Triangle

Triangles are three-sided closed figures which have three straight sides joined at three vertices and have three angles enclosed within the figure at the vertices. There are several types of triangles based on the lengths of its sides and the angles they contain. What many do not know is that a triangle is a three-sided polygon.

Summary

Things to Remember

The angle is mainly divided into the following types:

Acute angle: less than 90
Right angle: exactly 90
Obtuse angle: more than 90 but less than 180
Straight angle: exactly 180

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Triangle

Basic concepts of geometry

The basic concepts of geometry which have been dealt with previous classes and necessary to study in this class are listed below. A definite image which has no length, breadth and thickness is called a point. It is denoted by symbol dot (.) and capital letters A, B, C etc.

Straight line: The shortest locus which passes through two fixed points is called the straight line. It can be extended on both sides.

Line segment: A certain portion of a straight line is called the line segment. It has a fixed measurement.

Curved line: The line joining two fixed points without fixed direction is called the curved line.

Parallel lines: Any two or more than two straight lines which intersect each other after extending or the length of the perpendicular distance between them is always equal are parallel lines.

Angle: When any two straight lines or line segments meet at a point, they form a corner which is called an angle.

Relation between the pair of angles

1. Adjacent angles
A pair of angles having same vertex and a common side is called adjacent angles. If the exterior sides of both angles lie on a straight line then their sum is two right angles.

Adjacent angles

2 . Vertically opposite angles
When two straight line segments intersect at a point then a pair of angles formed in opposite each other is called vertically opposite angles.

Vertically Opposite angle

Classification of triangles on the basis of sides

1. Equilateral triangles

A triangle whose all sides are equal is known as an equilateral triangle. Each angle of an equilateral triangle are 60^o. So, an equilateral triangle is also called as equiangular i.e all there internal angles are also congruent to each other and are each 60^o. They are regular polygons.

2. Isosceles triangle

A triangle whose (at least) two sides are equal is called an isosceles triangle. Base angles of an isoceles triangle are equal. An isoceles triangle, therefore, has both two equal sides and two equal angles. In \(\triangle\)ABC, ∠B = ∠C ∴ \(\triangle\)ABC is an isosceles triangle.

3. Scalene triangle

A triangle whose all three sides are unequal is called a scalene triangle. In \(\triangle\)XYZ, none of the sides are equal. ∴ \(\triangle\)XYZ is a scalene triangle.

Classification of triangles on the basis of angles

1. Acute-angled triangle: A triangle whose all angles are acute angles (less than 90^o) is called an acute-angled triangle. In the given figure, all angles are less than 90^o so it is an acute-angled triangle.

2. Obtuse-angled triangle: A triangle whose one angle is obtuse (greater than 90^o) is called an obtuse-angled triangle. In the given figure, ∠Y is 120^o(greater than 90^o) ∴ \(\triangle\)XYZ is an obtuse-angled triangle.

3. Right-angled triangle: A triangle whose one angle is right angle (90^o) is called right-angled triangle. In \(\triangle\)ABC, ∠B = 90^o, so \(\triangle\)ABC, is a right-angled triangle.

Properties of triangles

The sum of interior angles of a triangle is two right angles or 180.
The exterior angle formed by producing a side of a triangle is equal to the sum of the two non-adjacent interior angles.
The sum of any two sides of the greater angle of a triangle is longer than the opposite side of the smaller angle.
The base angles of an isosceles triangle are equal
All angles of an equilateral triangle are equal.

Axioms and Postulates

Axoims

An axiom is a self-evident truth which is well-established, that accepted without controversy or question. Some of the axioms are presented in the following table with their conditions:

Axoims	Conditons
Equality of addition	When the equal quantities are added to both sides of equal quantities, the sum is also equal. e.g if a=b then a+c=b+c
Equality of subtraction	When the equal quantites are subtracted from both sides of equal quantities, the difference is also equal. e.g. if a=b then a-c=b-c
Equality of multiplication	When the equal quantites are multiplied by the same quantity, the product is also equal. e.g. if a=b then ac=bc
Equality of division	When the equal quantites are divided by the same quantity, the quotient is also equal. e.g. if a=b then a/c = b/c
Equality axiom	When two separate quantites are equal to a quantity the they is also equal to each other, If a=c and b=c then a=c
Whole part of axiom	A whole quantity is always equal to the sum of its all parts and the whole quantity is always greater than each of its parts. e.g. AD=AB+BC+CD and AD>AB or AD>BC or AD>CD
Substitution axiom	A quantity can be replaced by another equal quantity. This doesn't alter the final result. If, a=b then ax+c can be expressed as bx+c.

Postulates

A statement which is taken to be true without proof is called a postulate. Postulates are the basic structure from which lemmas and theorems are derived. Some of the postulates that we need in our geometry are listed below.

1. There is only one straight line that you can draw between any two points.

2. An infinite number of straight lines can be drawn through a point.

3. A line contains exactly at least two points.

4. Through any three noncollinear points, there exist exactly one plane.

5. Only one line can be the bisector of a given angle.

6. A straight line can be produced on either side infinitely.

7. Through one point, there is only line parallel to the first line.

8. The length of perpendicular means the distance between a point and a line.

Properties of angles when two parallel lines are intersected by a transversal line:

Let's draw at least two parallel lines AB and XY. The transversal line CD intersects AB at E and XY at F. We can establish the following relations among the angles formed:

a) Alternate angles are equal: If two parallel lines are intersected by a transversal line, then the alternate angles so formed are equal. ∴ ∠AEF = ∠EFY and ∠BEF = ∠EFX

b) Corresponding angles are equal: If two parallel lines are intersected by a transversal line, then the corresponding angles so formed are equal.

∴ ∠AED = ∠EFX, ∠AEF = ∠XFC

∠DEB = ∠EFY, ∠BEF = ∠YFC

c) The sum of co-interior angles is 180^o: If two parallel lines are intersected by a transversal line, then the sum of the co-interior angles so formed is 180^o.

∴ ∠AEF + ∠EFX = 180^o

∠BEF + ∠EFY = 180^o

d) The sum of co-exterior angles is 180^o: If two parallel lines are intersected by a transversl line, then the sum of the co-exterior angles so formed is 180^o.

∴ ∠AED + ∠XEC = 180^oand ∠DEB + ∠YEC = 180^o

Theorem 1 : The sum of the angles of any triangle is equal to two right angles.

Experimental Verification:

Step1: Let's draw three triangles of different orientations sizes. And name them as \(\triangle\)ABC each. (Draw the triangles in such a way that you can measure each angle by using a protractor.)

Step 2: Measure each angle of these three triangles with the help of protractor and fill up the following table.

Fig.	∠A	∠B	∠C	Results
(i)				∠A +∠B +∠C =
(ii)				∠A +∠B +∠C =
(iii)				∠A +∠B +∠C =

Conclusion :

Theoretical proof:

theorem 1

Given: ∠ABC, ∠BCA and ∠BAC are three angles of triangle ABC.

To proof: ∠ABC + ∠BCA + ∠BAC = 180^o

Construction: Let's draw a straight line parallel to BC through A.

Proof:

S.N.	Statements	Reasons
1	∠XAB = ∠ABC	XY \(\parallel\) BC and being alternate angles.
2	∠YAC = ∠ACB	XY \(\parallel\) BC and being alternate angles.
3	∠XAB + ∠BAC + ∠YAC = ∠XAY	According to whole-part axiom.
4	∠ABC + ∠BAC + ∠ACB = ∠XAY	From statement (1), (2) and (3).
5	∠XAY = 180⁰	A straight angle constitutes 180^o(two right angles)
6	∴ ∠ABC + ∠BAC + ∠ACB = 180^oor 2 right angles.	Substituting the value of ∠XAY into a statement (4).

Proved

Theorem 2: The exterior angle formed by extending a side of a triangle is equal to the sum of other two non-adjacent angles.

Experimental verification:

Step 1: Let's draw three triangles of different size in different orientations. Produce side BC to D in each figure. In each figure, ∠ACD is an exterior angle and ∠A and ∠B are two non-adjacent angles inside the triangle.

Step 2: Measure the size of exterior angle and other two non-adjacent angles in each figure and complete the table.

Fig.	Exterior angle ∠ACD	∠A	∠B	Results
(i)
(ii)
(iii)

Conclusion:

Theoretical proof:

theorem 2

Given: ABC is a triangle whose side BC is extended upto D such that ∠ACD ia an external angle.

To prove: ∠ACD = ∠ABC + ∠BAC

Proof:

S.N.	Statement	Reasons
1	∠ACD + ∠BCA = 180^o	The sum of adjacent angles in a straight line is 180^o.
2	∠ABC + ∠BCA + ∠BAC = 180^o	The sum of angles in a triangle is 180^o.
3	∠ACD + ∠BCA = ∠ABC + ∠BCA + ∠BAC	From statements (1) and (2).
4	∠ACD = ∠ABC + ∠BAC	Cancelling ∠BCA on both sides of the statement (3).

Proved
Alternative method:

Given: ABC is a triangle whose side BC is extended up to D such that ∠ACD in an external angle.

To prove: ∠ACD = ∠ABC + ∠BAC

Construction: Let us draw CE parallel to BA.

Proof:

S.N.	Statements	Results
1	∠ACD = ∠ACE + ∠DCE	Whole part axiom.
2	∠DCE = ∠ABC	CE \(\parallel\) BA being corresponding angles.
3	∠ACE = ∠BAC	CE \(\parallel\) BA being alternate angles.
4	∠ACD = ∠BAC + ∠ABC	From statements (1), (2) and (3).

Proved

Verification of properties of Isosceles triangle

Isosceles triangle is that triangle whose two sides are equal. Triangle is symmetric being two sides equal hence it possesses different properties. For example, the base angles of an isosceles triangle are equal. This kind of properties is proved as theoretical proof here which duly needs the conditions of congruency of triangles. Before this, we discuss the different conditions of congruency of triangles in a brief.

Congruency of triangles

There are 3 sides and 3 angles in a triangle. Two triangles are congruent when 3 parts (out of 6 parts) of one triangle are equal to 3 corresponding parts of another triangle

as per the following conditions. We take these conditions as axioms.

i) S.S.S. axiom:

s.s.s axiom

The two triangles are said to be congruent when three sides of a triangle are respectively equal to three corresponding sides of another triangle under S.S.S. axiom.

In \(\triangle\)ABC and \(\triangle\)MNO ,

a) AB = MN (S)

b) BC = NO (S)

c) AC = MO (S)

∴ \(\triangle\)ABC ≅ \(\triangle\)MNO [S.S.S. axiom]

Corresponding parts of congruent triangles are also equal. i.e. ∠A =∠M, ∠B = ∠N and ∠C =∠O.

ii) S.A.S. axiom:

S.A.S

The two triangles are said to be congruent when two sides and an angle made by them of a triangle are respectively equal to the corresponding sides and an angle of another triangle, under S.A.S. axiom.

In \(\triangle\)ABC and \(\triangle\)PQR,

a) AB = PQ (S)

b) ∠B = ∠Q (A)

c) BC = QR (S)

∴ \(\triangle\)ABC = \(\triangle\)PQR [S.A.S. axiom]

Now, ∠C = ∠R and ∠A = ∠P [corresponding angles of congruent triangles]

AC = PR [Corresponding sides of congruent triangles]

iii) A.S.A. axiom:

ASA

The two of are said to be congruent when two angles and their adjacent side of one triangle are respectively equal to the corresponding angles and side of another triangle, under A.S.A. axiom.

In \(\triangle\)ABC and \(\triangle\)DEF,

a) ∠B = ∠E (A)

b) BC = EF (S)

c) ∠C = ∠F (A)

∴ \(\triangle\)ABC ≅ \(\triangle\)DEF [A.S.A. axiom]

Now, AC = DF and AB = DE

[Correspondong sides of congruent triangles]

∠A =∠D [Corresponding angles of congruent triangles]

iv) R.H.S. axiom:

RHS

The two right-angled triangles are said to be congruent when the hypotenuse and one of the remaining sides of both triangles are respectively equal, under R.H.S. axiom.

In right angled \(\triangle\)ABC and \(\triangle\)MNO,

a)∠B = ∠N (R)

b) AC = MO (H)

c) BC = NO (S)

∴ \(\triangle\)ABC ≅ \(\triangle\)MNO [ R.H.S. axiom]

Now, ∠C = ∠O and ∠A = ∠M [Corresponding angles of congruent triangles]

AB = MN [Corresponding sides of congruent triangles]

v) S.A.A. axiom:

SAA

The two triangles are said to be congruent when two angles and a side of one triangle are respectively equal to the corresponding angles and side of another triangle, under S.A.A. axiom. This axiom can be verified by using A.S.A. axiom.

Here,

a) ∠A = ∠D [Given]

b) ∠B = ∠E [Given]

c) ∠A + ∠B + ∠C = ∠D + ∠E + ∠F [Sum of the angles of any triangle is 180^o]

d) ∠C = ∠F [From (a), (b) and (c)]

Now, in \(\triangle\)ABC and \(\triangle\)DEF,

e) ∠B = ∠F (A) [Given]

f) BC = EF (S) [Given]

g) ∠C = ∠F (A) [From (d)]

h) \(\triangle\)ABC ≅ \(\triangle\)DEF [A.S.A. axiom]

Now, AB = DE and AC = DF [Corresponding sides of congruent triangles]

Isosceles triangle

Theorem 3:

Experimental verification:

Step 1: Draw three isosceles triangle ABC with different shapes and sizes in different orientation where AB = AC.

Step 2: Measure the angles opposite to the equal sides of each triangle and tabulate them in the table.figure

Conclusion:

Theoretical proof:

Given: \(\triangle\)ABC is an isosceles triangle where AB = AC.

To prove: ∠ABC = ∠ACB

Construction: Let us draw AD⊥BC, from the vertex A.

Proof:

S.N.	Statement	Reasons
1.	In \(\triangle\)ABD and \(\triangle\)ACD
i)	∠ADB = ∠ADC (R)	Both angles are right angle.
ii)	AB = AC (H)	Given
iii)	AD = AD (S)	Common side
2.	∴ \(\triangle\)ABD ≅ \(\triangle\)ACD	By RHS axiom.
3.	∠ABD = ∠ACD i.e. ∠ABC = ∠ACB	Corresponding angles of congruent triangles are equal.

Proved

Theorem 4: Conversion of theorem 3

Experimental verification:

Step 1: Draw three line segments BC of different lengths in different positions.

Step2: Equal sizes of angles at both points B and C on each line segment is drawn. Mark the points as A where the arms of these angles meet. Now three \(\triangle\)ABC are formed.

Step 3: Measure the length of each side opposite to the equal angles in each triangle (i.e. AB and AC) and tabulate them in the table.

figure

Conclusion:

Theoretical proof:

Theorem 4

Given: In \(\triangle\)ABC, base angles are equal i.e. ∠B =∠C.

To prove: AB = AC, \(\triangle\)ABC is an isosceles triangle.

Construction: From the vertex A, let us draw AD⊥BC.

Proof:

S.N.	Statement	Reasons
1.	In \(\triangle\)ABD and \(\triangle\)ACD
i)	AD = AD (S)	Common side
ii)	∠ADB = ∠ADC (A)	By construction, AD⊥BC, both angles are equal.
iii)	∠ABD = ∠ACD (A)	Given
2.	∴ \(\triangle\)ABD ≅ \(\triangle\)ACD	By S.A.A. axiom.
3.	AB = AC i.e. \(\triangle\)ABC is an isosceles triangle.	Corresponding sides of the congruent triangles are equal.

Proved

Theorem 5:

Experimental verification:

Step 1: Draw three isosceles triangle ABC with different positions and sizes such that AB = AC in each triangle.

Step 2: Draw the bisector of the vertex angle ∠A in each triangle. The bisector meets BC at D.

Step 3: Measure the lengths of BD and DC and the angles ADB and ADC, then tabulate them in the table.

Figure	BD	DC	Result	∠ADB	∠ADC	Result
i)
ii)
iii)

Conclusion:

Theoretical proof:

Theorem-5

Given: \(\triangle\)ABC is an isosceles triangle where AB = AC. AD is the bisector of∠BAC.

To prove: AD⊥BC and BD = DC.

Proof:

S.N.	Statement	Reasons
1.	In \(\triangle\)ADB and \(\triangle\)ACD
i)	AB = AC (S)	Given
ii)	∠BAD = ∠CAD (A)	Given
iii)	AD = AD (S)	Common side
2.	∴ \(\triangle\)ABD ≅ \(\triangle\)ACD	S.A.S. axiom
3.	∠ADB =∠ADC	Corresponding angles of congruent triangles are equal.
4.	AD⊥BC	Equal adjacent angles in a linear pair mean the line is perpendicular.
5.	BD = DC	Corresponding sides of congruent triangles are equal.
6.	AD is the perpendicular bisector of BC.	From statement 4 and 5.

Proved

Theorem 6: Converse of theorem 5

Experimental verification:

Step 1: Draw three isosceles triangles ABC with different positions and size in a different position such that AB = AC in each triangle.

Step 2: Mark the mid-point of BC at D. Join A and D in each triangle.

Step 3: Measure ∠ADB, ∠ADC, ∠BAD and ∠CAD and tabulate them below:

Figure	∠ADB	∠ADC	Result	∠BAD	∠DAC	Result
i)
ii)
iii)

Conclusion:

Theoretical proof:

Theorem 6

Given: \(\triangle\)ABC is an isosceles triangle where AB = AC.

AD joins the vertex A and mid-point D of the base BC.

To prove: AD⊥BC and ∠BAD = ∠CAD.

S.N.	Statements	Reasons
1.	In \(\triangle\)ADB and \(\triangle\)ACD
i)	AB = AC (S)	Given
ii)	AD = AD (S)	Common side
iii)	BD = DC (S)	Given
2.	∴ \(\triangle\)ABD ≅ \(\triangle\)ACD	By S.S.S. axiom
3.	∠ADB = ∠ADC	Corresponding angles of congruent triangles are equal.
4.	AD⊥ BC	Each angle of adjacent angles in a linear pair are 90⁰ both.
5.	∠BAD = ∠CAD	Corresponding angles of congruent triangles are equal.

Proved

Relation between sides and angles of triangle:

Experiment no.1: The sum of two sides of a triangle is greater than the third side.

Experimental verification:

Step 1: Three triangles ABC with different positions and sizes in different orientation are drawn.

Step 2: Measure all three sides of each triangle and fill up the table.

Figure	AB	BC	CA	AB + BC	BC + CA	AB + CA	Result
i)
ii)
iii)

Conclusion:The sum of two sides of a triangle is greater than the third side.

Experiment no.2: In any triangle, the angle opposite to the longer side is greater than the angle opposite to the shorter side.

Experimental verification:

Step 1: Three triangles ABC with different positions and size in different orientations are drawn in such a way that BC is the longest and CA is the shortest side in each triangle.

Step 2: The angle opposite to the greater side BC (i.e.∠B) are measured and tabulate in the table:

Figure	Angle opposite to BC:∠A	Angle opposite to CA:∠B	Result
i)
ii)
iii)

Conclusion: In any triangle, the angle opposite to the longer side is greater than the angle opposite to the shorter side.

Angle - Side relation

Experiment no.3: In any triangle, the side opposite to the greater angle is longer than the side opposite to the smaller angle.

Experimental verification:

Step 1: Three triangle ABC with different positions and sizes in different orientations are drawn in such a way that∠A is the greatest and∠B is the smallest angles in each triangle.

Step 2: Measure the side BC opposite to the greatest single angle A and the side CA opposite to the smallest angle B and fill up the table below:

Figure	Angle opposite to ∠A (BC)	Angle opposite to ∠B (CA)	Result
i)
ii)
iii)

Conclusion: In any triangle, the side opposite to the greater angle is longer than the side opposite to the smaller angle.

Experimental no.4: Among all straight line segments down to a given line from a point outside it, the perpendicular is the shortest line.

Experimental verification:

Step 1: Three straight line segments XY of different length in different orientations are drawn.A point P is taken outside of each line segment. Three line segments PA, PB, PC and a perpendicular line PM are drawn from P to XY.

Step 2: Measure the lengths of each line segment PA, PB, PC and PM. Then tabulate them below:

Figure	PA	PB	PC	PM
i
ii
iii

Conclusion: Among all straight line segments down to a given line form a point outside it, the perpendicular is the shortest line.

Experiment no.5: The square of the hypotenuse in a right-angled triangle is equal to the sum of squares of its base and perpendicular.

Step 1: Draw three right-angled triangles of different dimensions.

Step 2: Fill up the given table.

Figure	AB	BC	CA	AB²	BC²	CA²	AB² + BC²	Results
i)
ii)
iii)

Conclusion:The square of hypotenuse in right-angled triangle is equal to the sum of squares of it's base and perpendicular.

Experiment no.5 is known as Pythagoras Theorem. Around 2500 years ago, Pythagoras was a Greek Mathematician. He invented a fact on the right-angled triangle which is named as Pythagoras Theorem. This theorem is used in all other fields of mathematics not only used in geometry.

Lesson

Geometry

Subject

Compulsory Maths

Grade

Grade 9

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