NCERT Solutions Class 10 Maths Chapter 6 — Triangles

Solution:

Step 1: The Basic Proportionality Theorem (BPT), also known as Thales Theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Step 2: In ΔABC, if DE || BC, then according to BPT, the ratio of the segments on AB is equal to the ratio of the segments on AC. [AD/DB = AE/EC]

Solution:

Step 1: First, calculate the ratios of the segments on sides PQ and PR.

Step 2: PS/SQ = 3 cm / 6 cm = 1/2.

Step 3: PT/TR = 4 cm / 8 cm = 1/2.

Step 4: Since PS/SQ = PT/TR (both are 1/2), by the converse of the Basic Proportionality Theorem, the line segment ST must be parallel to QR.

Solution:

Step 1: In ΔABC: ∠C = 180° - (∠A + ∠B) = 180° - (50° + 60°) = 180° - 110° = 70°.

Step 2: In ΔDEF: ∠E = 180° - (∠D + ∠F) = 180° - (50° + 70°) = 180° - 120° = 60°.

Step 3: Now, we have: ∠A = ∠D = 50°, ∠B = ∠E = 60°, and ∠C = ∠F = 70°.

Step 4: Since two corresponding angles (and thus all three) are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion.

Solution:

Step 1: Let's check the ratios of corresponding sides:

Step 2: PQ/XY = 6/3 = 2.

Step 3: QR/YZ = 8/4 = 2.

Step 4: PR/XZ = 10/5 = 2.

Step 5: Since the ratios of all three pairs of corresponding sides are equal (PQ/XY = QR/YZ = PR/XZ = 2), the triangles ΔPQR and ΔXYZ are similar by the SSS (Side-Side-Side) similarity criterion.

Solution:

Step 1: The SAS similarity criterion states that if two sides and the *included angle* of one triangle are proportional to two sides and the included angle of another triangle, then the two triangles are similar.

Step 2: In option A, the sides LM and MN are proportional to RS and ST, and the angle ∠M is the included angle between LM and MN, while ∠S is the included angle between RS and ST. This matches the SAS criterion.

Step 3: Options B, C, and D do not have the angle included between the proportional sides, hence they are incorrect.