NCERT Solutions Class 9 Maths Chapter 7 — Triangles

Solution:

Step 1: Congruent figures are those that can be perfectly superimposed on each other without any distortion.

Step 2: For this to happen, they must possess identical shapes and identical sizes.

Solution:

Step 1: When two triangles are congruent, their corresponding parts (sides and angles) are equal (CPCTC).

Step 2: From the congruence statement ΔPQR ≅ ΔXYZ, the corresponding parts are: PQ ↔ XY, QR ↔ YZ, PR ↔ XZ, ∠P ↔ ∠X, ∠Q ↔ ∠Y, ∠R ↔ ∠Z.

Step 3: Therefore, PQ = XY, QR = YZ, and ∠P = ∠X are all necessarily true. However, PR corresponds to XZ, not YZ. So, PR = YZ is not necessarily true.

Solution:

Step 1: The SSS congruence criterion states that if three sides of one triangle are equal to three corresponding sides of another triangle, then the triangles are congruent.

Step 2: Given AB = DE and BC = EF, the third pair of corresponding sides that must be equal is AC and DF.

Solution:

Step 1: The SAS congruence criterion requires two sides and the *included angle* (the angle formed between those two sides) of one triangle to be equal to two sides and the *included angle* of another triangle.

Step 2: In ΔPQR, for sides PQ and PR, the included angle is ∠P. Rohan used ∠Q, which is not included between PQ and PR.

Solution:

Step 1: We are given two angles and a side: ∠A = ∠X, AB = XY, and ∠B = ∠Y.

Step 2: The side AB is *included* between angles ∠A and ∠B in ΔABC. Similarly, XY is included between ∠X and ∠Y in ΔXYZ.

Step 3: Since two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, the ASA (Angle-Side-Angle) congruence criterion applies.