E, Z \( \leftrightarrow \) F. In simple words: When two triangles have two pairs of corresponding angles and the included side congruent (ASA test), they are congruent, meaning one can be perfectly superimposed on the other.

🎯 Exam Tip: The ASA (Angle-Side-Angle) congruence test requires the side to be *included* between the two angles. Carefully identify the included side to correctly apply this criterion. Drawing and overlaying can help visualize congruence.

Question 4. Draw two right angled triangles such that a side and the hypotenuse of one is congruent with the corresponding parts of the other. Copy one triangle on tracing paper and place it over the other. What do you notice? (Textbook pg. no. 84)
ℹ️ चित्र व्याख्या (Diagram Explanation): इस चित्र में दो समकोण त्रिभुज ABC और LMN दिखाए गए हैं। इसमें कोण B और कोण M समकोण हैं, कर्ण AC और LN बराबर हैं, और एक संगत भुजा (जैसे AB और LM) भी बराबर है। यह Hypotenuse-Side (HL) कसौटी को दर्शाता है।
Answer: We notice that the two triangles are congruent. (Students should draw figures and verify the answers.) In simple words: For right-angled triangles, if the hypotenuse and one side of one triangle are congruent to the hypotenuse and corresponding side of another triangle, then the triangles are congruent by the Hypotenuse-Side (HL) test.

🎯 Exam Tip: The Hypotenuse-Side (HL) test is a specific congruence criterion for right-angled triangles. Remember to always confirm that both triangles are right-angled and that the hypotenuse and a corresponding leg are congruent.