Maharashtra Board Class 9 Maths Part 2 Geometry Chapter 3 Set 3.2 Triangles Solutions

Question 1. In each of the examples given below, a pair of triangles is shown. Equal parts of triangles in each pair are marked with the same signs. Observe the figures and state the test by which the triangles in each pair are congruent.
ℹ️ चित्र व्याख्या (Diagram Explanation): पहले चित्र में, दो त्रिभुज ABC और PQR हैं। भुजा AB, PQ के बराबर; भुजा BC, QR के बराबर; और भुजा AC, PR के बराबर चिह्नित हैं।
(i) By SSS test
\(\triangle ABC \cong \triangle PQR\)
Answer: By SSS (Side-Side-Side) test, the triangles are congruent because all three corresponding sides are equal.
In simple words: When all three sides of one triangle are equal to the three corresponding sides of another triangle, they are congruent by the SSS test.

🎯 Exam Tip: The SSS congruence test is used when you have information about all three sides of two triangles and need to prove their congruence.

ℹ️ चित्र व्याख्या (Diagram Explanation): दूसरे चित्र में, दो त्रिभुज XYZ और LMN हैं। भुजा XY, LM के बराबर; कोण Y, कोण M के बराबर; और भुजा YZ, MN के बराबर चिह्नित हैं।
(ii) By SAS test
\(\triangle XYZ \cong \triangle LMN\)
Answer: By SAS (Side-Angle-Side) test, the triangles are congruent because two corresponding sides and the included angle are equal.
In simple words: If two sides and the angle between them in one triangle are equal to the corresponding two sides and included angle in another, the triangles are congruent by SAS.

🎯 Exam Tip: The SAS test requires the angle to be *included* between the two equal sides. Pay close attention to the position of the angle.

ℹ️ चित्र व्याख्या (Diagram Explanation): तीसरे चित्र में, दो त्रिभुज PRQ और STU हैं। कोण P, कोण S के बराबर; भुजा PR, ST के बराबर; और कोण R, कोण T के बराबर चिह्नित हैं।
(iii) By ASA test
\(\triangle PRQ \cong \triangle STU\)
Answer: By ASA (Angle-Side-Angle) test, the triangles are congruent because two corresponding angles and the included side are equal.
In simple words: When two angles and the side between them in one triangle are equal to the corresponding parts of another triangle, they are congruent by the ASA test.

🎯 Exam Tip: For ASA congruence, the side must be the one *included* between the two equal angles.

ℹ️ चित्र व्याख्या (Diagram Explanation): चौथे चित्र में, दो समकोण त्रिभुज LMN और PTR हैं। त्रिभुज LMN में कोण M समकोण है, और त्रिभुज PTR में कोण T समकोण है। कर्ण LN, PR के बराबर है और भुजा LM, PT के बराबर है।
(iv) By hypotenuse side test
\(\triangle LMN \cong \triangle PTR\)
Answer: By Hypotenuse-Side (RHS) test, the triangles are congruent because they are right-angled, their hypotenuses are equal, and one pair of corresponding sides are equal.
In simple words: For right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of another, they are congruent by RHS.

🎯 Exam Tip: The RHS congruence criterion is specifically for right-angled triangles and requires congruence of the hypotenuse and one other side.

Question 2. Observe the information shown in pairs of triangles given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.
ℹ️ चित्र व्याख्या (Diagram Explanation): पहले चित्र में, दो त्रिभुज ABC और PQR हैं। कोण B, कोण Q के बराबर; भुजा BC, भुजा QR के बराबर; और कोण C, कोण R के बराबर चिह्नित हैं।
(i) From the information shown in the figure,
In \(\triangle ABC\) and \(\triangle PQR\),
\(\angle ABC \cong \angle PQR\)
\(\text{seg BC} \cong \text{seg QR}\)
\(\angle ACB \cong \angle PRQ\)
\(\therefore \triangle ABC \cong \triangle PQR\) [ASA test]
\(\therefore \angle BAC \cong \angle QPR\) [Corresponding angles of congruent triangles]
\(\text{seg AB} \cong \text{seg PQ}\) and \(\text{seg AC} \cong \text{seg PR}\) [Corresponding sides of congruent triangles]
Answer: The triangles \(\triangle ABC\) and \(\triangle PQR\) are congruent by the ASA (Angle-Side-Angle) test. The remaining congruent parts are \(\angle BAC \cong \angle QPR\), \(\text{seg AB} \cong \text{seg PQ}\), and \(\text{seg AC} \cong \text{seg PR}\).
In simple words: Given two angles and the included side are equal, the triangles are congruent by ASA. This means all other corresponding angles and sides are also equal.

🎯 Exam Tip: After proving congruence using a specific test, remember to list all the remaining corresponding congruent parts to get full marks.

ℹ️ चित्र व्याख्या (Diagram Explanation): दूसरे चित्र में, दो त्रिभुज PTQ और STR हैं जो बिंदु T पर प्रतिच्छेद कर रहे हैं। भुजा PT, ST के बराबर; और भुजा TQ, TR के बराबर चिह्नित हैं। कोण PTQ और STR शीर्ष विपरीत कोण हैं।
(ii) From the information shown in the figure,
In \(\triangle PTQ\) and \(\triangle STR\),
\(\text{seg PT} \cong \text{seg ST}\)
\(\angle PTQ \cong \angle STR\) [Vertically opposite angles]
\(\text{seg TQ} \cong \text{seg TR}\)
\(\therefore \triangle PTQ \cong \triangle STR\) [SAS test]
\(\therefore \angle TPQ \cong \angle TSR\) and \(\angle TQP \cong \angle TRS\) [Corresponding angles of congruent triangles]
\(\text{seg PQ} \cong \text{seg SR}\) [Corresponding sides of congruent triangles]
Answer: The triangles \(\triangle PTQ\) and \(\triangle STR\) are congruent by the SAS (Side-Angle-Side) test. The remaining congruent parts are \(\angle TPQ \cong \angle TSR\), \(\angle TQP \cong \angle TRS\), and \(\text{seg PQ} \cong \text{seg SR}\).
In simple words: With two sides and the included vertically opposite angle proven equal, the triangles are congruent by SAS, making their other corresponding parts also equal.

🎯 Exam Tip: Always look for implied congruencies like vertically opposite angles or common sides when proving triangle congruence.

Question 3. From the information shown in the figure, state the test assuring the congruence of \(\triangle ABC\) and \(\triangle PQR\). Write the remaining congruent parts of the triangles.
ℹ️ चित्र व्याख्या (Diagram Explanation): चित्र में, दो समकोण त्रिभुज ABC और PQR हैं। कोण B और Q समकोण हैं। कर्ण AC, PR के बराबर; और भुजा BC, QR के बराबर चिह्नित हैं।
Solution:
In \(\triangle BAC\) and \(\triangle PQR\),
\(\text{seg BA} \cong \text{seg PQ}\)
\(\text{seg BC} \cong \text{seg PR}\)
\(\angle BAC = \angle PQR = 90^\circ\) [Given]
\(\therefore \triangle BAC \cong \triangle PQR\) [Hypotenuse side test]
\(\therefore \text{seg AC} \cong \text{seg QR}\) [c.s.c.t.]
\(\angle ABC \cong \angle QPR\) and \(\angle ACB \cong \angle QRP\) [c.a.c.t.]
Answer: The triangles \(\triangle ABC\) and \(\triangle PQR\) are congruent by the Hypotenuse-Side (RHS) test since both are right-angled, their hypotenuses are equal, and one pair of corresponding sides (\(\text{BC}\) and \(\text{QR}\)) are equal. The remaining congruent parts are \(\text{seg AC} \cong \text{seg QR}\), \(\angle ABC \cong \angle QPR\), and \(\angle ACB \cong \angle QRP\).
In simple words: For two right-angled triangles, if their hypotenuses and one pair of corresponding sides are equal, they are congruent by RHS, implying all other corresponding parts are also equal.

🎯 Exam Tip: Clearly state the given right angle and the equal hypotenuse-side pairs when using the RHS congruence test.

Question 4. As shown in the adjoining figure, in \(\triangle LMN\) and \(\triangle PNM\), LM = PN, LN = PM. Write the test which assures the congruence of the two triangles. Write their remaining congruent parts.
ℹ️ चित्र व्याख्या (Diagram Explanation): चित्र में, चतुर्भुज LPNM में, विकर्ण MN है जो दोनों त्रिभुजों LMN और PNM के लिए उभयनिष्ठ भुजा है। भुजा LM, PN के बराबर; और भुजा LN, PM के बराबर चिह्नित हैं।
Solution:
In \(\triangle LMN\) and \(\triangle PNM\),
\(\text{seg LM} = \text{seg PN}\)
\(\text{seg LN} \cong \text{seg PM}\) [Given]
\(\text{seg MN} = \text{seg NM}\) [Common side]
\(\therefore \triangle LMN = \triangle PNM\) [SSS test]
\(\therefore \angle LMN \cong \angle PNM\),
\(\therefore \angle MLN = \angle NPM\), and \(\angle LNM = \angle PMN\) [c.a.c.t.]
Answer: The triangles \(\triangle LMN\) and \(\triangle PNM\) are congruent by the SSS (Side-Side-Side) test, as \(\text{LM} = \text{PN}\), \(\text{LN} = \text{PM}\) (given), and \(\text{MN}\) is a common side. The remaining congruent parts are \(\angle LMN \cong \angle PNM\), \(\angle MLN \cong \angle NPM\), and \(\angle LNM \cong \angle PMN\).
In simple words: When all three sides of two triangles are equal, including a common side, they are congruent by SSS, leading to all corresponding angles being equal.

🎯 Exam Tip: Don't forget to identify and state any common sides or angles when proving congruence; they are crucial parts of the proof.

Question 5. In the adjoining figure, seg AB = seg CB and seg AD \(\cong\) seg CD. Prove that \(\triangle ABD \cong \triangle CBD\).
ℹ️ चित्र व्याख्या (Diagram Explanation): चित्र में, चतुर्भुज ABCD में, विकर्ण BD है। भुजा AB, CB के बराबर; और भुजा AD, CD के बराबर चिह्नित हैं।
Solution:
proof:
In \(\triangle ABD\) and \(\triangle CBD\),
\(\text{seg AB} = \text{seg CB}\)
\(\text{seg AD} = \text{seg CD}\) [Given]
\(\text{seg BD} \cong \text{seg BD}\) [Common side]
\(\therefore \triangle ABD = \triangle CBD\) [SSS test]
Answer: To prove \(\triangle ABD \cong \triangle CBD\), we observe that \(\text{seg AB} = \text{seg CB}\) and \(\text{seg AD} = \text{seg CD}\) are given. Also, \(\text{seg BD}\) is common to both triangles. Therefore, by the SSS (Side-Side-Side) test, \(\triangle ABD \cong \triangle CBD\).
In simple words: We prove the triangles congruent by SSS because all three corresponding sides (two given equal pairs and one common side) are equal.

🎯 Exam Tip: For proof questions, clearly list the given conditions, identify common elements, and then state the congruence test used for a complete answer.

Question 6. In the adjoining figure, \(\angle P \cong \angle R\), \(\text{seg PQ} = \text{seg RQ}\). Prove that \(\triangle PQT \cong \triangle RQS\).
ℹ️ चित्र व्याख्या (Diagram Explanation): चित्र में, दो त्रिभुज PQT और RQS हैं जो बिंदु Q पर उभयनिष्ठ कोण साझा करते हैं। कोण P, कोण R के बराबर; और भुजा PQ, भुजा RQ के बराबर चिह्नित हैं।
Proof:
In \(\triangle PQT\) and \(\triangle RQS\),
\(\angle P \cong \angle R\)
\(\text{seg PQ} = \text{seg RQ}\) [Given]
\(\angle Q = \angle Q\) [Common angle]
\(\therefore \triangle PQT = \triangle RQS\) [ASA test]
Answer: To prove \(\triangle PQT \cong \triangle RQS\), we are given \(\angle P \cong \angle R\) and \(\text{seg PQ} = \text{seg RQ}\). Additionally, \(\angle Q\) is a common angle to both triangles (\(\angle PQT = \angle RQS\)). Thus, by the ASA (Angle-Side-Angle) test, \(\triangle PQT \cong \triangle RQS\).
In simple words: Since two angles and the included side of one triangle are equal to the corresponding parts of the other, the triangles are congruent by ASA.

🎯 Exam Tip: When triangles share a common vertex, the angle at that vertex is often a "common angle" and can be used in congruence proofs like ASA or AAS.