The document contains information about triangles, including: 1) If two triangles have proportional sides and equal angles, they are similar triangles. 2) In a right triangle, a perpendicular line from the right angle to the hypotenuse divides it into two right triangles that are similar to each other and to the original triangle. 3) A line dividing two sides of a triangle proportionally is parallel to the third side.

1. STANDARD IX
Part – 2
2013-2014

2. Jana-gana-mana-adhinayaka, jaya he
Bharata-bhagya-vidhata,
Punjab-Sindh-Gujarat-Maratha
Dravida-Utkala-Banga
Vindhya-Himachala-Yamuna-Ganga
Uchchala-Jaladhi-taranga.
Tava Shubha name jage,
Tava Shudbha asisa mage,
Gahe tava jaya gatha,
Jana-gana-mangala-dayaka jaya he
Bharata-bhagya-vidhata,
Jaya he, jaya he, jaya he,
Jaya he, jaya he, jaya he,

3. India is my country. All Indians are my brothers and sisters.
I love my country, and I am proud of its rich and varied
Heritage. I shall always strive to be worthy of it.
I shall give respect to my parents, teachers and all elders
And treat everyone with courtesy.
I pledge my devotion to my country and my people. In their
Well-being and prosprity alone likes my happiness.

7. Congruency of two triangles
 If three sides of a triangle are equal to the three sides of another triangle, then these triangles are
are congruent.
 If two sides of a triangle and their included angle are equal to two sides of another triangle and their
included angle, then these triangles are congruent.
 Just because two sides and some angles of a triangle are equal to two sides and some angle of another
triangle, the two triangles need not be congruent.
 If one side and the two angles on it of a triangle are equal to one side and the two angles on it of
another triangle then these triangles are congruent.
 If the hypotenuse and one another side of a right angled triangle are equal to the hypotenuse and one
other Side of another right angled triangle, then these two triangles are congruent.

8. If all the angles of a triangle are equal to the angles of another
triangle, then all the pairs of sides opposite equal angles have the
same ratio.
c b z y
a x
x/a = y/b = z/c

9. This can be shortened a bit more :
If all the angles of a triangle are equal to the angles of
another triangle, then the sides opposite equal angles are
proportional.

10. Can you find out the proportional sides in the following figures?
A P
a)
500 700 500 700
B C Q
Ans : A = Q ; C = R ; A = P
The pairs of proportional sides
AB,PQ ; AC,PR ; BC,QR

11. D P
700
800 800 300
F Q R
Ans: E = Q F = R D = P
The pairs of proportional sides:
EF, QR ; DF, PR ; DE, PQ

12. In the figures below, we have ABC and some other triangles with the
same angle. Write against each, the names of the equal angles and the
lengths of the equal sides.
A K
B C (i) L M
K = A L = B
M = ………….. KL = ……………….
LM = 12 cm
MK = ………………………………….

13. Ans : K = A , L= B , M= C
BC/LM = AC/MK = AB/KL ( If all the angles of a triangle are equal to the angles of another then the sides opposite equal angles are proportional)
BC = 6 cm, AB = 4 cm, AC =5 cm, LM = 12 cm
Therefore BC/LM = ½
That is, BC/LM = AC/MK = AB/KL = ½
AC/MK = ½
5/MK = ½
MK = 10 cm
AB/KL = ½
4/KL = ½
KL = 8 CM

14. Give a triangle, there are several ways to draw another one with
the same angles, but of different size. Look at this triangle:
A
B C
Suppose we want to enlarge it without altering the angles. We
can extend the left and right sides as much as we want ant this won’t
change the top angle. A
B C
D E

15. We get a triangle, however we close the extended sides:
A
B C
If the two bottom angles are also to be equal, how should we draw the
bottom line?
The bottom line should be parallel to the line just above.
A
B C
Can you prove that if the lines at the bottom are parallel, then
the triangles would have equal angles.
Ans : Angle B and angle D are corresponding angles formed, when

16. AD cuts the parallel lines BC and DE.
B = D (When a pair of parallel lines is cut by a third lines, each pair of
corresponding angles are equal)
C and E are corresponding angles formed, when AE cuts the parallel
lines BC and DE.
C = E (When a pair of parallel lines is cut by a third line, each pair of
corresponding angles are equal)
A is a common angle to both ABC and ADE have equal angles.

17. Irrational problem
If in two triangles withn the same three angles, the ratio of one pair of sides
opposite equal angles can be expressed in terms of natural numbers, then we can show that the
other pairs of sides also have the same ratio, by dividing the triangles into smaller ones, as we
have shown.
But there are instances where the ratio of sides cannot be expressed in terms of
rational numbers . For example, draw an isosceless right angled triangle with the lenghts of
the perpendicular sides 1 and another isosceless right angled triangle with the lengths of the
perpendicular sides 2 .
2 2
1 2
1
2

18. The angles of both triangles are 450 , 450 ,900. But the ratio of sides opposite
equal angles is 1: 2 .
However small we divide one of the perpendicular sides of one triangle, we
cannot completely fill with it, perpendicular side of the other triangle. The same is true for
the hypotenuse also.
So, the method of dissection of the triangles to prove equality of ratios will not
work in cases such as this.
From this we have understood that the instances where the ratio of the sides
opposite angles cannot be expressed in natural numbers, then we can’t prove that the other
sides also have the same ratio dividing the triangle.

19. REVERSE QUESTION

20. A P X
c b z y z y
B C Q R Y Z
a x x
Thus the three sides of PQR are equal to the three sides of
XYZ and so these triangles are congruent. So, the angles opposite
their pairs of equal sides are also equal:
X = P , Y= Q , Z = R
P = A, Q = B , R = C
X = A , Y = B, Z = C

21. Triangle speciality
If the angles of a triangle are equal to angles of another triangle, then their sides are
proportional; on the other hand, if the sides of two triangles are proportional, then they have the
same angles. Among polygons, only triangles have this pecularity.
For example, all angles of a square and a rectangle which is not a square, are right
angles; but the sides are not proportional.
On the other hand, a square and a rhombus which is not a square have proportional
sides; but the angles are not equal.

22. similarity
We saw that if the angles of a triangle are all equal to the angles of
another triangle, then the sides of the two triangles are proportional; and on the
other hand, if the lenghths of the sides of a triangle are proportional to the
lengths of the sides of another triangle, then the angles of one triangle are equal
to the angles of the other.

23. In ABC shown below, A is a right angle
A
Draw the perpendicular from A to BC . Now we get two small right
angled triangles also.
B C
P
What can we say about the angles of these?
Let’s write B = X0 for convenience.
A
B C
P
?
B C
A

24. (One angle is right angle, Third angle = 180-(90+x)= 90-x)
Similarly, using the right angled triangles ABP and ACP, we can write other
angles in terms of x.
A
(90-x)0
X0 (90-x)0
B P C
Thus the angles of the triangles ABP and ACP are 900 , X0 and (90-x)0. So , they are
similar.
The angles of our original triangle ABC are also these. So, this triangle is also similar to
ABP and ACP.
Thus in a right angled triangle, the perpendicular to the hypotenuse from the opposite
vertex divides, it into two right angled triangles, which are similar to each other; they
are also similarly to the original triangle.

25. If two sides of a
triangle are proportional to
two sides of another
triangle and if their
included angles are equal,
then the triangles are
similar.

26. P X
A
c b y z y
B a C Q x R Y x Z
X = P , Y = Q , Z = R
P = A , Q = B , R = C
X = A , Y = B , Z = C

27. Similarity and Congruence
If two triangles are congruent, then they are also similar. (Congruent triangles have
the same angles; also the ratio of each pair of sides opposite equal angles is 1:1)
But two similar triangles may not be congruent.
Look at a comparison of these two concepts :
 If two triangles have their sides proportional, then they are similar; they are
congruent only if sides are equal.
 If two triangles have two pairs of angles equal, then they are similar; they are
congruent only if two pairs of angles and their common side are equal.
 If two triangles have two pairs of sides proportional and their included angles
equal, then they are smilar; they are congruent only if two pairs of sides are
equal and their included angles are equal.

28. ? The circles shown below have the same centre O.
P
Q
A
O
B
Prove that OAB and OPQ are similar.

29. Ans: Consider OAB and OPQ.
O is common to both the triangles.
Therefore AOB = POB ……………….(1)
In OAB, OA= OB(Radii of the circle)
That is, OA/OB = 1 …………………..(2)
In OPQ, OP=OQ (Radii of the circle)
That is, OP/OQ = 1 ……………………(3)
From (2) and (3), OA/OB= OP/OQ ……………..(4)
If a pair of sides are proportional and their included angles are equal, then also the
triangles are similar.
From (1) and (4), OAB and OPQ are similar triangles.

30. Prove that in a triangle, a line dividing two sides
proportionally is parallel to the third side.
Ans : In the figure, line PQ cuts the sides AB and AC of ABC
proportionally. A
P Q
B C
i.e., AP/PB = AQ/QC ……………(1)

31. We want to prove that the line PQ is parallel to BC
From the data given,
We can say PB/AP = QC/AQ
Adding 1 to both sides
1 + PB/AP = 1+ QC/AQ
Now 1+ PB/AP = (AP + PB)/AP = AB/AP
1 + QC/AQ = (AQ+QC)/AQ = AC/AQ
Therefore AB/AP = AC/AQ
Noq the sides AB and AC of ABC and AP and AQ of APQ
are proportional and the included angles of both the triangle are A
itself.
Therefore the triangles are similar.

32. Since the angles opposite to the proportional sides of the simi9lar
triangles are equal APQ = ABC
Since they are corresponding angles, PQ || BC

33. END