CLASS 10 CBSE

1. CH 6
TRIANGLES
EX 6.2 CONTINUE
+
CONCEPT OF CONGRUENCY AND SIMILARITY
Ex 6.1 and Ex 6.3

2. 5. In the figure, DE || OQ and DF || OR. Show that EP || QR
In∆ PQR
∴ E and F are two distinct points on PQ and PR respectively and
𝑃𝐸
𝐸𝑄
=
𝑃𝐹
𝐹𝑅
;E and F are dividing the
two sides PQ and PR in the same ratio in ΔPQR.
∴ EF || QR
Sol. In ΔPQO
DE || OQ [Given]
∴ Using the Basic Proportionality Theorem, we have:
FROM (1) AND (2)

3. 6. In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR.
Show that BC || QR.
Sol. In ΔPQR,
O is a point OP, OQ and OR and joined. We have A, B and C on
OP, OQ. and OR respectively such that AB || PQ and AC || OR.
Now, in OPQ,
AB || PQ [Given]

10. For example,
all circles are similar
to each other,
all squares are
similar to each other,
and all equilateral
triangles are similar
to each other. On the
other
hand, ellipses are not
all similar to each
other, rectangles are
not all similar to each
other, and isosceles
triangles are not all
similar to each other.

11. But we don't need to know all three sides and all three
angles ...two or three out of the six is usually enough.
Two triangles are similar if they have:
•all their angles equal
•corresponding sides are in the same ratio
There are three ways to find if two triangles are similar: AA, SAS and SSS
AA
AA stands for "angle, angle" and means
that the triangles have two of their
angles equal.
If two triangles have two of their angles
equal, the triangles are similar.

12. SAS
SAS stands for "side, angle, side" and means that we have two triangles where:
•the ratio between two sides is the same as the ratio between another two sides
•and we also know the included angles are equal.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal,
then the triangles are similar.
In this example we can see that:
•one pair of sides is in the ratio of 21 : 14 = 3 : 2
•another pair of sides is in the ratio of 15 : 10 = 3 : 2
•there is a matching angle of 75° in between them
So there is enough information to tell us that the two triangles are
similar.

13. SSS
SSS stands for "side, side, side" and means that we have two triangles with all three pairs of
corresponding sides in the same ratio.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
In this example, the ratios of sides are:
•a : x = 6 : 7.5 = 12 : 15 = 4 : 5
•b : y = 8 : 10 = 4 : 5
•c : z = 4 : 5
These ratios are all equal, so the two triangles are similar.

14. EXERCISE 6.1
1. Fill in the blanks using the correct b given in brackets:
(i) All circles are ……… (congruent, similar)
(ii) All squares are ……… (similar, congruent)
(iii) All ……… triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if
(a) their corresponding angles are ……… and
(b) their corresponding sides are ……… (equal, proportional).
Sol. (i) All circles are similar.
(ii) All squares are similar.
(iii) All equilateral triangles are similar.
(iv) Two polygons of the same number of sides are similar if:
(a) Their corresponding angles are equal and
(b) Their corresponding sides are proportional.
2. Give two different examples of pair of.
(i) similar figures (ii) non-similar figures.
Sol. (i) (a) Any two circles are similar figures.
(b) Any two squares are similar figures.
(ii) (a) A circle and a triangle are non-similar figures.
(b) An isosceles triangle and a scalene triangle are non-similar figures.

15. 3. State whether the following quadrilaterals are similar or not:
Sol. On observing the given figures, we find that:
Their corresponding sides are proportional but their corresponding angles are Dot equal.
∴ The given figures are not similar.
• Similar Triangles
Triangles are a special type of polygons. The study of their similarity is important.
Two triangles are said to be similar if:
(i) Their corresponding sides are proportional, and,
(ii) Their corresponding angles are equal.

16. EXERCISE 6.3
1. State which pairs of triangles in the figures, are similar. Write the similarity criteria used by you
for answering the question and also write the pairs of similar triangles in the symbolic form:
Sol. (i) In ΔABC and ΔPQR
We have:
∠A = ∠P = 60°
∠B = ∠Q = 80°
∠C = ∠R = 40°
∴ The corresponding angles are equal,
∴ Using the AAA similarity rule,
ΔABC ~ ΔPQR

18. 2. In the figure, ΔODC ~ ΔOBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC, ∠DCO and ∠OAB.

19. Sol. We have:
∠BOC = 125° and ∠CDO = 70°
Thus, from (1), (2) and (3)
∠DOC = 55°, ∠DCO = 55° and ∠OAB = 55°.
since, ∠DOC + ∠BOC = 180° [Linear Pair]
⇒ ∠DOC 180° – 125° = 55° ...(1)
In ΔDOC
Using the angle sum property, we get
∠DOC + ∠ODC + ∠DCO = 180°
⇒ 55° + 70° + ∠DCO = 180°
⇒∠DCO =180° – 55° – 70° = 55° ...(2)
Again,
ΔODC ~ ΔOBA [Given]
∴ Their corresponding angles are equal
And ∠OCD = ∠OAB = 55° ...(3)

20. 3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using
a similarity criterion for two triangles, show that
Sol. We have a trapezium ABCD in which AB || DC. The
diagonals AC and BD intersect at O.
In ΔOAB and ΔOCD
AB || DC [Given]
and BD intersects them
∴∠OBA = ∠ODC ...(1)
[Alternate angles]
similarly,
∠OAB = ∠OCD ...(2)
∴Using AA similarity rule,
ΔOAB ~ ΔOCD