Exercise 6.1 answers: Triangles

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(i) All circles are ________ . (congruent, similar)

1. A circle is fixed by just one number, its radius.
2. Any two circles can always be scaled one onto the other, so they are always similar — but they need equal radii to be congruent.

Answersimilar

(ii) All squares are ________ . (similar, congruent)

1. A square is fixed by just one number, its side.
2. Any two squares always have equal angles (90° each) and sides in one ratio, so they are always similar — but they need equal sides to be congruent.

Answersimilar

(iii) All ________ triangles are similar. (isosceles, equilateral)

1. Every equilateral triangle has all three angles equal to 60°.
2. So any two equilateral triangles always have matching angles and sides in one ratio — they are similar.
3. An isosceles triangle's apex angle can be anything, so two isosceles triangles need not have matching angles at all.

Answerequilateral

(iv) Two polygons of the same number of sides are similar, if …

1. This is exactly the definition of similar polygons used in this chapter.
2. Similar polygons must have (a) corresponding angles equal, and (b) corresponding sides in the same ratio (proportional).

Answer(a) equal, (b) proportional

Watch this explained “Circles”, 1:49 into What similarity asks for that congruence does not

Open NCERT p. 78Checked by computerAnswers can differ: one example

(i) similar figures.

1. Many answers are possible; here are two.
2. Two squares of sides 2 cm and 5 cm: every angle of each is 90°, and every side of the bigger square is 5/2 times the matching side of the smaller one. Equal angles and sides in one ratio, so they are similar.
3. Two circles of radii 2 cm and 5 cm: enlarging the smaller circle by 5/2 gives exactly the bigger one, so they are similar (all circles are similar).

AnswerTwo squares of sides 2 cm and 5 cm; two circles of radii 2 cm and 5 cm.

(ii) non-similar figures.

1. Again many answers are possible; here are two.
2. A square of side 2 cm and a 2 cm by 4 cm rectangle: all angles are 90° in both, but the side ratios are 2/2 = 1 and 2/4 = 1/2, not one ratio, so they are not similar.
3. An equilateral triangle of side 3 cm and a square of side 3 cm: one has 3 sides and the other has 4, and enlarging a figure never changes its number of sides, so they are not similar.

AnswerA square of side 2 cm and a 2 cm by 4 cm rectangle; an equilateral triangle of side 3 cm and a square of side 3 cm.

Watch this explained “But not every family”, 3:15 into What similarity asks for that congruence does not

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1. ABCD is a square: all four sides are 3 cm and all four angles are marked 90°.
2. PQRS has all four sides 1.5 cm, but its corners carry no right-angle mark and the figure is drawn leaning over — it is a general rhombus, not a square, so its angles are not 90°.
3. The sides ARE in one ratio: every side of PQRS is exactly half the matching side of ABCD.
4. But the angles are NOT equal (PQRS's angles are not 90° like ABCD's), so one of the two conditions needed for similar polygons fails.

AnswerNo, ABCD and PQRS are not similar.

Watch this explained “Sides alone is not enough”, 8:09 into The two conditions polygons must both meet