Class 7 Maths Notes for Triangles & Its Properties
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Page 1
NOTES
CLASS 7
MATHS
Study Material
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CLASS 7
MATHS
Study Material
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Page 2
THE TRIANGLE AND ITS PROPERTIES
I. Important Concepts/ Result
1. The six elements of a triangle are its three angles and the three sides.
2. The line segment joining a vertex of a triangle to the mid point of its opposite side is
called a median of the triangle. A triangle has 3 medians.
3. 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.
4. An exterior angle of a triangle is formed, when a side of a triangle is produced. At
each vertex, you have two ways of forming an exterior angle.
5. A property of exterior angles: The measure of any exterior angle of a triangle is equal to
the sum of the measures of its interior opposite angles.
6. The angle sum property of a triangle: The total measure of the three angles of a triangle is
180°.
7. A triangle is said to be equilateral, if each one of its sides has the same length. In an
equilateral triangle, each angle has measure 60°.
8. A triangle is said to be isosceles, if atleast any two of its sides are of same length.
The non-equal side of an isosceles triangle is called its base; the base angles of an
isosceles triangle have equal measure.
9. Property of the lengths of sides of a triangle:
10. The sum of the lengths of any two sides of a triangle is greater than the length of the third
side. The difference between the lengths of any two sides is smaller than the length of the
third side.
II Some illustrations/Examples
(i) MCQs
1. How many medians can a triangle have?
(a) 2 (b) 1 (c) 3 (d) 0
Answer: (c) 3
2. Which is the longest side of a right triangle?
(a) Perpendicular (b) Base (c) Hypotenuse (d) none of these.
Answer: (c) hypotenuse
3. Find the value of x in given figure.
(a) 180° (b) 55° (c) 90° (d) 60°
Solution: Sum of interior opposite angles = Exterior
angle or 60° + x = 120° or x = 60°
4. Which of the following are the angles of an isosceles right-angled triangle.
a) 60°, 60°, 60° b) 45°, 45°, 90°
c) 50°, 50°, 80° d) 30°, 30°, 120°
ii) Case study
54
I. Important Concepts/ Result
1. The six elements of a triangle are its three angles and the three sides.
2. The line segment joining a vertex of a triangle to the mid point of its opposite side is
called a median of the triangle. A triangle has 3 medians.
3. 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.
4. An exterior angle of a triangle is formed, when a side of a triangle is produced. At
each vertex, you have two ways of forming an exterior angle.
5. A property of exterior angles: The measure of any exterior angle of a triangle is equal to
the sum of the measures of its interior opposite angles.
6. The angle sum property of a triangle: The total measure of the three angles of a triangle is
180°.
7. A triangle is said to be equilateral, if each one of its sides has the same length. In an
equilateral triangle, each angle has measure 60°.
8. A triangle is said to be isosceles, if atleast any two of its sides are of same length.
The non-equal side of an isosceles triangle is called its base; the base angles of an
isosceles triangle have equal measure.
9. Property of the lengths of sides of a triangle:
10. The sum of the lengths of any two sides of a triangle is greater than the length of the third
side. The difference between the lengths of any two sides is smaller than the length of the
third side.
II Some illustrations/Examples
(i) MCQs
1. How many medians can a triangle have?
(a) 2 (b) 1 (c) 3 (d) 0
Answer: (c) 3
2. Which is the longest side of a right triangle?
(a) Perpendicular (b) Base (c) Hypotenuse (d) none of these.
Answer: (c) hypotenuse
3. Find the value of x in given figure.
(a) 180° (b) 55° (c) 90° (d) 60°
Solution: Sum of interior opposite angles = Exterior
angle or 60° + x = 120° or x = 60°
4. Which of the following are the angles of an isosceles right-angled triangle.
a) 60°, 60°, 60° b) 45°, 45°, 90°
c) 50°, 50°, 80° d) 30°, 30°, 120°
ii) Case study
54
Page 3
Triangles: Three friends Ram, Hari & Neelesh are standing at position A, B & C
respectively as shown in figure (i). Ram wants to join the line through Hari &
Neelesh. Hari suggest Ram to move along AD & take the position at D as in figure
(ii). Neelesh suggest Ram to move along AE & take the position at E as in figure (iii).
i. In figure (ii) AD is ______________ (median/altitude).
ii. In figure (iii) AE is ______________ (median/altitude).
iii. If Ram wants to join line BC soon, then whose suggestion is fair & why?
iv. Following statements are true or false-
S.No. Statement True/False
1 ΔABD is isosceles
2 BE=EC
3 ΔABC is right angled
(ii) Short answer type questions
1. Find the value of x in given figure.
Solution: By angle sum property of a triangle,
∠U + 30° + 122° = 180°
∠U= 180° – 152° = 28°
2. 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?
3. Solution: We know that the sum of two sides of a
triangle is always greater than
the third.
Therefore, third side has to be less than the sum of the two sides. The third side is thus,
less than 8 + 6 = 14 cm. The side cannot be less than the difference of the two sides.
Thus, the third side has to be more than 8 – 6 = 2 cm.
The length of the third side could be any length greater than 2 and less than 14 cm.
4. State if these numbers 5, 7, 9 could possibly be the lengths of the sides of a triangle:
Solution: Yes, these numbers can be the lengths of the sides of a triangle because the
sum of any two sides of a triangle is always greater than the third side.
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5
5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is
right-angled triangle.
55
respectively as shown in figure (i). Ram wants to join the line through Hari &
Neelesh. Hari suggest Ram to move along AD & take the position at D as in figure
(ii). Neelesh suggest Ram to move along AE & take the position at E as in figure (iii).
i. In figure (ii) AD is ______________ (median/altitude).
ii. In figure (iii) AE is ______________ (median/altitude).
iii. If Ram wants to join line BC soon, then whose suggestion is fair & why?
iv. Following statements are true or false-
S.No. Statement True/False
1 ΔABD is isosceles
2 BE=EC
3 ΔABC is right angled
(ii) Short answer type questions
1. Find the value of x in given figure.
Solution: By angle sum property of a triangle,
∠U + 30° + 122° = 180°
∠U= 180° – 152° = 28°
2. 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?
3. Solution: We know that the sum of two sides of a
triangle is always greater than
the third.
Therefore, third side has to be less than the sum of the two sides. The third side is thus,
less than 8 + 6 = 14 cm. The side cannot be less than the difference of the two sides.
Thus, the third side has to be more than 8 – 6 = 2 cm.
The length of the third side could be any length greater than 2 and less than 14 cm.
4. State if these numbers 5, 7, 9 could possibly be the lengths of the sides of a triangle:
Solution: Yes, these numbers can be the lengths of the sides of a triangle because the
sum of any two sides of a triangle is always greater than the third side.
Here, 5 + 7 > 9, 5 + 9 > 7, 9 + 7 > 5
5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is
right-angled triangle.
55
Page 4
Solution: In the given triangle, the largest side is 6 cm.
We know that in a right angled triangle, the sum of the squares of the smaller sides
should be equal to the square of the largest side.
Therefore, 32 + 42 = 9 + 16 = 25
But, 62 = 36
32 + 42 = 25 which is not equal to 62
Hence, the given triangle is not a right angled triangle.
6. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles.
Solution: Given angles of the triangle are in the ratio 1: 2: 3
So take first angle as x, second angle as 2x and third angle as 3x
We know that sum of all the angles of a triangle = 180o
x + 2x + 3x = 180o
6x = 180o
x = 30o
2x = 30o × 2 = 60o 3x = 30o × 3 = 90o
Therefore the first angle is 30o, second angle is 60o and third angle is 90o.
7. In the given figure, the sides BC, CA and BA of a
△ABC are produced to D, E and F respectively. If
∠ACD = 105° and ∠EAF = 45°; find all the
angles of the △ABC.
Solution: In a △ABC, ∠BAC and ∠EAF are
vertically opposite angles.
Hence, ∠BAC = ∠EAF = 45°
Considering the exterior angle property, we have ∠BAC + ∠ABC = ∠ACD = 105°
On rearranging we get ∠ABC = 105°– 45°= 60°
We know that the sum of angles in a triangle is 180°
∠ABC + ∠ACB +∠BAC = 180°
∠ACB = 75°
Therefore, the angles are 45°, 60° and 75°
8. In △ABC, ∠A = 50o and BC is produced to a point D. The bisectors of ∠ABC and ∠ACD
meet at E. Find ∠E.
Solution: In the given triangle,
56
We know that in a right angled triangle, the sum of the squares of the smaller sides
should be equal to the square of the largest side.
Therefore, 32 + 42 = 9 + 16 = 25
But, 62 = 36
32 + 42 = 25 which is not equal to 62
Hence, the given triangle is not a right angled triangle.
6. If the angles of a triangle are in the ratio 1: 2: 3, determine three angles.
Solution: Given angles of the triangle are in the ratio 1: 2: 3
So take first angle as x, second angle as 2x and third angle as 3x
We know that sum of all the angles of a triangle = 180o
x + 2x + 3x = 180o
6x = 180o
x = 30o
2x = 30o × 2 = 60o 3x = 30o × 3 = 90o
Therefore the first angle is 30o, second angle is 60o and third angle is 90o.
7. In the given figure, the sides BC, CA and BA of a
△ABC are produced to D, E and F respectively. If
∠ACD = 105° and ∠EAF = 45°; find all the
angles of the △ABC.
Solution: In a △ABC, ∠BAC and ∠EAF are
vertically opposite angles.
Hence, ∠BAC = ∠EAF = 45°
Considering the exterior angle property, we have ∠BAC + ∠ABC = ∠ACD = 105°
On rearranging we get ∠ABC = 105°– 45°= 60°
We know that the sum of angles in a triangle is 180°
∠ABC + ∠ACB +∠BAC = 180°
∠ACB = 75°
Therefore, the angles are 45°, 60° and 75°
8. In △ABC, ∠A = 50o and BC is produced to a point D. The bisectors of ∠ABC and ∠ACD
meet at E. Find ∠E.
Solution: In the given triangle,
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∠ACD = ∠A + ∠B. (Exterior angle is equal to the sum of two opposite interior angles.)
We know that the sum of all three angles of a triangle is 180o.
Therefore, for the given triangle, we know that the sum of the angles = 180o
∠ABC + ∠BCA + ∠CAB = 180o
∠A + ∠B + ∠BCA = 180o
∠BCA = 180o – (∠A + ∠B)
But we know that EC bisects ∠ACD
Therefore ∠ECA = ∠ACD/2
∠ECA = (∠A + ∠B)/2 [∠ACD = (∠A + ∠B)]
But EB bisects ∠ABC
∠EBC = ∠ABC/2 = ∠B/2
∠EBC = ∠ECA + ∠BCA
∠EBC = (∠A + ∠B)/2 + 180o – (∠A + ∠B)
Similarly, in △EBC, we get,
∠B/2 + (∠A + ∠B)/2 + 180o – (∠A + ∠B) + ∠BEC = 180o
∠BEC = ∠A + ∠B – (∠A + ∠B)/2 – ∠B/2
∠BEC = ∠A/2
∠BEC = 50o/2 = 25o
9. In Fig.39, AD and CF are respectively perpendiculars
to sides BC and AB of △ABC. If ∠FCD = 50o, find
∠BAD
Solution: We know that the sum of all angles of a triangle is 180o
Therefore, for the given △FCB, we have
∠FCB + ∠CBF + ∠BFC = 180o
50o + ∠CBF + 90o = 180o
∠CBF = 180o – 50o – 90o = 40o
Using the above steps for △ABD, we can say that:
∠ABD + ∠BDA + ∠BAD = 180o
∠BAD = 180o – 90o – 40o = 50o
10. A man goes 15 m due west and then 8 m due north. How far is he from the starting point?
Solution: Given a man goes 15 m due
west and then 8 m due north. Let O be the
starting point and P be the final point.
Then OP becomes the hypotenuse in the
triangle.
So by using the Pythagoras theorem, we
can find the distance OP.
57
We know that the sum of all three angles of a triangle is 180o.
Therefore, for the given triangle, we know that the sum of the angles = 180o
∠ABC + ∠BCA + ∠CAB = 180o
∠A + ∠B + ∠BCA = 180o
∠BCA = 180o – (∠A + ∠B)
But we know that EC bisects ∠ACD
Therefore ∠ECA = ∠ACD/2
∠ECA = (∠A + ∠B)/2 [∠ACD = (∠A + ∠B)]
But EB bisects ∠ABC
∠EBC = ∠ABC/2 = ∠B/2
∠EBC = ∠ECA + ∠BCA
∠EBC = (∠A + ∠B)/2 + 180o – (∠A + ∠B)
Similarly, in △EBC, we get,
∠B/2 + (∠A + ∠B)/2 + 180o – (∠A + ∠B) + ∠BEC = 180o
∠BEC = ∠A + ∠B – (∠A + ∠B)/2 – ∠B/2
∠BEC = ∠A/2
∠BEC = 50o/2 = 25o
9. In Fig.39, AD and CF are respectively perpendiculars
to sides BC and AB of △ABC. If ∠FCD = 50o, find
∠BAD
Solution: We know that the sum of all angles of a triangle is 180o
Therefore, for the given △FCB, we have
∠FCB + ∠CBF + ∠BFC = 180o
50o + ∠CBF + 90o = 180o
∠CBF = 180o – 50o – 90o = 40o
Using the above steps for △ABD, we can say that:
∠ABD + ∠BDA + ∠BAD = 180o
∠BAD = 180o – 90o – 40o = 50o
10. A man goes 15 m due west and then 8 m due north. How far is he from the starting point?
Solution: Given a man goes 15 m due
west and then 8 m due north. Let O be the
starting point and P be the final point.
Then OP becomes the hypotenuse in the
triangle.
So by using the Pythagoras theorem, we
can find the distance OP.
57
Page 6
OP2 = 152 + 82
OP2 = 225 + 64
OP2 = 289
OP = 17
Hence, the required distance is 17 m.
III .Questions for Practice:
MCQs
1. How many altitudes can a triangle have?
(a) 1 (b) 2 (c) 3 (d) 4
2. A/an …………….. connects a vertex of a triangle to the midpoint of the opposite side.
(a) altitude (b) median (c) vertex (d) none of these
3. Find the value of x in the adjoining figure.
(a) 50° (b) 70° (c) 120° (d) 180°
4. In the Pythagoras property, the triangle must be
(a) acute angled (b) right angled
(c) obtuse angled (d) none of these
5. Find the value of x in this figure.
(a) 40° (b) 60° (c) 35° (d) 180°
6. ∆PQR is right angled at P. If PQ = 3 cm and PR = 4 cm, find QR.
(a) 7 cm (b) 5 cm (c) 2 cm (d) none of these.
7. Which is the longest side in ∆PQR right angled at P?
(a) PQ (b) QR (c) PR (d) none of these.
8. The hypotenuse of a right triangle is 17 cm long. If one of the remaining two sides is 8
cm in length, then the length of the other side is:
(a) 15 cm (b) 12 cm (c) 13 cm (d) none of these.
9. How many acute angles can a right triangle have?
(a) 1 (b) 2 (c) 3 (d) 0
10. The acute angles of right triangle are in the ratio 2 : 1. Find the measure of each of these
angles.
(a) 55° and 35° (b) 60° and 30° (c) 50° and 40° (d) 45° and 45°
11. One of the angles of a triangle is 100° and the other two angles are equal. Find the
measure of each of these equal angles.
(a) 45° (b) 40° (c) 41° (d) 42°
ii) Short answer type questions
1. One of the exterior angles of a triangle is 120°, and the
interior opposite angles are equal to each other. What is the
measure of these two angles?
2. Find the value of x in given figure.
58
OP2 = 225 + 64
OP2 = 289
OP = 17
Hence, the required distance is 17 m.
III .Questions for Practice:
MCQs
1. How many altitudes can a triangle have?
(a) 1 (b) 2 (c) 3 (d) 4
2. A/an …………….. connects a vertex of a triangle to the midpoint of the opposite side.
(a) altitude (b) median (c) vertex (d) none of these
3. Find the value of x in the adjoining figure.
(a) 50° (b) 70° (c) 120° (d) 180°
4. In the Pythagoras property, the triangle must be
(a) acute angled (b) right angled
(c) obtuse angled (d) none of these
5. Find the value of x in this figure.
(a) 40° (b) 60° (c) 35° (d) 180°
6. ∆PQR is right angled at P. If PQ = 3 cm and PR = 4 cm, find QR.
(a) 7 cm (b) 5 cm (c) 2 cm (d) none of these.
7. Which is the longest side in ∆PQR right angled at P?
(a) PQ (b) QR (c) PR (d) none of these.
8. The hypotenuse of a right triangle is 17 cm long. If one of the remaining two sides is 8
cm in length, then the length of the other side is:
(a) 15 cm (b) 12 cm (c) 13 cm (d) none of these.
9. How many acute angles can a right triangle have?
(a) 1 (b) 2 (c) 3 (d) 0
10. The acute angles of right triangle are in the ratio 2 : 1. Find the measure of each of these
angles.
(a) 55° and 35° (b) 60° and 30° (c) 50° and 40° (d) 45° and 45°
11. One of the angles of a triangle is 100° and the other two angles are equal. Find the
measure of each of these equal angles.
(a) 45° (b) 40° (c) 41° (d) 42°
ii) Short answer type questions
1. One of the exterior angles of a triangle is 120°, and the
interior opposite angles are equal to each other. What is the
measure of these two angles?
2. Find the value of x in given figure.
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Page 7
3. Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is a right-
angled triangle.
4. If the measures of the angles of a triangle are (2p)°, (3p − 15)° and (4p − 12)°. Find
the value of p.
5. Find the value of x an y in the figure.
6. In the isosceles triangle, the vertical angle is 30°
more than each of its base angles. Find all the
angles of the triangle.
7. Find the value of x in the figure
8. If one angle of a triangle is 60° and the other two
angles are in the ratio 1 : 2, then find the angles.
9. Mohini walks 1200m due East and then 500m due North.
How far is she from her starting point?
10. Find the value of 'x' and 'y' in the given figure.
11. In a △ABC, AD is the altitude from A such that AD = 12
cm. BD = 9 cm and DC = 16 cm. Examine if △ABC is
right angled at A.
12. A student when asked to measure two exterior angles of
△ABC observed that the exterior angles at A and B are of
103o and 74o respectively. Is this triangle possible? Why or why not?
iii) Long answer type questions
1. Find the value of 'x' and 'y' in the below given figure.
2. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their
feet is 12 m. Find the distance between their tops.
3. In the figure ABC is a right triangle right angled at A. D
lies on BA produced and DE perpendicular to BC
intersecting AC at F. If ∠AFE = 130o, find
(i) ∠BDE (ii) ∠BCA (iii) ∠ABC
4. ABC is an equilateral triangle with side a. AD is an
altitude. Find the value of AD2.
59
angled triangle.
4. If the measures of the angles of a triangle are (2p)°, (3p − 15)° and (4p − 12)°. Find
the value of p.
5. Find the value of x an y in the figure.
6. In the isosceles triangle, the vertical angle is 30°
more than each of its base angles. Find all the
angles of the triangle.
7. Find the value of x in the figure
8. If one angle of a triangle is 60° and the other two
angles are in the ratio 1 : 2, then find the angles.
9. Mohini walks 1200m due East and then 500m due North.
How far is she from her starting point?
10. Find the value of 'x' and 'y' in the given figure.
11. In a △ABC, AD is the altitude from A such that AD = 12
cm. BD = 9 cm and DC = 16 cm. Examine if △ABC is
right angled at A.
12. A student when asked to measure two exterior angles of
△ABC observed that the exterior angles at A and B are of
103o and 74o respectively. Is this triangle possible? Why or why not?
iii) Long answer type questions
1. Find the value of 'x' and 'y' in the below given figure.
2. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their
feet is 12 m. Find the distance between their tops.
3. In the figure ABC is a right triangle right angled at A. D
lies on BA produced and DE perpendicular to BC
intersecting AC at F. If ∠AFE = 130o, find
(i) ∠BDE (ii) ∠BCA (iii) ∠ABC
4. ABC is an equilateral triangle with side a. AD is an
altitude. Find the value of AD2.
59
Page 8
5. Find the value of 'x' and 'y' in the below given figure.
6. Find the value of 'p', 'q', and 'r' in the below given figure.
7. In the below given figure, ∠ADB = 90°, AB = 17 cm, AC = 25 cm and AD = 15 cm. Find
the value of BD.
8. A ladder 50 dm long when set against the wall of a house just reaches a window at a
height of 48 dm. How far is the lower end of the ladder from the base of the wall?
iv) Case study:
Case study 1: BRIDGE
During an educational excursion, the Mathematics
teacher observed triangles in a bridge over a river
and showed them to the students. To emphasise
the relatedness of math in daily life he just drew
its simplified figure (side view) as shown here
with following description:
AD=AE=BE=BF=CF=CG=10 m.
60
6. Find the value of 'p', 'q', and 'r' in the below given figure.
7. In the below given figure, ∠ADB = 90°, AB = 17 cm, AC = 25 cm and AD = 15 cm. Find
the value of BD.
8. A ladder 50 dm long when set against the wall of a house just reaches a window at a
height of 48 dm. How far is the lower end of the ladder from the base of the wall?
iv) Case study:
Case study 1: BRIDGE
During an educational excursion, the Mathematics
teacher observed triangles in a bridge over a river
and showed them to the students. To emphasise
the relatedness of math in daily life he just drew
its simplified figure (side view) as shown here
with following description:
AD=AE=BE=BF=CF=CG=10 m.
60
Page 9
and∠A=∠B=∠C=900.
(i) Count the no. of isosceles triangles in figure. A student Raman said that ΔADE,
ΔBEF & ΔCFG are congruent.
Is his statement true?
(ii) Value of ∠D is
(iii) Find the length of DE2.
(iv) State following statements are
true or false-
S.No. Statement True/False
1 Arun said : AM is altitude of ΔADE
2 Deksha said : DF = AC
3 Rakesh said : AC = 20 cm.
Case study 2: GOLF HITS
A golfer David hits three balls from same point P
one by one in different directions as shown in
figure:
First hit towards south with range PA= 90 m.
Second hit towards east with range PB=120 m.
Third hit towards 600 north of east with range
PC=120 m.
(i) Type of ΔPBC is ___________.
(isosceles/equilateral)
(ii) Name the right angle in figure (if any).
(iii) How far is ball 1 from ball 2 ?
(iv) How far is ball 2 from ball 3 ?
(v) Use the appropriate sign in statement
AC____ AB+BC.
(a) > (b) < (c) = (d) +
ANSWERS:
MCQs
1. (c) 3 2. (b) median 3. (c) 120° 4. (b) right angled 5.(b) 60°
6.(b) 5 cm 7.(a) 15 cm 8. (b) 2 9. (b) 60° and 30° 10. (b) 40°
ii) Short answer type questions
1. 30° 2. 80° 3. Yes the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is
a right-angled triangle.
4.p=23 5. x = 120° , y = 70° 6. 50°, 50° and 80° 7. 30°, 60° and 90°
8. 40° and 80° 9. 1300 m 10. x = 80° , y = 75°
61
(i) Count the no. of isosceles triangles in figure. A student Raman said that ΔADE,
ΔBEF & ΔCFG are congruent.
Is his statement true?
(ii) Value of ∠D is
(iii) Find the length of DE2.
(iv) State following statements are
true or false-
S.No. Statement True/False
1 Arun said : AM is altitude of ΔADE
2 Deksha said : DF = AC
3 Rakesh said : AC = 20 cm.
Case study 2: GOLF HITS
A golfer David hits three balls from same point P
one by one in different directions as shown in
figure:
First hit towards south with range PA= 90 m.
Second hit towards east with range PB=120 m.
Third hit towards 600 north of east with range
PC=120 m.
(i) Type of ΔPBC is ___________.
(isosceles/equilateral)
(ii) Name the right angle in figure (if any).
(iii) How far is ball 1 from ball 2 ?
(iv) How far is ball 2 from ball 3 ?
(v) Use the appropriate sign in statement
AC____ AB+BC.
(a) > (b) < (c) = (d) +
ANSWERS:
MCQs
1. (c) 3 2. (b) median 3. (c) 120° 4. (b) right angled 5.(b) 60°
6.(b) 5 cm 7.(a) 15 cm 8. (b) 2 9. (b) 60° and 30° 10. (b) 40°
ii) Short answer type questions
1. 30° 2. 80° 3. Yes the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is
a right-angled triangle.
4.p=23 5. x = 120° , y = 70° 6. 50°, 50° and 80° 7. 30°, 60° and 90°
8. 40° and 80° 9. 1300 m 10. x = 80° , y = 75°
61
Page 10
11.Yes △ABC is right angled at A.
12.No this triangle is not possible because then the two interior angles will be 77° and 106° and the
sum of interior angles in a triangle is equal to180°
iii) Long answer type questions
1. x = 60°, y = 90°
2. the distance between their tops is 13 m.
3. (i) ∠BDE = 40° (ii) ∠BCA = 40° (iii) ∠ABC = 50°
2 2
4. AD = 3 a /4
5. x = 110°, y = 80°
6. q = 40°, r = 60° , p = 80°
7. BD = 8 cm
8. Distance of the ladder from the base of the wall is 14 dm
iv) Case study:
Case study 1
(i) no. of isosceles triangles in figure is 5. Yes ΔADE,ΔBEF &ΔCFG are congruent
(ii) Value of ∠D is 450
(iii) length of DE2 = 200 m
(iv)
S.No. Statement True/False
1 Arun said : AM is altitude of True
ΔADE
2 Deksha said : DF = AC True
3 Rakesh said : AC = 20 cm. False
Case study 2
GOLF HITS
(i) ΔPBC is isosceles triangle ( ii) Angle APB is the right angle
(ii)Distance of ball 1 from ball 2 =150 m
(iii)How far Distance of ball 2 from ball 3 = 120 m iv) AC< AB+BC.
Chapter Test
Time Allowed 1 hour. Maximum Marks: 20
General Instructions:
1. This Question Paper has 4 Sections A - D.
2. Section A has 4 MCQs carrying 1 mark each
3. Section B has 4 questions carrying 02 marks each.
5. Section C has 1 question carrying 04 marks.
6. Section D has 1 case based integrated units of assessment (04 marks each) with sub-parts of
the values of 1, 1 and 2 marks each respectively.
7. All Questions are compulsory.
8. Draw neat figures wherever required.
62
12.No this triangle is not possible because then the two interior angles will be 77° and 106° and the
sum of interior angles in a triangle is equal to180°
iii) Long answer type questions
1. x = 60°, y = 90°
2. the distance between their tops is 13 m.
3. (i) ∠BDE = 40° (ii) ∠BCA = 40° (iii) ∠ABC = 50°
2 2
4. AD = 3 a /4
5. x = 110°, y = 80°
6. q = 40°, r = 60° , p = 80°
7. BD = 8 cm
8. Distance of the ladder from the base of the wall is 14 dm
iv) Case study:
Case study 1
(i) no. of isosceles triangles in figure is 5. Yes ΔADE,ΔBEF &ΔCFG are congruent
(ii) Value of ∠D is 450
(iii) length of DE2 = 200 m
(iv)
S.No. Statement True/False
1 Arun said : AM is altitude of True
ΔADE
2 Deksha said : DF = AC True
3 Rakesh said : AC = 20 cm. False
Case study 2
GOLF HITS
(i) ΔPBC is isosceles triangle ( ii) Angle APB is the right angle
(ii)Distance of ball 1 from ball 2 =150 m
(iii)How far Distance of ball 2 from ball 3 = 120 m iv) AC< AB+BC.
Chapter Test
Time Allowed 1 hour. Maximum Marks: 20
General Instructions:
1. This Question Paper has 4 Sections A - D.
2. Section A has 4 MCQs carrying 1 mark each
3. Section B has 4 questions carrying 02 marks each.
5. Section C has 1 question carrying 04 marks.
6. Section D has 1 case based integrated units of assessment (04 marks each) with sub-parts of
the values of 1, 1 and 2 marks each respectively.
7. All Questions are compulsory.
8. Draw neat figures wherever required.
62
Page 11
SECTION A
Q Section A consists of 4 questions of 1 mark each. M
1 How many altitudes can a triangle have? 1
(a) 1 (b) 2 (c) 3 (d) 4
2 In right △ABC, right angled at B, BA = 8 cm and BC= 15 cm. Find the length of 1
the hypotenuse AC
(a) 7 cm (b) 17 cm (c) 23 cm (d) 20 cm
3 In each of the following, there are three positive numbers. Which of these pairs 1
could possibly be the lengths of the sides of a triangle:
(a) 5, 7, 9 (b) 2, 10, 15 (c) 2, 5, 7 (d) 5, 8, 20
4 In each of the following, the measures of three angles are given. State in which 1
case the angles can possibly be those of a triangle:
(a) 63o, 37 o , 80 o (b) 45 o , 61 o , 73 o
(c) 59 o , 72 o , 61 o (d) 30 o , 20 o , 125 o
SECTION B
o
5 In△ ABC, ∠A = 100 , AD bisects ∠A and AD ⊥ BC. 2
Find ∠B
6 ∠CBX is an exterior angle of ∆ABC at B. Name 2
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ∠CBX
Also, name the interior opposite angles to an exterior angle at A.
7 One of the exterior angles of a triangle is 80o, and the interior opposite angles are 2
equal to each other. What is the measure of each of these two angles?
8 The angles of a triangle are arranged in ascending order of magnitude. If the 2
difference between two consecutive angles is 10o. Find the three angles.
SECTION C
9 (a)Find the unknown length x in the following figure 1. 4
(b)Find the values of the unknown angles x in figure 2 and figure 3:
SECTION D
63
Q Section A consists of 4 questions of 1 mark each. M
1 How many altitudes can a triangle have? 1
(a) 1 (b) 2 (c) 3 (d) 4
2 In right △ABC, right angled at B, BA = 8 cm and BC= 15 cm. Find the length of 1
the hypotenuse AC
(a) 7 cm (b) 17 cm (c) 23 cm (d) 20 cm
3 In each of the following, there are three positive numbers. Which of these pairs 1
could possibly be the lengths of the sides of a triangle:
(a) 5, 7, 9 (b) 2, 10, 15 (c) 2, 5, 7 (d) 5, 8, 20
4 In each of the following, the measures of three angles are given. State in which 1
case the angles can possibly be those of a triangle:
(a) 63o, 37 o , 80 o (b) 45 o , 61 o , 73 o
(c) 59 o , 72 o , 61 o (d) 30 o , 20 o , 125 o
SECTION B
o
5 In△ ABC, ∠A = 100 , AD bisects ∠A and AD ⊥ BC. 2
Find ∠B
6 ∠CBX is an exterior angle of ∆ABC at B. Name 2
(i) The interior adjacent angle
(ii) The interior opposite angles to exterior ∠CBX
Also, name the interior opposite angles to an exterior angle at A.
7 One of the exterior angles of a triangle is 80o, and the interior opposite angles are 2
equal to each other. What is the measure of each of these two angles?
8 The angles of a triangle are arranged in ascending order of magnitude. If the 2
difference between two consecutive angles is 10o. Find the three angles.
SECTION C
9 (a)Find the unknown length x in the following figure 1. 4
(b)Find the values of the unknown angles x in figure 2 and figure 3:
SECTION D
63
Page 12
10 CASE STUDY : Golden Triangle
A golden triangle is an isosceles triangle in which the
ratio of the longer side to the smaller side is
1.618(approximately)
Δ ABC is a golden triangle. AB = 4cm. CD is a median
of the triangle.
Q 1. Find the length of AD and AC approximately
Q2 What are the values of the three angles of a golden
triangle?
Q3 Is Δ ADC a right triangle? Give reason.
OR
AE is bisector of angle BAC. Is ΔABE a golden triangle? Give reason
64
A golden triangle is an isosceles triangle in which the
ratio of the longer side to the smaller side is
1.618(approximately)
Δ ABC is a golden triangle. AB = 4cm. CD is a median
of the triangle.
Q 1. Find the length of AD and AC approximately
Q2 What are the values of the three angles of a golden
triangle?
Q3 Is Δ ADC a right triangle? Give reason.
OR
AE is bisector of angle BAC. Is ΔABE a golden triangle? Give reason
64
Page 13
CHAPTER TEST
Time Allowed 1 ½ Hrs. Maximum Marks: 30
General Instructions:
1. This Question Paper has 5 Sections A-E.
2. Section A has 5 MCQs carrying 1 mark each
3. Section B has 4 questions carrying 02 marks each.
4. Section C has 3 questions carrying 03 marks each.
5. Section D has 1 questions carrying 04 mark.
6. Section E has 1 case based integrated unit of assessment (04 marks) with sub-parts of the
values of 1, 1 and 2 marks each respectively.
7. All Questions are compulsory.
8. Draw neat figures wherever required.
SECTION A
Q Section A consists of 5 questions of 1 mark each. M
No ks
1 Measures of each of the angles of an equilateral triangle is ……………… 1
(a) 90° (b) 30° (c) 60° (d) 45°
2 The acute angles of right triangle are in the ratio 2: 1. Which of the following is 1
the measure of each of these angles.
(a) 55° and 35° (b) 60° and 30° (c) 50° and 40° (d) 45° and 45°
3 Find the value of x in the adjoining figure. 1
(a) 50°
(b) 70°
(c) 120°
(d) 180°
4 Which is the longest side in the triangle PQR right angled at P? 1
(a) PQ (b) QR (c) PR (d) none of these.
5 PQR is a triangle right angled at P. If PQ = 3 cm and PR = 4 cm, find QR. 1
(a) 7 cm (b) 5 cm (c) 1 cm (d) 25 cm
SECTION B
Section B consists of 4 questions of 2 marks each.
6 Find the value of x in given figure. 2
7 Find the value of x an y: 2
8 The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two 2
numbers can length of the third side fall?
9 Determine whether the triangle whose lengths of sides are 6 cm, 8 cm, 10 cm is a 2
right-angled triangle.
65
Time Allowed 1 ½ Hrs. Maximum Marks: 30
General Instructions:
1. This Question Paper has 5 Sections A-E.
2. Section A has 5 MCQs carrying 1 mark each
3. Section B has 4 questions carrying 02 marks each.
4. Section C has 3 questions carrying 03 marks each.
5. Section D has 1 questions carrying 04 mark.
6. Section E has 1 case based integrated unit of assessment (04 marks) with sub-parts of the
values of 1, 1 and 2 marks each respectively.
7. All Questions are compulsory.
8. Draw neat figures wherever required.
SECTION A
Q Section A consists of 5 questions of 1 mark each. M
No ks
1 Measures of each of the angles of an equilateral triangle is ……………… 1
(a) 90° (b) 30° (c) 60° (d) 45°
2 The acute angles of right triangle are in the ratio 2: 1. Which of the following is 1
the measure of each of these angles.
(a) 55° and 35° (b) 60° and 30° (c) 50° and 40° (d) 45° and 45°
3 Find the value of x in the adjoining figure. 1
(a) 50°
(b) 70°
(c) 120°
(d) 180°
4 Which is the longest side in the triangle PQR right angled at P? 1
(a) PQ (b) QR (c) PR (d) none of these.
5 PQR is a triangle right angled at P. If PQ = 3 cm and PR = 4 cm, find QR. 1
(a) 7 cm (b) 5 cm (c) 1 cm (d) 25 cm
SECTION B
Section B consists of 4 questions of 2 marks each.
6 Find the value of x in given figure. 2
7 Find the value of x an y: 2
8 The lengths of two sides of a triangle are 6 cm and 8 cm. Between which two 2
numbers can length of the third side fall?
9 Determine whether the triangle whose lengths of sides are 6 cm, 8 cm, 10 cm is a 2
right-angled triangle.
65
Page 14
SECTION C
Section C consists of 3 questions of 3 marks each.
10 Find the value of 'p' and 'q' in the adjoining figure. 3
11 In the isosceles triangle, the vertical angle is 30° more than each of its base 3
angles. Find all the angles.
12 In a △ABC, AD is the altitude from A such that AD = 12 cm. BD = 9 cm and DC 3
= 16 cm. Examine if △ABC is right angled at A.
SECTION D
Section D consists of 1 question of 4 mark.
13 The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m 4
above the ground. 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?
SECTION E
14 CASE STUDY
Length of a pond
Mr. Pillai has a
triangular field as
shown in the picture.
AB+BC = 34m. On
side AC there is a
pond.
Q1. What is the
measure of angle A?
Q2. What is the length of the side AC?
Q3. How can he find the length of the pond without going into the water? What
is the length of the pond?
66
Section C consists of 3 questions of 3 marks each.
10 Find the value of 'p' and 'q' in the adjoining figure. 3
11 In the isosceles triangle, the vertical angle is 30° more than each of its base 3
angles. Find all the angles.
12 In a △ABC, AD is the altitude from A such that AD = 12 cm. BD = 9 cm and DC 3
= 16 cm. Examine if △ABC is right angled at A.
SECTION D
Section D consists of 1 question of 4 mark.
13 The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m 4
above the ground. 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?
SECTION E
14 CASE STUDY
Length of a pond
Mr. Pillai has a
triangular field as
shown in the picture.
AB+BC = 34m. On
side AC there is a
pond.
Q1. What is the
measure of angle A?
Q2. What is the length of the side AC?
Q3. How can he find the length of the pond without going into the water? What
is the length of the pond?
66
Page 15
Study Materials
Notes
Model Papers Class 6 Notes
Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes
Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats
Maps of India / World Class 12 Notes
Books and Solutions
NCERT Books
NCERT Book Solutions
HC Verma Chapter Wise Solutions
RD Sharma Solutions
CGBSE Solutions
Notes
Model Papers Class 6 Notes
Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes
Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats
Maps of India / World Class 12 Notes
Books and Solutions
NCERT Books
NCERT Book Solutions
HC Verma Chapter Wise Solutions
RD Sharma Solutions
CGBSE Solutions
Document Details
| Board / Org | Aglasem |
| Exam | Class 7 |
| Type | Notes |
| Pages | 15 |
| Language | english |
| Updated | 30 Apr 2026 |