Welcome to AssamSchool. This article contains 30 Extra Questions from Triangles from Class 9 Mathematics Chapter 7. These Triangles Class 9 Extra Questions are helpful to test your understanding and knowledge of the chapter. These additional questions will not only help you to excel in the chapter but also will give you a strong foundation in Mathematics.
We have also provided step-by-step answers to the questions for reference. So, let’s start…
30 Triangles Class 9 Extra Questions
1. Define congruent triangles and state the SAS congruence rule.
2. In \(∆ABC\) and \(∆PQR\), if \(AB = PQ\), \(BC = QR\), and \(∠B = ∠Q\), are the triangles congruent? Which criterion applies?
3. True or False: AAA is a valid congruence criterion for triangles. Justify.
4. What does CPCT stand for? Why is it important in triangle congruence?
5. If \(∆ABC ≅ ∆DEF\) under ASA criterion, which angles and sides must be equal?
6. In an isosceles triangle with \(AB = AC\), name the angles that are equal.
7. State the RHS congruence rule with an example.
8. If two sides of a triangle are \(5 cm\) and \(7 cm\), can the third side be \(3 cm\)? Why or why not?
9. What is the measure of each angle in an equilateral triangle?
10. In \(∆ABC\), if \(AB = BC\), what type of triangle is it?
11. Prove that the bisector of the vertical angle of an isosceles triangle bisects the base.
12. In Fig. 7.16, \(AC = AD\) and \(AB\) bisects \(∠A\). Show \(BC = BD\).
13. In \(∆ABC\) and \(∆DEF\), \(∠A = ∠D\), \(∠B = ∠E\), and \(BC = EF\). Are the triangles congruent? If yes, state the criterion.
14 In Fig. 7.30, \(AD\) is the perpendicular bisector of \(BC\) Prove \(AB = AC\).
15. If two medians of a triangle are equal, prove the triangle is isosceles.
16. In \(∆ABC\), \(AB = AC\), and \(BD\) and \(CE\) are altitudes. Prove \(BD = CE\).
17. In Fig. 7.34, \(AD = AB\) and \(AB = AC\). Prove \(∠BCD = 90°\).
18. If \(∆ABC ≅ ∆PQR\) by SSS rule, and \(AB = 6 cm\), \(BC = 7 cm\), \(CA = 8 cm\), find \(PQ\), \(QR\), and \(RP\).
19. Prove that the altitudes of an equilateral triangle are equal.
20. In Fig. 7.39, if \(AP\) is the perpendicular bisector of \(BC\), show \(∠ABP = ∠ACP\).
21. In \(∆ABC\), \(D\) and \(E\) are midpoints of \(AB\) and \(AC\). Prove \(DE || BC\) and \(DE = ½ BC\).
22. Prove that a triangle with two equal medians is isosceles.
23. In Fig. 7.40, if \(AM\) and \(PN\) are medians of \(∆ABC\) and \(∆PQR\), and \(AB = PQ\), \(BC = QR\), prove \(∆ABC ≅ ∆PQR\).
24. If two triangles have equal perimeters and are congruent by SSS, are their areas equal? Prove.
25. In Fig. 7.38, if \(PB = PC\) and \(PB ⊥ l\), \(PC ⊥ m\), prove \(AP\) bisects \(∠BAC\).
26. Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side.
27. In \(∆ABC\), \(AB = AC\). A line through \(B\) is parallel to \(AC\), and a line through \(C\) is parallel to \(AB\). Prove they intersect at a point equidistant from \(A\), \(B\), and \(C\).
28. If the bisector of \(∠A\) of \(∆ABC\) is perpendicular to \(BC\), prove \(AB = AC\).
29. In Fig. 7.23, if \(CM = DM\) and \(∠DBC = 90°\), prove \(CM = ½ AB\).
30. Prove that the sum of any two sides of a triangle is greater than the third side using congruence.
Answers
- Answer:
Congruent triangles have equal sides and angles. SAS: Two sides and included angle equal. - Answer:
Yes, by SAS congruence rule. - Answer:
False. AAA ensures similarity, not congruence. - Answer:
CPCT = Corresponding Parts of Congruent Triangles. It ensures equal parts after congruence is proven. - Answer:
\(∠A = ∠D\), \(∠B = ∠E\), side \(BC = EF\). - Answer:
\(∠B = ∠C\). - Answer:
RHS: Hypotenuse and one side equal in right triangles. Example: Hypotenuse \(= 5 cm\), side \(= 3 cm\). - Answer:
No. \(5 + 3 > 7\) is false. Violates triangle inequality. - Answer:
\(60°\) each. - Answer:
Isosceles triangle. - Answer:
Let \(∆ABC\) with \(AB = AC\). Bisect \(∠A\) to meet \(BC\) at \(D\). By SAS, \(∆ABD ≅ ∆ACD ⇒ BD = DC\). - Answer:
\(∆ABC ≅ ∆ABD\) (SAS), so \(BC = BD\) (CPCT). - Answer:
Yes, by AAS criterion. - Answer:
\(∆ADB ≅ ∆ADC\) (SAS), so \(AB = AC\). - Answer:
Let medians \(BE = CF\). Use SSS to prove \(∆BEC ≅ ∆CFB ⇒ AB = AC\). - Answer:
\(∆BDC ≅ ∆CEB\) (RHS), so \(BD = CE\). - Answer:
\(∆ABC\) is isosceles. \(AD = AB ⇒ ∆ABD\) is isosceles. Use angle sum property to show \(∠BCD = 90°\). - Answer:
\(PQ = 6 cm\), \(QR = 7 cm\), \(RP = 8 cm\). - Answer:
All altitudes in an equilateral triangle are equal due to symmetry and congruence. - Answer:
\(∆ABP ≅ ∆ACP\) (SAS) \(⇒ ∠ABP = ∠ACP\). - Answer:
Use Midpoint Theorem. \(∆ADE ≅ ∆ABC ⇒ DE || BC\) and \(DE = ½ BC\). - Answer:
Let medians \(BE = CF\). Prove \(∆BEC ≅ ∆CFB ⇒ AB = AC\). - Answer:
\(∆ABM ≅ ∆PQN (SSS) ⇒ ∠ABC = ∠PQR\). Then \(∆ABC ≅ ∆PQR\) (SAS). - Answer:
Yes. Congruent triangles have equal areas. - Answer:
\(∆PAB ≅ ∆PAC\) (RHS) \(⇒ ∠PAB = ∠PAC\). - Answer:
Use Midpoint Theorem or coordinate geometry to show parallelism. - Answer:
Let lines intersect at \(D\). Prove \(AD = BD = CD\) using parallelogram properties. - Answer:
\(∆ABD ≅ ∆ACD\) (ASA) \(⇒ AB = AC\). - Answer:
\(∆AMC ≅ ∆BMD\) (SAS) \(⇒ ∠DBC = 90°\). By midpoint theorem, \(CM = ½ AB\). - Answer:
Assume contradiction. Use triangle inequality and congruence to show impossibility.
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The Bottom Line
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You can also check the related MCQs for this chapter here: MCQ on Triangles
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Abdur Rohman is an Electrical Engineer from Charaideo, Assam, who wears multiple hats as a part-time teacher, blogger, entrepreneur, and digital marketer. Passionate about education, he founded The Assam School blog to provide free, comprehensive textbook solutions, MCQs (Multiple Choice Questions), and other academic content for students from Class V to XII.