Class 9 Exam > Class 9 Notes > Mathematics (Maths) Class 9 > RD Sharma Solutions: Triangle and its Angles- 1
Triangle and its Angles- 1 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download
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Page 1
Question:1
In a ? ABC, if ?A = 55°, ?B = 40°, find ?C.
Solution:
?A + ?B + ?C = 180° [The sum of three angles of a triangle is 180°. ] ? 55°+40°+ ?C = 180° ? 95°+ ?C = 180° ? ?C = 180°-95° ? ?C = 85°
Question:2
If the angles of a triangle are in the ratio 1 : 2 : 3, determine three angles.
Solution:
Let the angles of the given triangle be of xº, 2xº and 3xº. Then,
? x +2x +3x = 180 [The sum of three angles of a triangle is 180°] ? 6x = 180 ? x = 30
Hence, the angles of the triangle are 30º, 60º and 90º.
Question:3
The angles of a triangle are (x - 40)°, (x - 20)° and
1
2
x -10
°
. Find the value of x.
Solution:
Given angles are
? (x -40)+(x -20)+
1
2
x -10 = 180 ?
5
2
x = 180 +70 ?
5
2
x = 250 ? x =
250×2
5
? x = 100
Hence, the value of x is 100°.
Question:4
Two angles of a triangle are equal and the third angle is greater than each of those angles by 30°. Determine all the angles of the triangle.
Solution:
Let the two equal angles are x°, then the third angle will be (x + 30)°.
? x +x +(x +30) = 180 [Sum of the three angles of a triangle is 180°] ? 3x +30 = 180 ? 3x = 150 ? x = 50
Therefore, the angles of the given triangle are 50°, 50° and 80°.
Question:5
If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.
Solution:
Let ABC be a triangle such that
?A = ?B + ?C [Since, one angle is sum of the other two] ? ?A + ?B + ?C = 180° [Sum of the three angles of a triangle is 180°] ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
Hence, the given triangle is a right angled triangle.
Question:6
Can a triangle have:
i
Two right angles?
ii
Two obtuse angles?
iii
Two acute angles?
iv
All angles more than 60°?
v
All angles less than 60°?
vi
All angles equal to 60°?
Solution:
i Let a triangle ABC has two angles equal to . We know that sum of the three angles of a triangle is 180°.
( )
( )
Page 2
Question:1
In a ? ABC, if ?A = 55°, ?B = 40°, find ?C.
Solution:
?A + ?B + ?C = 180° [The sum of three angles of a triangle is 180°. ] ? 55°+40°+ ?C = 180° ? 95°+ ?C = 180° ? ?C = 180°-95° ? ?C = 85°
Question:2
If the angles of a triangle are in the ratio 1 : 2 : 3, determine three angles.
Solution:
Let the angles of the given triangle be of xº, 2xº and 3xº. Then,
? x +2x +3x = 180 [The sum of three angles of a triangle is 180°] ? 6x = 180 ? x = 30
Hence, the angles of the triangle are 30º, 60º and 90º.
Question:3
The angles of a triangle are (x - 40)°, (x - 20)° and
1
2
x -10
°
. Find the value of x.
Solution:
Given angles are
? (x -40)+(x -20)+
1
2
x -10 = 180 ?
5
2
x = 180 +70 ?
5
2
x = 250 ? x =
250×2
5
? x = 100
Hence, the value of x is 100°.
Question:4
Two angles of a triangle are equal and the third angle is greater than each of those angles by 30°. Determine all the angles of the triangle.
Solution:
Let the two equal angles are x°, then the third angle will be (x + 30)°.
? x +x +(x +30) = 180 [Sum of the three angles of a triangle is 180°] ? 3x +30 = 180 ? 3x = 150 ? x = 50
Therefore, the angles of the given triangle are 50°, 50° and 80°.
Question:5
If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.
Solution:
Let ABC be a triangle such that
?A = ?B + ?C [Since, one angle is sum of the other two] ? ?A + ?B + ?C = 180° [Sum of the three angles of a triangle is 180°] ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
Hence, the given triangle is a right angled triangle.
Question:6
Can a triangle have:
i
Two right angles?
ii
Two obtuse angles?
iii
Two acute angles?
iv
All angles more than 60°?
v
All angles less than 60°?
vi
All angles equal to 60°?
Solution:
i Let a triangle ABC has two angles equal to . We know that sum of the three angles of a triangle is 180°.
( )
( )
Hence, if two angles are equal to , then the third one will be equal to zero which implies that A, B, C is collinear, or we can say ABC is not a triangle
A triangle can’t have two right angles.
ii Let a triangle ABC has two obtuse angles
This implies that sum of only two angles will be equal to more than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have two obtuse angles.
iii Let a triangle ABC has two acute angles .
This implies that sum of two angles will be less than . Hence third angle will be the difference of 180° and sum of both acute angles
Therefore, a triangle can have two acute angles.
iv Let a triangle ABC having angles are more than 60°.
This implies that the sum of three angles will be more than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have all angles more than .
v Let a triangle ABC having angles are less than 60°.
This implies that the sum of three angles will be less than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have all angles less than 60°.
vi Let a triangle ABC having angles all equal to 60°.
This implies that the sum of three angles will be equal to 180° which satisfies the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can have all angles equal to 60°.
Question:7
The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10°, find the three angles.
Solution:
Let the angles of a triangle are
Since, thedifferencebetweentwoconsceutiveanglesis10°
? x +(x +10)+(x +20) = 180 [Sum of the three angles of a triangle is 180°] ? 3x +30 = 180 ? 3x = 150 ? x = 50
Therefore, the angles of the given triangle are 50°,
50 +10° and
50 +20° i.e. 50°, 60° and 70°.
Question:8
ABC is a triangle in which ?A = 72°, the internal bisectors of angles B and C meet in O. Find the magnitude of ?BOC.
Solution:
Since OB and OC are the angle bisector of
?B and ?C
?A + ?B + ?C = 180° ? 72°+2 ?OBC +2 ?OCB = 180° [Sum of the three angles of a triangle is 180°] ? 2( ?OBC + ?OCB) = 108° ? ?OBC + ?OCB = 54° ? 180°- ?BOC = 54° [Since
Hence magnitude of
?BOC is 126°.
Question:9
The bisectors of base angles of a triangle cannot enclose a right angle in any case.
Solution:
Let ABC be a triangle and BO and CO be the bisectors of the base angle respectively.
We know that if the bisectors of angles ?ABC and ?ACB of a triangle ABC meet at a point O, then
?BOC = 90°+
1
2
?A
Page 3
Question:1
In a ? ABC, if ?A = 55°, ?B = 40°, find ?C.
Solution:
?A + ?B + ?C = 180° [The sum of three angles of a triangle is 180°. ] ? 55°+40°+ ?C = 180° ? 95°+ ?C = 180° ? ?C = 180°-95° ? ?C = 85°
Question:2
If the angles of a triangle are in the ratio 1 : 2 : 3, determine three angles.
Solution:
Let the angles of the given triangle be of xº, 2xº and 3xº. Then,
? x +2x +3x = 180 [The sum of three angles of a triangle is 180°] ? 6x = 180 ? x = 30
Hence, the angles of the triangle are 30º, 60º and 90º.
Question:3
The angles of a triangle are (x - 40)°, (x - 20)° and
1
2
x -10
°
. Find the value of x.
Solution:
Given angles are
? (x -40)+(x -20)+
1
2
x -10 = 180 ?
5
2
x = 180 +70 ?
5
2
x = 250 ? x =
250×2
5
? x = 100
Hence, the value of x is 100°.
Question:4
Two angles of a triangle are equal and the third angle is greater than each of those angles by 30°. Determine all the angles of the triangle.
Solution:
Let the two equal angles are x°, then the third angle will be (x + 30)°.
? x +x +(x +30) = 180 [Sum of the three angles of a triangle is 180°] ? 3x +30 = 180 ? 3x = 150 ? x = 50
Therefore, the angles of the given triangle are 50°, 50° and 80°.
Question:5
If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right triangle.
Solution:
Let ABC be a triangle such that
?A = ?B + ?C [Since, one angle is sum of the other two] ? ?A + ?B + ?C = 180° [Sum of the three angles of a triangle is 180°] ? ?A + ?A = 180° ? 2 ?A = 180° ? ?A = 90°
Hence, the given triangle is a right angled triangle.
Question:6
Can a triangle have:
i
Two right angles?
ii
Two obtuse angles?
iii
Two acute angles?
iv
All angles more than 60°?
v
All angles less than 60°?
vi
All angles equal to 60°?
Solution:
i Let a triangle ABC has two angles equal to . We know that sum of the three angles of a triangle is 180°.
( )
( )
Hence, if two angles are equal to , then the third one will be equal to zero which implies that A, B, C is collinear, or we can say ABC is not a triangle
A triangle can’t have two right angles.
ii Let a triangle ABC has two obtuse angles
This implies that sum of only two angles will be equal to more than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have two obtuse angles.
iii Let a triangle ABC has two acute angles .
This implies that sum of two angles will be less than . Hence third angle will be the difference of 180° and sum of both acute angles
Therefore, a triangle can have two acute angles.
iv Let a triangle ABC having angles are more than 60°.
This implies that the sum of three angles will be more than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have all angles more than .
v Let a triangle ABC having angles are less than 60°.
This implies that the sum of three angles will be less than 180° which contradicts the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can’t have all angles less than 60°.
vi Let a triangle ABC having angles all equal to 60°.
This implies that the sum of three angles will be equal to 180° which satisfies the theorem sum of all angles in a triangle is always equals 180°.
Therefore, a triangle can have all angles equal to 60°.
Question:7
The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10°, find the three angles.
Solution:
Let the angles of a triangle are
Since, thedifferencebetweentwoconsceutiveanglesis10°
? x +(x +10)+(x +20) = 180 [Sum of the three angles of a triangle is 180°] ? 3x +30 = 180 ? 3x = 150 ? x = 50
Therefore, the angles of the given triangle are 50°,
50 +10° and
50 +20° i.e. 50°, 60° and 70°.
Question:8
ABC is a triangle in which ?A = 72°, the internal bisectors of angles B and C meet in O. Find the magnitude of ?BOC.
Solution:
Since OB and OC are the angle bisector of
?B and ?C
?A + ?B + ?C = 180° ? 72°+2 ?OBC +2 ?OCB = 180° [Sum of the three angles of a triangle is 180°] ? 2( ?OBC + ?OCB) = 108° ? ?OBC + ?OCB = 54° ? 180°- ?BOC = 54° [Since
Hence magnitude of
?BOC is 126°.
Question:9
The bisectors of base angles of a triangle cannot enclose a right angle in any case.
Solution:
Let ABC be a triangle and BO and CO be the bisectors of the base angle respectively.
We know that if the bisectors of angles ?ABC and ?ACB of a triangle ABC meet at a point O, then
?BOC = 90°+
1
2
?A
From the above relation it is very clear that if is equals 90° then must be equal to zero.
Now, if possible let is equals zero but on other hand it represents that A, B, C will be collinear, that is they do not form a triangle.
It leads to a contradiction.
Hence, the bisectors of base angles of a triangle cannot enclose a right angle in any case.
Question:10
If the bisectors of the base angles of a triangle enclose an angle of 135°, prove that the triangle is a right triangle.
Solution:
Let ABC be a triangle and Let BO and CO be the bisectors of the base angle respectively.
We know that if the bisectors of angles ?ABC and ?ACB of a triangle ABC meet at a point O, then
?BOC = 90°+
1
2
?A
? 135° = 90°+
1
2
?A ? 45° =
1
2
?A ? ?A = 90°
Hence the triangle is a right angled triangle.
Question:11
In a ? ABC, ?ABC = ?ACB and the bisectors of ?ABC and ?ACB intersect at O such that ?BOC = 120°. Show that ?A = ?B = ?C = 60°.
Solution:
Let ABC be a triangle and BO and CO be the bisectors of the base angle respectively.
We know that if the bisectors of angles ?ABC and ?ACB of a triangle ABC meet at a point O, then
?BOC = 90°+
1
2
?A
? 120° = 90°+
1
2
?A ? 30° =
1
2
?A ? ?A = 60°
are equal as it is given that .
?A + ?B + ?C = 180° [Sum of three angles of a triangle is 180°] ? 60°+2 ?B = 180° [ ? ?ABC = ?ACB] ? ?B = 60°
Hence, .
Question:12
If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.
Solution:
Let a triangle ABC having angles .
It is given that the sum of two angles are less than third one.
We know that the sum of all angles of a triangle equal to 180°.
Similarly we can prove that
Since, all angles are less than 90°.
Hence, triangle is acute angled.
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FAQs on Triangle and its Angles- 1 RD Sharma Solutions - Mathematics (Maths) Class 9
| 1. What is the sum of the angles in a triangle? |  |
Ans. The sum of the angles in a triangle is always 180 degrees. This is a fundamental property of triangles, and it holds true for all types of triangles, whether they are equilateral, isosceles, or scalene.
| 2. How do you find the measure of an angle in a triangle? | |
Ans. To find the measure of an angle in a triangle, you need to have information about the other two angles. If you know the measures of two angles in a triangle, you can subtract their sum from 180 degrees to find the measure of the third angle. For example, if one angle measures 40 degrees and another angle measures 60 degrees, the measure of the third angle would be 180 - (40 + 60) = 80 degrees.
| 3. Are the angles of an equilateral triangle always equal? | |
Ans. Yes, the angles of an equilateral triangle are always equal. Since an equilateral triangle has three congruent sides, it also has three congruent angles. Each angle in an equilateral triangle measures 60 degrees. This property holds true for all equilateral triangles.
| 4. Can a triangle have two right angles? | |
Ans. No, a triangle cannot have two right angles. The sum of the angles in a triangle is always 180 degrees, and if two angles are already right angles (each measuring 90 degrees), the third angle would have to be 180 - (90 + 90) = 0 degrees. However, a 0-degree angle is not possible in a triangle.
| 5. What is the difference between an acute triangle and an obtuse triangle? | |
Ans. An acute triangle is a triangle in which all three angles are less than 90 degrees. In contrast, an obtuse triangle is a triangle that has one angle greater than 90 degrees. The other two angles in an obtuse triangle are always acute angles, meaning they are less than 90 degrees.
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Nov 24, 2024 Last updated |
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