Geometry Chapter 1 Angles
Angle
When two line segments intersect at a point, an angle is formed. Those two line segments are called the arms of that angle and the point is called the vertex of the angle.
The line segments AB and AC have intersected at point A and the angle, ∠BAC has been formed. AB and AC are the two arms of the angle and A is the vertex.
If we assume a point D on AB and another point E on AC then ∠DAE and ∠BAC will be of the same measure.
Adjacent Angles
If two angles have the same vertex and one common arm and if the two angles are on opposite sides of the common arm then the two angles are called adjacent angles.
∠POQ and ∠QOR are adjacent angles because the vertex of both angles is
O and their common arm are OQ and the two angles are on opposite sides of this common arm.
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Perpendicular And Right Angle
If a straight line stands on another straight line in such a way that, two adjacent angles are equal then one of the straight lines is called a perpendicular to the other. Each of the two adjacent angles is called a right angle.
The straight line OC is perpendicular to AB. Both∠AOC and∠BOC are right angles.
1 right angle = 90°.
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WBBSE Class 8 Angles Notes
Straight angle
A straight line AB is drawn on a piece of paper. If point C is taken on the straight line AB then ∠ACB will be a straight angle.
1 straight angle = 180° = 2 right angles.
Acute angle, Obtuse angle, and Reflex angle
Acute angle: An angle that is less than a right angle is acute.
For example, 30°, 44°, 70°, etc., are acute angles.
Obtuse angle :
An angle that is greater than one right angle but less than two right angles is called an obtuse angle.
For example, 95°, 110°, 145°, etc., are obtuse angles.
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Reflex angle: An angle that is greater than two right angles but less than four right angles is called a reflex angle.
For example, 190°, 210°, 300°, etc., are reflex angles.
Complementary angle and supplementary angle
Complementary angle: If the sum of the two angles is equal to 90° or one right angle then each angle is called the complementary angle of the other angle.
For example, 20° and 70° are complementary angles. In ∠AOB + ∠BOC = 90°.
So, angles ∠AOB and ∠BOC are complementary angles.
We say that each of the angles ∠AOB and ∠BOC complement the other.
Supplementary angle: If the sum of the two angles is equal to 180° or two right angles then each angle is called the supplementary angle of the other angle.
For example, 100° and 80° are supplementary angles. In the figure ∠AOB +∠BOC = 180°. So, angles ∠AOB and ∠BOC are supplementary angles. We say that each of the angles ∠AOB and ∠BOC is a supplement of the other.
External angle: If any arm of an angle is extended in the direction opposite to the arm, then the angle formed by it with the other arm is called the external angle of that angle.
Let ∠BAC be an angle.
The arm BA is produced in such a way that ∠CAD is formed. Therefore, ∠CAD is the external angle of ∠BAC.
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Note that, ∠CAD is adjacent to ∠CAB and is supplementary to ∠CAB.
Internal bisector of an angle
If a straight line bisects an angle, then it is called the internal bisector of that angle.
External bisector of an angle
If a straight line bisects the external angle of a given angle, then it is called the external bisector of that angle.
In the figure, BD is the internal bisector, and BE is the external bisector of ∠ABC.
It can be proved that ∠EBD = 90°.
Vertically opposite angles
If two straight lines intersect each other, two pairs of angles are formed on the opposite sides of the intersecting point. Then any angle of a pair of angles is called a vertically opposite angle of the other.
The straight lines AB and CD intersect at point 0.
∠BOD is the vertically opposite angle of AOC and ∠AOD is the vertically opposite angle of ∠BOC.
The vertically opposite angles are always of the same measure.
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:. ∠AOC = ∠BOD and ∠BOC = ∠AOD.
Understanding Types of Angles
Transversal, Exterior angles, Interior angles, Interior opposite angles, Alternate angles, Corresponding angles
If a straight line cuts two other straight lines, the former straight line is called the transversal of those two straight lines. In the straight line, EF cuts the two straight lines AB and CD. Therefore, the straight line EF is the transversal of the straight lines AB and CD.
When a straight line cuts two other straight lines then eight angles are
formed. Among them, four angles are in the inside region of the two straight lines. These four angles are called interior angles and the other four angles are called exterior angles.
The interior angles are ∠AGH, ∠GHC, ∠GHD, and ∠HGB. The exterior angles are ∠EGA, ∠EGB, ∠CHF, and ∠FHD.
The further-off interior angle, with respect to an exterior angle, is called the interior opposite angle. For example, ∠GHD is the interior opposite angle, in respect of ∠EGB. ,
The interior angle adjacent to one exterior angle and the interior angle adjacent to a further off interior angle in respect of the same exterior angle are called alternate angles to each other.
In the figure, ∠GHD is the alternate angle of ∠AGH, and ∠GHC is the alternate angle of ∠BGH.
An exterior angle and an interior opposite angle on the same side of the transversal are called corresponding angles.
In the figure, the pair of angles (∠EGB, ∠GHD), (∠EGA, ∠GHC), (∠AGH, ∠CHF), and (∠BGH, ∠DHF) are corresponding angles.
Interior angles on the same side of the transversal
∠BGH and ∠GHD and also ∠AGH and ∠GHC are the interior angles on the same side of the transversal.
Classification of triangle
Triangle: A plane figure bounded by three line segments is called a triangle.
A triangle has three sides and three angles.
Three angular points are called the vertices of the triangle.
ABC is a triangle. Its three sides are AB, BC, and AC.
Its three angles are ∠ABC, ∠BCA, and ∠CAB. Its three vertices are A, B, and C.
The sum of the three angles of a triangle is equal to 2 right angles or 180°.
In any triangle, the sum of the two sides is always greater than the third side. Again, the difference between any two sides is always less than the third side.
If any angular point of a triangle is taken as a vertex then its opposite side is called its base.
The angle opposite to the base of a triangle is called its vertical angle.
If BC is taken as the base then ∠BAC will be its vertical angle.
The triangles are classified based on
1. sides and
2. angles.
1. based on sides, there are three types of triangles: equilateral triangle, isosceles triangle, and scalene triangle.
Step-by-Step Guide to Measuring Angles
Equilateral triangle: If the lengths of the three sides of a triangle are the same then the triangle is called an equilateral triangle.
In the figure, ΔABC is an equilateral triangle.
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Isosceles triangle: If the lengths of the two sides of a triangle are the same then the triangle is called an isosceles triangle. In the ADEF is an isosceles triangle.
Scalene triangle: If the lengths of the three sides of a triangle are unequal then the triangle is called a scalene triangle. The APQR is a scalene triangle.
2. On the basis of angles there are three types of triangles: Acute-angled triangle, obtuse-angled triangle, and right-angled triangle.
Acute-angled triangle: If each of the three angles of a triangle is acute then the triangle is called an acuteangled triangle. In the figure below ΔABC is an acuteangled triangle.
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Obtuse-angled triangle: If any one angle of a triangle is obtuse then the triangle is called an obtuse-angle triangle.
The ADEF is an obtuse-angle triangle.
Right-angled triangle: If any one angle of a triangle is a right angle then the triangle is called a right-angled triangle.
The APQR is a right-angle triangle
The median of a triangle
The line segment obtained by joining the middle point of any side of a triangle to the opposite vertex is called the median of the triangle.
The line segment AD has been obtained by joining the midpoint D of the side BC of the triangle ABC to the opposite vertex A.
AD is the median of the triangle ABC. There are always three medians of a triangle.
Height of a triangle
The line segment obtained by drawing the perpendicular from any vertex of a triangle to the opposite side is called the height of the triangle.
AD has been drawn perpendicular from the vertex a
A to the opposite side BC of the triangle ABC.
So AD is the height of the triangle ABC.
In this case, BC is the base of the triangle.
If AC is taken as the base of the triangle then perpendicular from B on AC will be the height of the triangle.
If AB is taken as the base of the triangle then perpendicular from C on AB will be the
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Classification of quadrilateral
Quadrilateral: A plane figure enclosed by four line segments is called a quadrilateral.
A quadrilateral has four sides and four angles. Four angular points are called the four vertices of a quadrilateral.
The line segment joining any two opposite vertices of a quadrilateral is called the diagonal of the quadrilateral.
There are two diagonals of a quadrilateral.
ABCD is a quadrilateral. Its four sides are AB, BC, CD, and DA.
Its four angles are ∠ABC, ∠BCD, ∠CDA, and ∠DAB.
Its four angular points are A, B, C, and D.
Its two diagonals are AC and BD.
the sum of the four angles of a quadrilateral = 360º.
Parallelogram: If the opposite sides of a quadrilateral are parallel then it is called a parallelogram.
Rectangle: If one angle of a parallelogram is a right angle then it is called a rectangle.
Square: If the lengths of the two adjacent sides of a rectangle are equal then it is called a square.
Key Terms Related to Angles in Geometry
Trapezium: If only one pair of opposite sides of a quadrilateral are parallel then it is called a trapezium.
Isosceles trapezium: If the lengths of the non-parallel sides (i.e., oblique sides) of a trapezium are equal then it is called an isosceles trapezium.
Rhombus: If the lengths of the four sides of a quadrilateral are equal but none of the angles is a right angle then it is called a rhombus.
Axioms
Some geometrical properties are obtained through activities. They are called axioms. These geometrical properties do not require any logical proof. Great mathematician Euclid named the axioms, ‘Common notions or conceptions of thought.’ Axioms are of two types— General and Geometric. Let us state below some axioms.
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1. General axioms :
1. The quantities which are equal to the same quantity are equal.
2. If equal quantities are added with equal quantities, the sums are equal.
3. If equal quantities are subtracted from equal quantities, the remainder are equal.
4. If equal quantities are added with unequal quantities, the sums are unequal.
5. If equal quantities are subtracted from unequal quantities, the remainders are unequal.
6. The same multiples of equal quantities are equal.
7. The same fractions of equal quantities are equal.
8. Every object is greater than its part.
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2. Geometric axioms:
1. If any geometrical figure (for example, a line or angle, or triangle) coincides with another geometrical figure then they are called congruent.
2. Two straight lines cannot enclose any plane surface.
3. All right angles are of equal measure.
4. Both the two intersecting straight lines cannot be parallel to a third straight line. (It is Playfair’s axiom).
5. When a straight line cuts two other straight lines, those other two straight lines are parallel if a pair of corresponding angles are equal.
6. Only one straight line can be drawn through a given point that is parallel to a given straight line.
7. Of the two triangles, if two sides and their included angle of one, are respectively equal to the two sides and their included angle of the other, then the triangles are congruent. (It is called side-angle-side congruence or SAS congruence).
8. Of the two triangles, if the two angles and one side of one, are respectively equal to the two angles and one side of the other, then the triangles are congruent. (It is called angle-angle-side congruence or ΔAS congruence.)
Some Examples
Example 1
Find the measure of the angles of an equilateral triangle.
Solution: Each angle is equal to 60°.
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Example 2
Find the measure of the angles of a right-angled isosceles triangle.
Solution: 45°, 45° and 90°.
Example 3
If one angle of a right-angled triangle is 65° then find other angles of the triangle.
Solution :
Given:
one angle of a right-angled triangle is 65°
Since the triangle is right-angled therefore one angle is 90°, and one other angle
= 180° – (90° + 65°)
= 180° – 155°
= 25°.
∴ 90° and 25°.
Other angles of the triangle 90° and 25°.
Conceptual Questions on Applications of Angles
Example 4
By what name do we call the greatest side of a right-angled triangle?
Solution: Hypotenuse.
Example 5
Find the minimum and maximum number of acute angles of a triangle.
Solution: A triangle may have at least two and almost three acute angles.
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Example 6
If one angle of a triangle is twice the sum of the other two angles then find the measure of that angle.
Solution :
Given:
one angle of a triangle is twice the sum of the other two angles.
Let, the measure of that angle – x°.
Then, the sum of the other two angles
= (180 – x)
According to the question, x°
= 2(180 – x)°
or, x = 2(180 – x)
or, x = 360 – 2x
or, x + 2x= 360
or, 3x – 360
The measure of that angle is 120°.
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Example 7
The ratio of the four angles of a quadrilateral is 5: 6: 13: 12. Find the measure of the greatest angle.
Solution :
Given:
The ratio of the four angles of a quadrilateral is 5: 6: 13: 12.
Let, the measures of the angles of that quadrilateral be 5x°, 6x°, 13x°, and 12x°.
Hence, 5x° + 6x° + 13xº + 12x° = 360°
or, 36x° = 360°
or, 36x = 360
or, x = 360 / 36
= 10
Hence, the greatest angle of the quadrilateral
= 13 x 10°
= 130°.
The greatest angle of the quadrilateral is 130°.
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Example 8
Is it possible that the three sides of a triangle are, a – b, 2a, and a + b?
Solution :
Given:
a – b, 2a, And a + b.
We know that the sum of any two sides of a triangle is greater than the third side. But, in this case, (a – b) + (a + b) = a – b + a + b = 2a.
Hence, the sum of the two sides is equal to the third.
Hence, in this case, the given three lengths cannot be the sides of a triangle.
It is not possible.
Examples of Complementary and Supplementary Angles
Example 9
In ΔABC, if ∠A + ∠B = 135º and ∠B + ∠C = 90° then find the nature of ΔABC
Solution:
Given:
In ΔABC, if ∠A + ∠B = 135º and ∠B + ∠C = 90°
AC = (∠A+ ∠B + ∠C) – (∠A + ∠B)
= 180°- 135°
= 45°
∠A = (∠A + ∠B + ∠C)-(∠B + ∠C)
= 180° – 90°
= 90°
∠B = (∠A +∠B + ∠C) – (∠C + ∠A)
= 180° – (45° + 90°)
= 180°- 135°
= 45°
In ΔABC,
∠A = 90°,
∠B = 45°,
∠C = 45°.
.’. ΔABC is a right-angled isosceles triangle.
ΔABC is a right-angled isosceles triangle.
Example 10
In ΔABC, ∠BAC – ∠ABC = 10° and∠ACB = 50°. Find the measure of ∠ABC.
Solution :
Given:
In ΔABC, ∠BAC – ∠ABC = 10° and∠ACB = 50°
Since, ∠ACB = 50°,
therefore, ∠BAC + ∠ABC – 180º – 50° = 130°.
Now,∠BAC + ∠ABC = 130° ———-(1)
∠BAC -∠ABC = 10° ———–(2)
By (1) – (2) we get, 2∠ABC = 120°
or, ∠ABC = 120º / 2
= 60º
The measure of ∠ABC is 60°.
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Example 11
Each of the two base angles of an isosceles triangle is twice its vertical angle. Find the measure of the vertical angle.
Solution :
Given:
Each of the two base angles of an isosceles triangle is twice its vertical angle.
Let, the vertical angle be x°.
Then, each of the two base angles is 2x°.
So, x° + 2x° + 2x° = 180°
or, 5x° = 180°
or, x = 180º / 5
= 36°.
The vertical angle is 36°.
Practice Problems on Angles for Class 8
Example 12
The vertical angle A of the isosceles triangle is three times another angle B of it. What is the measure of angle A?
Solution :
Given:
The vertical angle A of the isosceles triangle is three times another angle B of it.
Let, ∠B = x°
then∠C =x° and ∠A = 3x°
∴ 3x° + x° + x° = 180°
or, 5x° = 180°
or, x° = 180° / 5
= 36°
∴ ∠A = 3x°
= 3 x 36°
= 108°
The measure of ∠A = 108°.
Example 13
In the triangle ABC, ∠A + ∠C = 140 and ∠A + 3∠B = 180°. Find the measure of the angles of the triangle.
Solution :
Given:
In the triangle ABC, ∠A + ∠C = 140 and ∠A + 3∠B = 180°.
∠B = 180° – (∠A + ∠C)
= 180° – 140° = 40°
∴ ∠A + 3 x 40° = 180°
or,∠A + 120° = 180°
or, ∠A = 180° – 120° = 60°
∴ ∠C = 180° – (∠A + ∠B)
= 180° – (60° + 40°)
= 180° – 100° = 80°
∴ ∠A = 60°, ∠B = 40°, ∠C = 80°
Example 14
The angles of a triangle are such that, 1/2 of one angle = 1/3rd of another angle = 1/4th of another angle. Find the angles of the triangle.
Solution :
Given:
The angles of a triangle are such that, 1/2 of one angle = 1/3rd of another angle = 1/4th of another angle.
Let, the equal portions be x°.
the angles are 2x°, 3x° and 4x°.
∴ 2x° + 3x° + 4x° = 180°
or, 9x° = 180°
∴ angles of the triangle are 2 x 20°, 3 x 20°, 4 x 20°
i.e., 40°, 60° and 80°.
Example 15
In the triangle, ABC, AB2 + BC2 = AC2 and AC = 72 BC. Find the nature of the triangle.
Solution :
Given:
In the triangle, ABC, AB2 + BC2 = AC2 and AC = 72 BC
AB2 + BC2 = AC2
or, AB2 + BC2 = (72 BC)2
or, AB2 + BC2 = 2 BC2
or, AB2 = 2BC2 – BC2
or, AB2 = BC2
or, AB = BC
Hence, the triangle is a right-angled isosceles triangle whose ∠ABC = 90° and AB = BC.
∴ The triangle is isosceles right-angled.