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The Triangle and its Properties
Main Concepts and Results
• The six elements of a triangle are its three angles and the three sides.
• The line segment joining a vertex of a triangle to the midpoint of its opposite side is called
a median of the triangle. A triangle has 3 medians.
• The perpendicular line segment from a vertex of a triangle to its opposite side is called an
altitude of the triangle. A triangle has 3 altitudes.
• An exterior angle of a triangle is formed, when a side of a triangle is produced.
• The measure of any exterior angle of a triangle is equal to the sum of the measures of its
two interior opposite angles.
• The sum of the three angles of a triangle is 180°.
• A triangle is said to be equilateral, if each of its sides has the same length.
• In an equilateral triangle, each angle has measure 60°.
• A triangle is said to be isosceles if at least two of its sides are of same length.
• The sum of the lengths of any two sides of a triangle is always greater than the length of
the third side.
• The difference of the lengths of any two sides of a triangle is always smaller than the
length of the third side.
• In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and
the other two sides are called its legs or arms.
• In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares
on its legs.
• Two plane figures, say, F1 and F2 are said to be congruent, if the trace-copy of F1 fits
exactly on that of F2. We write this as F1 ≅ F2.
• Two line segments, say AB and CD, are congruent, if they have equal lengths. We write
this as ≅ AB CD. However, it is common to write it as AB= CD.
• Two angles, say ∠ABC and ∠ PQR, are congruent, if their measures are equal. We write
this as ∠ ABC ≅ ∠ PQR or as m ∠ ABC = m ∠PQR or simply as ∠ ABC = ∠ PQR.
• Under a given correspondence, two triangles are congruent, if the three sides of the one
are equal to the three sides of the other (SSS).
• Under a given correspondence, two triangles are congruent if two sides and the angle
included between them in one of the triangles are equal to the two sides and the angle
included between them of the other triangle (SAS).
• Under a given correspondence, two triangles are congruent if two angles and the side
included between them in one of the triangles are equal to the two angles and the side
included between them of the other triangle (ASA).
• Under a given correspondence, two right-angled triangles are congruent if the hypotenuse
and a leg (side) of one of the triangles are equal to the hypotenuse and one of the leg (side)
of the other triangle (RHS).
Exercise
1) Write the six elements (i.e., the 3 sides and the 3 angles) of ΔABC.
2) Write the:
(i)
Side opposite to the vertex Q of ΔPQR
(ii)
Angle opposite to the side LM of ΔLMN
(iii)
Vertex opposite to the side RT of ΔRST
3) Look at Fig.1 and classify each of the triangles according to its
(a) Sides (b) Angles
Fig.1
4) How many medians can a triangle have?
5) Does a median lie wholly in the interior of the triangle? (If you think that this is not
true, draw a figure to show such a case).
6) How many altitudes can a triangle have?
7) Draw rough sketches of altitudes from A to BC for the following triangles (Fig.2):
Fig.2
8) Will an altitude always lie in the interior of a triangle? If you think that this need not
be true, draw a rough sketch to show such a case.
9) Can you think of a triangle in which two altitudes of the triangle are two of its sides?
10) Can the altitude and median be same for a triangle?
11) In Δ PQR, D is the mid-point of QR .
12) PM is _________________.
PD is _________________.
Is QM = MR?
13) Draw rough sketches for the following: (a) In ΔABC, BE is a median. (b) In ΔPQR, PQ
and PR are altitudes of the triangle. (c) In ΔXYZ, YL is an altitude in the exterior of the
triangle.
14) Verify by drawing a diagram if the median and altitude of an isosceles triangle can be
same.
15) Exterior angles can be formed for a triangle in many ways. Three of them are shown
here (Fig.3)
Fig.3
There are three more ways of getting exterior angles. Try to produce those rough
sketches.
16) Are the exterior angles formed at each vertex of a triangle equal?
17) What can you say about the sum of an exterior angle of a triangle and its adjacent
interior angle?
18) What can you say about each of the interior opposite angles, when the exterior
angle is
(i) a right angle? (ii) an obtuse angle? (iii) an acute angle?
19) Can the exterior angle of a triangle be a straight angle?
20) An exterior angle of a triangle is of measure 70° and one of its interior opposite
angles is of measure 25°. Find the measure of the other interior opposite angle.
21) The two interior opposite angles of an exterior angle of a triangle are 60° and 80°.
Find the measure of the exterior angle.
22) Is something wrong in this diagram (Fig 4)? Comment.
Fig.4
23) Find the value of the unknown exterior angle x in the following diagrams:
24) Find the value of the unknown interior angle x in the following figures:
25) Take a piece of paper and cut out a triangle, say, ΔABC (Fig 5). Make the altitude AM
by folding ΔABC such that it passes through A. Fold now the three corners such that
all the three vertices A, B and C touch at M.
Fig.6
26) Draw any three triangles, say ΔABC, ΔPQR and ΔXYZ in your notebook. Use your
protractor and measure each of the angles of these triangles. Tabulate your results.
27) In the given figure (Fig 7) find m∠P.
Fig.7
28) Find the value of the unknown x in the following diagrams:
29) Find the values of the unknowns x and y in the following diagrams:
30) Two angles of a triangle are 30º and 80º. Find the third angle.
31) One of the angles of a triangle is 80º and the other two angles are equal. Find the
measure of each of the equal angles.
32) The three angles of a triangle are in the ratio 1:2:1. Find all the angles of the triangle.
Classify the triangle in two different ways.
33) Can you have a triangle with two right angles?
34) Can you have a triangle with two obtuse angles?
35) Can you have a triangle with two acute angles?
36) Can you have a triangle with all the three angles greater than 60°?
37) Can you have a triangle with all the three angles equal to 60°?
38) Can you have a triangle with all the three angles less than 60°?
39) Find angle x in each figure:
40) Find angles x and y in each figure.
41) Draw any three triangles, say ΔABC, ΔPQR and ΔXYZ in your notebook
Use your ruler to find the lengths of their sides and then tabulate your results as follows:
42) Is there a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm?
43) The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two
numbers can length of the third side fall?
44) Is it possible to have a triangle with the following sides?
(i)
2 cm, 3 cm, 5 cm (ii) 3 cm, 6 cm, 7 cm (iii) 6 cm, 3 cm, 2 cm
45) Take any point O in the interior of a triangle PQR. Is
(i)
OP + OQ > PQ? (ii) OQ + OR > QR? (iii) OR + OP > RP?
45) AM is a median of a triangle ABC. Is AB + BC + CA > 2 AM? (Consider the sides of
triangles ΔABM and ΔAMC.)
46) ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD? 5. ABCD is quadrilateral. Is
AB + BC + CD + DA < 2 (AC + BD)?
47) The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two
measures should the length of the third side fall?
48) Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is a
right-angled triangle.
49) Δ ABC is right-angled at C. If AC = 5 cm and BC = 12 cm find the length of AB.
50) Find the unknown length x in the following figures (Fig.8):
Fig.8
51) PQR is a triangle, right-angled at P. If PQ = 10cm and PR = 24 cm, find QR.
52) ABC is a triangle, right-angled at C. If AB = 25 cm and AC = 7 cm, find BC.
53) A 15 m long ladder reached a window 12 m high from the ground on placing it
against a wall at a distance a. Find the distance of the foot of the ladder from the
wall.
54) Which of the following can be the sides of a right triangle? (i) 2.5 cm,6.5 cm, 6 cm.
(ii) 2 cm, 2 cm, 5 cm. (iii) 1.5 cm, 2cm, 2.5 cm. In the case of right-angled triangles,
identify the right angles.
55) A tree is broken at a height of 5 m from the ground and its top touches the ground at
a distance of 12 m from the base of the tree. Find the original height of the tree.
56) Angles Q and R of a ΔPQR are 25º and 65º. Write which of the following is true: (i)
PQ2 + QR2 = RP2 (ii) PQ2 + RP2 = QR2 (iii) RP2 + QR2 = PQ2
57) Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.
58) The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.
59) Which is the longest side in the triangle PQR, right-angled at P?
60) Which is the longest side in the triangle ABC, right-angled at B?
61) Which is the longest side of a right triangle? 4. ‘The diagonal of a rectangle produce
by itself the same area as produced by its length and breadth’– This is Baudhayan
Theorem. Compare it with the Pythagoras property.
62) In Fig. 9, find x and y .
Fig.9
(i)
(ii)
If AD = DC? Why?
In given problem, can ∠ B be 85° instead of 60°? If yes find the values of x
and y in that case.
(iii)
What type of triangle is ∆ ADC?
63) The measure of three angles of a triangle are in the ratio 5 : 3 : 1. Find the measures
of these angles.
64) In Fig. 10, find the value of x .
Fig.10
65) In Fig. 11 (i) and (ii), find the values of a , b and c.
Fig.11
66) In triangle XYZ, the measure of angle X is 30° greater than the measure of angle Y
and angle Z is a right angle. Find the measure of ∠ Y.
67) In a triangle ABC, the measure of angle A is 40° less than the measure of angle B
and 50° less than that of angle C. Find the measure of ∠ A
68) I have three sides. One of my angle measures 15°. Another has a measure of 60°.
What kind of a polygon am I? If I am a triangle, then what kind of triangle am I?
69) Jiya walks 6 km due east and then 8 km due north. How far is she from her starting
place?
70) Jayanti takes shortest route to her home by walking diagonally across a rectangular
park. The park measures 60 metres × 80 metres. How much shorter is the route
across the park than the route around its edges?
71) In ∆ PQR of Fig. 12, PQ = PR. Find the measures of ∠ Q and ∠ R.
72) In Fig. 13, find the measures of ∠ x and ∠ y.
73) In Fig. 14, find the measures of ∠ PON and ∠ NPO
74) In Fig.15, QP || RT. Find the values of x and y.
75) Find the measure of ∠ A in Fig. 16.
74) In a right-angled triangle if an angle measures 35°, then find the measure of the
third angle.
75) Each of the two equal angles of an isosceles triangle is four times the third angle.
Find the angles of the triangle.
76) The angles of a triangle are in the ratio 2 : 3 : 5. Find the angles.
77) If the sides of a triangle are produced in an order, show that the sum of the exterior
angles so formed is 360°.
78) In ∆ ABC, if ∠ A = ∠ C, and exterior angle ABX = 140°, then find the angles of the
triangle.
79) 81) Find the values of x and y in Fig. 17.
Fig.17
80) Find the value of x in Fig. 18.
81) The angles of a triangle are arranged in descending order of their magnitudes. If the
difference between two consecutive angles is 10°, find the three angles.
82) In ∆ ABC, DE || BC (Fig. 19). Find the values of x , y and z .
83) In Fig. 20, find the values of x, y and z
84) If one angle of a triangle is 60° and the other two angles are in the ratio 1 : 2, find the
angles.
85) In ∆ PQR, if 3∠P = 4∠ Q = 6∠ R, calculate the angles of the triangle.
86) In ∆ DEF, ∠ D = 60°, ∠ E = 70° and the bisectors of ∠ E and ∠F meet at O. Find (i) ∠F
(ii) ∠ EOF.
87) In Fig. 21, ∆PQR is right-angled at P. U and T are the points on line QRF. If QP || ST
and US || RP, find ∠ S.
Fig.21
88) In each of the given pairs of triangles of Fig. 22, applying only ASA congruence
criterion, determine which triangles are congruent. Also, write the congruent
triangles in symbolic form.
Fig.22
89) In each of the given pairs of triangles of Fig. 23, using only RHS congruence criterion,
determine which pairs of triangles are congruent. In case of congruence, write the
result in symbolic form:
Fig.23
90) In Fig. 24, if RP = RQ, find the value of x.
Fig.24
91) In Fig. 25, if ST = SU, then find the values of x and y.
Fig.25
92) Check whether the following measures (in cm) can be the sides of a right-angled
triangle or not.
1.5, 3.6, 3.9
93) Height of a pole is 8 m. Find the length of rope tied with its top from a point on the
ground at a distance of 6 m from its bottom.
94) In Fig. 26, if y is five times x, find the value of z
Fig.26
95) The lengths of two sides of an isosceles triangle are 9 cm and 20 cm. What is the
perimeter of the triangle? Give reason.
96) Without drawing the triangles write all six pairs of equal measures in each of the
following pairs of congruent triangles.
(a) ∆ STU ≅ ∆DEF
(b) ∆ ABC ≅ ∆ LMN
(c) ∆ YZX ≅ ∆PQR
(d) ∆ XYZ ≅ ∆ MLN
97) In the following pairs of triangles of Fig. 27, the lengths of the sides are indicated along
the sides. By applying SSS congruence criterion, determine which triangles are congruent. If
congruent, write the results in symbolic form.
Fig.27
98) ABC is an isosceles triangle with AB = AC and D is the mid-point of base BC (Fig. 28).
(a) State three pairs of equal parts in the triangles ABD and ACD.
(b) Is ∆ ABD ≅ ∆ ACD. If so why?
99) In Fig. 29, it is given that LM = ON and NL = MO
(i)
(ii)
State the three pairs of equal parts in the triangles NOM and MLN.
Is ∆ NOM ≅ ∆ MLN. Give reason?
Fig.29
100) Triangles DEF and LMN are both isosceles with DE = DF and LM = LN, respectively. If DE
= LM and EF = MN, then, are the two triangles congruent? Which condition do you use?
If ∠ E = 40°, what is the measure of ∠ N?
101) If ∆ PQR and ∆ SQR are both isosceles triangle on a common base QR such that P and S
lie on the same side of QR. Are triangles PSQ and PSR congruent? Which condition do you
use?
102) In Fig. 30, which pairs of triangles are congruent by SAS congruence criterion
(condition)? If congruent, write the congruence of the two triangles in symbolic form
Fig.30
103) In Fig. 31, PQ = PS and ∠ 1 = ∠ 2.
(i)
(ii)
Is ∆PQR ≅ ∆ PSR? Give reasons.
Is QR = SR? Give reasons.
Fig.31
104) In Fig. 32, DE = IH, EG = FI and∠ E = ∠ I. Is ∆ DEF ≅ ∆ HIG? If yes, by which congruence
criterion?
Fig.33
105) In Fig. 33, ∠ 1 = ∠ 2 and ∠ 3 = ∠ 4. (i) Is ∆ ADC ≅ ∆ ABC? Why ? (ii) Show that AD = AB
and CD = CB.
Fig.33
106) Observe Fig. 34 and state the three pairs of equal parts in triangles ABC and DBC. (i) Is
∆ ABC ≅ ∆ DCB? Why? (ii) Is AB = DC? Why? (iii) Is AC = DB? Why?
Fig.35
107) In Fig. 35, QS ⊥ PR, RT ⊥ PQ and QS = RT. (i) Is ∆ QSR ≅ ∆ R TQ? Give reasons. (ii) Is ∠
PQR = ∠ PRQ? Give reasons.
Fig.35
108) Points A and B are on the opposite edges of a pond as shown in Fig. 36. To find the
distance between the two points, the surveyor makes a right-angled triangle as shown. Find
the distance AB.
Fig.36
109) Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between
the tops is 13 m, find the distance between their feet.
110) The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above
the ground, (a) Find the length of the ladder. (b) If the ladder is shifted in such a way that its
foot is 8 m away from the wall, to what height does its top reach?
111) In Fig. 37, state the three pairs of equal parts in ∆ ABC and ∆ EOD. Is ∆ ABC ≅ ∆ EOD?
Why?
Fig.37
112) Draw an equilateral triangle of side 6 cm, an isosceles triangle of base 3 cm and equal
sides 6 cm each and a scalene triangle of sides 3 cm, 6 cm and 7cm. Now draw a median and
an altitude in each triangle from the top vertex, measure and tabulate the lengths of all the
medians and altitude’s of respective triangles. What can you conclude from this activity
(This activity can also be done by paper folding)?
113) Draw two triangles which have a pair of corresponding sides equal but are not
congruent.
114) Draw two triangles which have two pairs of corresponding sides equal but are not
congruent.
115) Draw two triangles, which have one pair of corresponding angles equal and one pair of
corresponding sides equal but are not congruent.
116) Draw two triangles which have three pairs of corresponding angles equal but are not
congruent. Solve the given cross number/word and then fill up the given boxes in activities 6
and 7. Clues are given below for across as well as downward fillings. For across and
downward clue numbers are written at the corner of boxes. Answers of clues have to fill up
in their respective boxes.
Multiple choice questions
1) In Fig. 38, side QR of a ∆ PQR has been produced to the point S. If ∠ PRS = 115° and ∠ P =
45°, then ∠Q is equal to,
(a) 70° (b) 105° (c) 51° (d) 80°
Fig.38
2) In an equilateral triangle ABC (Fig. 39), AD is an altitude. Then 4AD2 is equal to
(a) 2BD2 (b) BC 2 (c) 3AB2 (d) 2DC2
Fig.39
3) Which of the following cannot be the sides of a triangle?
(a) 3 cm, 4 cm, 5 cm (b) 2 cm, 4 cm, 6 cm
(c) 2.5 cm, 3.5 cm, 4.5 cm (d) 2.3cm, 6.4 cm, 5.2 cm
4) Which one of the following is not a criterion for congruence of two triangles?
(a) ASA (b) SSA (c) SAS (d) SSS
5) In Fig. 40, PS is the bisector of ∠ P and PQ = PR. Then ∆PRS and ∆ PQS are congruent by
the criterion
(a) AAA (b) SAS (c) ASA (d) both (b) and (c)
Fig.40
6) The sides of a triangle have lengths (in cm) 10, 6.5 and a, where a is a whole number. The
minimum value that a can take is
(a) 6 (b) 5 (c) 3 (d) 4
7) Triangle DEF of Fig. 41 is a right triangle with ∠ E = 90°.
What type of angles are ∠ D and ∠ F?
(a) They are equal angles
(b) They form a pair of adjacent angles
(c) They are complementary angles
(d) They are supplementary angles
Fig.41
8) In Fig. 42, PQ = PS. The value of x is
(a) 35° (b) 45° (c) 55° (d) 70°
Fig.42
9) In a right-angled triangle, the angles other than the right angle are
(a) (a) obtuse (b) right (c) acute (d) straight
10) In an isosceles triangle, one angle is 70°. The other two angles are of
(i)
55° and 55° (ii) 70° and 40° (iii) any measure
In the given option(s) which of the above statement(s) are true?
(a) (i) only (b) (ii) only (c) (iii) only (d) (i) and (ii)
11) In a triangle, one angle is of 90°. Then
(i)
(ii)
(iii)
The other two angles are of 45° each
In remaining two angles, one angle is 90° and other is 45°
Remaining two angles are complementary
In the given option(s) which is true?
(a) (i) only (b) (ii) only (c) (iii) only (d) (i) and (ii)
12) Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is
(a) Obtuse angled triangle (b) Acute-angled triangle
(c) Right-angled triangle (d) An Isosceles right triangle
13) In Fig. 43, PB = PD. The value of x is
(a) 85° (b) 90°(c) 25° (d) 35°
Fig.43
9. In ∆PQR,
(a) PQ – QR > PR
(b) PQ + QR < PR
(c) PQ – QR< PR
(d) PQ + PR< QR
10. In ∆ ABC,
(a) AB + BC > AC (b) AB + BC < AC
(c) AB + AC < BC (d) AC + BC < AB
11) The top of a broken tree touches the ground at a distance of 12 m from its base. If the
tree is broken at a height of 5 m from the ground then the actual height of the tree is
(a) 25 m (b) 13 m (c) 18 m (d) 17 m
12 The triangle ABC formed by AB = 5 cm, BC = 8 cm, AC = 4 cm is
(b)
(c)
(d)
(e)
an isosceles triangle only
a scalene triangle only
an isosceles right triangle
scalene as well as a right triangle
13) Two trees 7 m and 4 m high stand upright on a ground. If their bases (roots) are 4 m
apart, then the distance between their tops is
(f) 3 m (b) 5 m (c) 4 m (d) 11 m
14) If in an isosceles triangle, each of the base angles is 40°, then the triangle is
(a) Right-angled triangle (b) Acute angled triangle (c) Obtuse angled triangle (d)
Isosceles right-angled triangle
15) If two angles of a triangle are 60° each, then the triangle is
(g) Isosceles but not equilateral (b) Scalene (c) Equilateral (d) Right-angled
16) The perimeter of the rectangle whose length is 60 cm and a diagonal is 61 cm is
(a) (a) 120 cm (b) 122 cm (c) 71 cm (d) 142 cm
17) In ∆ PQR, if PQ = QR and ∠Q = 100°, then ∠ R is equal to
(a) 40° (b) 80° (c) 120° (d) 50°
18) Which of the following statements is not correct?
(h)
(i)
(j)
(k)
The sum of any two sides of a triangle is greater than the third side
A triangle can have all its angles acute
A right-angled triangle cannot be equilateral
Difference of any two sides of a triangle is greater than the third side
19) In Fig. 44, BC = CA and ∠ A = 40. Then, ∠ ACD is equal to
(a) 40° (b) 80° (c) 120° (d) 60
20)
Figh.44