Congruence & Symmetry MCQ Test — Class 7 CBSE

Solution:

The definition of congruence in geometry means that two figures are exactly alike.

This implies they must have both the same shape and the same size.

If one figure can be moved (translated, rotated, reflected) and perfectly overlap the other, they are congruent. Option A provides the most complete and accurate definition.

Solution:

Congruence refers to figures having the same shape and size.

For line segments, their 'shape' is always a straight line, so the crucial factor for congruence is their 'size'.

The 'size' of a line segment is its length. Therefore, two line segments are congruent if they have the same length.

Solution:

Angles are defined by the amount of rotation between two rays meeting at a common vertex.

The 'size' or 'opening' of an angle is determined by its measure, typically in degrees.

For two angles to be congruent, their measures must be exactly the same. The lengths of the arms do not affect the angle's measure.

Solution:

The problem states that all three corresponding sides of ΔABC and ΔXYZ are equal: AB = XY, BC = YZ, and CA = ZX.

When three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent by the SSS (Side-Side-Side) criterion.

Therefore, the SSS congruence criterion applies.

Solution:

We are given two sides (PQ and QR) and the included angle (∠PQR) of ΔPQR.

Similarly, we are given two sides (DE and EF) and the included angle (∠DEF) of ΔDEF.

Since PQ = DE, ∠PQR = ∠DEF, and QR = EF, the two triangles are congruent by the SAS (Side-Angle-Side) criterion, as the angle is included between the two sides.

Solution:

The ASA (Angle-Side-Angle) congruence criterion states that if two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

For ΔMNO and ΔRST, the included side between ∠M and ∠N is MN. The included side between ∠R and ∠S is RS.

Therefore, for ASA, we need ∠M = ∠R, MN = RS, and ∠N = ∠S. Option A matches this.

Solution:

The congruence statement ΔXYZ ≅ ΔPQR means that corresponding vertices are in the same order.

This implies: Vertex X corresponds to Vertex P, Vertex Y corresponds to Vertex Q, and Vertex Z corresponds to Vertex R.

Therefore, corresponding angles are equal: ∠X = ∠P, ∠Y = ∠Q, ∠Z = ∠R. And corresponding sides are equal: XY = PQ, YZ = QR, XZ = PR.

Looking at the options, ∠Y = ∠Q is always true based on the corresponding vertices.

Solution:

A line of symmetry is a line that divides a figure into two mirror-image halves.

An equilateral triangle has three equal sides and three equal angles.

Each median (line segment from a vertex to the midpoint of the opposite side) acts as a line of symmetry. Since there are three vertices, an equilateral triangle has 3 lines of symmetry.

Solution:

A square has four equal sides and four right angles.

If you rotate a square by 90°, it looks exactly the same. It will also look the same at 180°, 270°, and 360° (back to original).

It looks identical 4 times during a full 360° rotation.

Therefore, the order of rotational symmetry for a square is 4.

Solution:

Rohan used AB = PQ (Side), ∠A = ∠P (Angle), and BC = QR (Side).

The SAS (Side-Angle-Side) congruence criterion requires the angle to be *included* between the two sides.

In ΔABC, the sides given are AB and BC, but the angle given is ∠A. ∠A is not the included angle between sides AB and BC (which would be ∠B). Similarly for ΔPQR, ∠P is not included between PQ and QR.

Therefore, Rohan made a mistake because the given angle was not the included angle between the two given sides.