KUMAR ONLINE CLASSCBSE(NCERT): CLASS IX MATHS

ByM. S . Kumar SwamyTGT(Maths)KV Gachibowli

TRIANGLES

ASSERTION & REASONING QUESTIONS

ASSERTION & REASONING QUESTIONSDIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:(a)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation

of assertion (A).(b)Both assertion (A) and reason (R) are true but reason (R) is not the correct

explanation of assertion (A).(c)Assertion (A) is true but reason (R) is false.(d)Assertion (A) is false but reason (R) is true.

1. Assertion : In the given figure, BO and CO are the bisectors of ∠B and ∠C respectively. If ∠A = 50° then ∠BOC = 115°Reason : The sum of all the interior angles of a triangle is 1800

Ans: We know that the sum of all the interior angles of a triangle is 1800

So, Reason is correct.Now, In ΔABC, we have:∠A + ∠B + ∠C=180° [Sum of the angles of a triangle]⇒ 50° + ∠B + ∠C = 180°⇒∠B + ∠C = 130°In ΔOBC, we have:∠OBC + ∠OCB + ∠BOC = 180°⇒ ∠B + ∠C + ∠BOC = 180° [Using (i)]⇒ 65°+ ∠BOC = 180°Hence, Assertion is also correct Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

⇒ ∠B + ∠C = 65° ...(i)

⇒∠BOC=115°

2. Assertion : In ∆ABC, ∠C = ∠A, BC = 4 cm and AC = 5 cm. Then, AB = 4 cmReason : In a triangle, angles opposite to two equal sides are equal.

Ans: We know that “In a triangle, angles opposite to two equal sides are equal.”So, Reason is correct.In ∆ABC, ∠C = ∠A (Given)Therefore, BC = AB (Sides opposite to equal angles.)⇒ BC = AB = 4 cm So, Assertion is also correct But reason (R) is not the correct explanation of assertion (A).Correct option is (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

3. Assertion : In ∆ABC, BC = AB and ∠B = 80°. Then, ∠A = 50°Reason : In a triangle, angles opposite to two equal sides are equal.

Ans: We know that “In a triangle, angles opposite to two equal sides are equal.”So, Reason is correct.In ∆ABC, AB = BC⇒∠A = ∠C (Angles opposite to equal sides)Let ∠A = ∠C = xUsing angle sum property of a triangle,∠A + ∠B + ∠C = 180°⇒ 2x = 100°So, Assertion is also correct Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

⇒ x + 80°+ x = 180° ⇒ 2x = 180°− 80°⇒ x = 50° ⇒∠A = 50°

4. Assertion : In ∆ABC, D is the midpoint of BC. If DL ⊥ AB and DM ⊥ AC such that DL = DM, then BL = CMReason : If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent.Ans: We know that “If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent.” - This is ASA Congruence Rule. So, Reason is correct.In △BDL and △CDM, we haveBD = CD (D is midpoint)DL = DM (Given)and ∠BLD = ∠CMD (90° each)∴ △BDL ≅△CDM (RHS criterion) So, Assertion is also correct Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

⇒ BL = CM (CPCT)

5. Assertion : In the adjoining figure, X and Y are respectively two points on equal sides AB and AC of ∆ABC such that AX = AY then CX = BY.Reason : If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruentAns: We know that “If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent” - This is SAS Congruence Rule. So, Reason is correct.In △AXC and △AYB, we have AC = AB (Given)AX = AY (Given)and ∠BAC = ∠CAB (Common)∴ △AXC ≅△AYB (SAS criterion)So, Assertion is also correct Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

⇒ CX = BY (CPCT)

6. Assertion : In the given figure, ABCD is a quadrilateral in which AB || DC and P is the midpoint of BC. On producing, AP and DC meet at Q then DQ = DC + AB.Reason : If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruentAns: We know that “If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent” - This is SAS Congruence Rule. So, Reason is correct.In △ABP and △QCP, we have∠BPA = ∠CPQ (Vertically opposite angle)∠PAB = ∠PQC (Alternate angles)and PB = PC (P is the midpoint)∴ △ABP ≅△QCP (AAS criterion) Now, DQ = DC + CQ So, Assertion is also correct Correct option is (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

⇒ AB = CQ (CPCT)⇒ DQ = DC + AB (AB = CQ prove above)

7. Assertion : Angles opposite to equal sides of a triangle are not equal.Reason : Sides opposite to equal angles of a triangle are equal.

Ans: We know that Angles opposite to equal sides of a triangle are equal.So, Assertion is not correct.Also, we know that Sides opposite to equal angles of a triangle are equal.So, Reason is correct.Correct option is (d) Assertion (A) is false but reason (R) is true.

(a)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b)Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(c)Assertion (A) is true but reason (R) is false.(d)Assertion (A) is false but reason (R) is true.

8. Assertion : In ∆ABC, AB = AC and ∠B = 50⁰, then ∠C is 50⁰.Reason : Angles opposite to equal sides of a triangle are equal.

Ans: We know that Angles opposite to equal sides of a triangle are equal.So, Reason is correct.Now, In ∆ABC, AB = AC ∠B = ∠C (Angles opposite to equal sides)⇒ ∠C = 50⁰ So, Assertion is also correct.Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(a)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b)Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(c)Assertion (A) is true but reason (R) is false.(d)Assertion (A) is false but reason (R) is true.

9. Assertion : ΔABC and ΔDBC are two isosceles triangles on the same base BC andvertices A and D are on the same side of BC. If AD is extended to intersect BC at E, thenΔABD≅ ΔACDReason : If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent.Ans: We know that “If in two right triangles, hypotenuse and one side ofa triangle are equal to the hypotenuse and one side of other triangle,then the two triangles are congruent.” – This is RHS Congruence RuleSo, Reason is correctIn ΔABD andΔACD,BD = CD (Given)AB = AC (Given)and AD = AD (Common side)∴ By SSS congruence criteria, ΔABD≅ ΔACDSo, Assertion is also correct.Correct option is (b) Both assertion (A) and reason (R) are true andreason (R) is not the correct explanation of assertion (A).

10. Assertion : In triangles ABC and PQR,∠A = ∠P, ∠C = ∠R and AC = PR. The two triangles are congruent by ASA congruence.Reason : If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent.Ans: We know that “If two angles and the included side of one triangle are equalto two angles and the included side of the other triangle, then the two trianglesare congruent.” – This is ASA Congruence RuleSo, Reason is correctNow, In triangles ABC and PQR,∠A = ∠P, ∠C = ∠R and AC = PR.∴ By ASA congruence criteria,Δ ABC≅Δ PQRSo, Assertion is also correct.Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) isthe correct explanation of assertion (A).

11. Assertion : In ∆ABC and ∆PQR, AB = PQ, AC = PR and ∠BAC = ∠QPR then ∆ABC ≅ ∆PQRReason : Both the triangles are congruent by SSS congruence.

Ans: In ∆ABC and ∆PQR, AB = PQ, AC = PR and ∠BAC = ∠QPR (Given)then ∆ABC ≅ ∆PQR by ASA Congruence RuleSo, Assertion (A) is true.But Reason (R) is false.Correct option is (c) Assertion (A) is true but reason (R) is false.

(a)Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b)Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(c)Assertion (A) is true but reason (R) is false.(d)Assertion (A) is false but reason (R) is true.

12. Assertion: In the given figure, BE and CF are two equal altitudes of ΔABC then ΔABE ≅ ΔACFReason: If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.Ans: We know that “If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.” – This is AAS Congruence RuleSo, Reason (R) is true.In △ABE and △ACF, we haveBE = CF (Given)∠BEA = ∠CFA = 90°∠A = ∠A (Common)∴ △ABE ≅△ACF (By AAS Congruence rule)So, Assertion (A) is also true.Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

13. Assertion: In ABC, ∠A = ∠ C and BC = 4 cm and AC = 3 cm then the length of side AB = 3 cm.Reason: Sides opposite to equal angles of a triangle are equal.Ans: We know that Sides opposite to equal angles of a triangle are equal.So, Reason is correct.Now, In ∆ABC, ∠A = ∠C (Given)AB = CB (Sides opposite to equal sides)⇒ AB = 4 cmSo, Assertion is not correct.Correct option is (d) Assertion (A) is false but reason (R) is true.

14. Assertion : If the altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is an isosceles triangle. Reason: If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.Ans: We know that “If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.” – This is AAS Congruence Rule So, Reason is correctIn △ABL and △ACM,BL = CM (Given)∠BAL = ∠CAM (Common angle)∠ALB = ∠AMC = 90°∴ △ABL ≅△ACM (AAS criterion)Hence, ∆ABC is an isosceles triangle.So, Assertion (A) is also true.Correct option is (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

⇒ AB = AC (CPCT)

15. Assertion: Two angles measures a – 60° and 123° – 2a. If each one is opposite to equal sides of an isosceles triangle, then the value of a is 61°.Reason: Sides opposite to equal angles of a triangle are equal.Ans: We know that Sides opposite to equal angles of a triangle are equal.So, Reason is correct.Since angles opposite to equal sides of an isosceles triangle are equal,therefore a – 60° = 123° – 2a⇒ 3a = 123° + 60° = 183°⇒ a = = 61°So, Assertion is also correct.But reason (R) is not the correct explanation of assertion (A)Correct option is (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).

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