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CBSE(NCERT): CLASS IX MATHS

ASSERTION & REASONING QUESTIONS

TRIANGLES

By M. S. Kumar Swamy TGT(Maths) KV Gachibowli

ASSERTION & REASONING QUESTIONS

DIRECTION : 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 ABC, we have: A + B + C=180? [Sum of the angles of a triangle] 50? + B + C = 180? B + C = 130? B + C = 65? ...(i) In OBC, we have: OBC + OCB + BOC = 180?

B + C + BOC = 180? [Using (i)]

65?+ BOC = 180? BOC=115? 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).

2. Assertion : In ABC, C = A, BC = 4 cm and AC = 5 cm. Then, AB = 4 cm 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, 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 = x Using angle sum property of a triangle, A + B + C = 180? x + 80?+ x = 180? 2x = 180?- 80? 2x = 100? x = 50? A = 50? 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).

4. Assertion : In ABC, D is the midpoint of BC. If DL AB and DM AC such that DL = DM, then BL = CM 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 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 have BD = CD (D is midpoint) DL = DM (Given) and BLD = CMD (90? each) BDL CDM (RHS criterion) BL = CM (CPCT) 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).

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 congruent

Ans: 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) CX = BY (CPCT) 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).

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 congruent

Ans: 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) AB = CQ (CPCT) Now, DQ = DC + CQ DQ = DC + AB (AB = CQ prove above) 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).

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