CBSE Class 7 Mathematics Geometric Twins MCQs Set 11

Answer: (b) congruent figures

Explanation:
1. Figures that are exact copies of each other, with the same shape and the same size, are called congruent figures.
2. Such figures can be placed one over the other so that they fit exactly.

Teacher's Note:
1. Congruent means same shape and same size.
2. Same shape alone is not enough.

Answer: (c) three sides of the triangle

Explanation:
1. Each S in SSS stands for a Side.
2. So SSS means three sides: if all three sides of one triangle equal the three sides of another, the triangles are congruent.

Teacher's Note:
1. S stands for side and A stands for angle.
2. SSS needs no angle at all.

Answer: (d) hypotenuse

Explanation:
1. The side facing the right angle is the longest side of a right-angled triangle.
2. It is called the hypotenuse; it is the H in the RHS condition.

Teacher's Note:
1. The hypotenuse always faces the right angle.
2. It is the longest side of the triangle.

Answer: (a) 60°

Explanation:
1. All three sides are equal, so all three angles are equal.
2. The angles add up to 180°, so each angle = 180° ÷ 3 = 60°.

Teacher's Note:
1. Equal sides give equal opposite angles.
2. Angle sum of a triangle is 180°.

Answer: (a) SSA

Explanation:
1. ASA, RHS and SAS are all conditions that guarantee congruence.
2. In SSA the angle is not between the two sides, and two different triangles can be drawn from the same data.
3. So SSA does not guarantee congruence.

Teacher's Note:
1. Check where the angle lies with respect to the sides.
2. SSA is the arrangement that fails.

Answer: (c) Both the statements are true

Explanation:
1. Congruent figures fit exactly on superimposition, so Statement I is true.
2. Turning or flipping does not change shape or size, so it is allowed while checking. Statement II is true.
3. So both the statements are true.

Teacher's Note:
1. Superimposition is the test for congruence.
2. Turning or flipping only changes position.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. All sides of an equilateral triangle are equal, so by R all three angles are equal.
2. With the angle sum 180°, each angle is 180° ÷ 3 = 60°. So A is true.
3. R is true and gives exactly the reasoning for A, so R explains A.

Teacher's Note:
1. R gives both facts needed for A.
2. Equal sides lead to equal angles.

Answer: (a) its length and its breadth

Explanation:
1. All angles of a rectangle are right angles, so only the two side lengths can change.
2. A single length, the perimeter or one diagonal can belong to many different rectangles.
3. So we must take its length and its breadth.

Teacher's Note:
1. A rectangle is fixed by two measurements.
2. Many rectangles share the same perimeter.

Answer: (c) QR

Explanation:
1. In the statement, the letters correspond in order: A with P, B with Q, C with R.
2. So side BC corresponds to side QR, and BC = QR.

Teacher's Note:
1. Match letters by their position.
2. Second and third letters give the side BC ↔ QR.

Answer: (b) SSS

Explanation:
1. Single marks: AB = PQ. Double marks: BC = QR. Triple marks: CA = RP.
2. All three pairs of sides are equal and no angle is marked.
3. So the condition is SSS.

Teacher's Note:
1. Same number of marks means equal sides.
2. Three pairs of equal sides means SSS.

Answer: (c) A is true, but R is false.

Explanation:
1. Rotating or flipping is allowed while checking congruence, so A is true.
2. Turning or flipping changes only the position of a figure, never its shape or size, so R is false.
3. So A is true but R is false.

Teacher's Note:
1. Turning a figure does not stretch or shrink it.
2. That is exactly why turning is allowed.

Answer: (a) (i)-(q), (ii)-(r), (iii)-(s), (iv)-(p)

Explanation:
1. SSS uses three sides: (i)-(q). SAS uses two sides and the angle between them: (ii)-(r).
2. ASA uses two angles and the side between them: (iii)-(s). RHS uses a right angle, the hypotenuse and one more side: (iv)-(p).
3. So the match is (i)-(q), (ii)-(r), (iii)-(s), (iv)-(p).

Teacher's Note:
1. Read each letter of the name in order.
2. The middle letter is the included part.

Answer: (d) 70°

Explanation:
1. In △ABC, ∠C = 180° − (65° + 45°) = 180° − 110° = 70°.
2. C corresponds to F, so ∠F = ∠C = 70°.

Teacher's Note:
1. Find the third angle first.
2. Third letters correspond: C ↔ F.

Answer: (b) have the same shape but need not have the same size

Explanation:
1. Equal angles fix only the shape of a triangle, not its size.
2. Many triangles of different sizes can be drawn with angles 30°, 70° and 80°.
3. So they have the same shape but need not have the same size.

Teacher's Note:
1. Angles alone cannot fix size.
2. A photo and its enlargement have the same angles.

Answer: (a) Both the statements are true

Explanation:
1. In SSA the arc can cut the ray at two points, giving two different triangles, so Statement I is true.
2. A right angle, the hypotenuse and one more side form the RHS condition, which guarantees congruence. Statement II is true.
3. So both the statements are true.

Teacher's Note:
1. RHS is the special case where SSA works.
2. The right angle makes the difference.

Answer: (d) A is false, but R is true.

Explanation:
1. ∠B is not between AB and AC (the angle between them is ∠A), so the data is SSA.
2. SSA can give two non-congruent triangles, so A is false.
3. R correctly describes why SSA fails, so R is true. Hence A is false but R is true.

Teacher's Note:
1. Find the angle between the two given sides.
2. If the given angle is not that one, it is SSA.

Answer: (c) △ ABD ≅ △ CDB

Explanation:
1. In the rectangle, AB = CD and AD = CB (opposite sides), and BD = DB is common.
2. So A must match C, B must match D, and D must match B.
3. The correct statement is △ ABD ≅ △ CDB.

Teacher's Note:
1. Match each side with its equal side first.
2. The order of letters must follow the matching.

Answer: (b) 70°

Explanation:
1. AB = AC, so the angles opposite them are equal: ∠B = ∠C.
2. ∠B + ∠C = 180° − 40° = 140°, so ∠B = 140° ÷ 2 = 70°.

Teacher's Note:
1. Equal sides give equal base angles.
2. Halve what is left after the top angle.

Answer: (b) (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)

Explanation:
1. (i) Two sides with the angle A between them: SAS, so (i)-(s).
2. (ii) Two angles with the side BC between them: ASA, so (ii)-(p).
3. (iii) Two angles and BC, which is not between ∠A and ∠B: AAS, so (iii)-(q).
4. (iv) Right angle, hypotenuse AC and side BC: RHS, so (iv)-(r). The match is (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r).

Teacher's Note:
1. Check whether the side lies between the two angles.
2. A right angle with the hypotenuse points to RHS.

Answer: (c) △ EHN ≅ △ IBG

Explanation:
1. The matching is H ↔ B, E ↔ I, N ↔ G.
2. In (c), E is under I, H is under B and N is under G, so every pair is kept.
3. So △ EHN ≅ △ IBG is correct; the others break the matching.

Teacher's Note:
1. Rearrange both names in the same way.
2. Check each letter pair one by one.

Answer: (d) A is false, but R is true.

Explanation:
1. R states the correct rule, so R is true.
2. In △QRP ≅ △XYZ, Q would match X; but Q actually matches Y. So A is false.
3. The correct rewriting is △QRP ≅ △YZX. Hence A is false but R is true.

Teacher's Note:
1. Reorder both names together, never just one.
2. Check the first letters: Q and Y must go together.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. O is the midpoint of both segments, so AO = OD and BO = OC; ∠AOB = ∠DOC (vertically opposite). R is true.
2. By SAS, △AOB ≅ △DOC, so the corresponding sides AB and DC are equal. A is true.
3. R gives the reason for A, so R is the correct explanation.

Teacher's Note:
1. Vertically opposite angles are always equal.
2. Congruence then gives equal corresponding sides.

Answer: (b) Only Statement I is true

Explanation:
1. Statement I is the ASA condition, so it is true.
2. Two angles and a side not between them is the AAS condition, which also guarantees congruence (the third angle is fixed by the angle sum). So Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. With two angles known, the third is known too.
2. So AAS works as well as ASA.

Answer: (d) RHS

Explanation:
1. ∠ADB = ∠ADC = 90°, so both triangles are right-angled at D.
2. Their hypotenuses AB and AC are equal, and AD is a common side.
3. Right angle, hypotenuse and one side: RHS.

Teacher's Note:
1. The side opposite the right angle is the hypotenuse.
2. A common side is equal to itself.

Answer: (d) AB = BC

Explanation:
1. AB = AD, CB = CD and AC is common, so △ABC ≅ △ADC by SSS. (a) is correct.
2. Then ∠BAC = ∠DAC and ∠BCA = ∠DCA, so (b) and (c) are correct.
3. Nothing in the data makes AB equal to BC, so (d) is not a correct conclusion.

Teacher's Note:
1. Use only what the data gives.
2. Congruence gives equal corresponding parts, not any parts.

Answer: (d) A is false, but R is true.

Explanation:
1. Two right triangles can have the same hypotenuse but different other sides (for example 5 cm with legs 3 cm, 4 cm, or with a leg of 1 cm). So A is false.
2. R correctly states what RHS needs, so R is true.
3. Hence A is false but R is true.

Teacher's Note:
1. The hypotenuse alone does not fix the triangle.
2. RHS needs one more equal side.

Answer: (a) Both the statements are true

Explanation:
1. With AB = AC, right angles at D and the common side AD, RHS gives △ADB ≅ △ADC. Statement I is true.
2. ∠B and ∠C are corresponding angles of these congruent triangles, so ∠B = ∠C. Statement II is true.
3. So both the statements are true.

Teacher's Note:
1. This is how the isosceles triangle property is proved.
2. The altitude makes two right triangles.

Answer: (c) Sides 5 cm and 6 cm with an angle of 50° that is not between them

Explanation:
1. (a) is SSS, (b) is SAS and (d) is ASA; each fixes the triangle.
2. (c) is SSA: the arc of the second side can cut the ray at two points, giving two different triangles.
3. So the answer is (c).

Teacher's Note:
1. Name the arrangement for each option.
2. Only SSA can give two different triangles.

Answer: (c) (i)-(q), (ii)-(p), (iii)-(s), (iv)-(r)

Explanation:
1. (i) Three pairs of sides: SSS, so (i)-(q).
2. (ii) ∠A, ∠C and side AB, which is not between A and C: AAS, so (ii)-(p).
3. (iii) ∠B is not between AB and AC: SSA, congruence not guaranteed, so (iii)-(s).
4. (iv) ∠A lies between AB and AC: SAS, so (iv)-(r). The match is (i)-(q), (ii)-(p), (iii)-(s), (iv)-(r).

Teacher's Note:
1. Locate the given angle between or outside the given sides.
2. SSA is the only failing case here.

Answer: (b) 30°

Explanation:
1. AB and AC are radii, so AB = AC and the angles opposite them are equal: ∠ABC = ∠ACB.
2. ∠ABC = (180° − 120°) ÷ 2 = 60° ÷ 2 = 30°.

Teacher's Note:
1. Two radii make an isosceles triangle.
2. Base angles share what is left of 180°.