Download Triangle

Triangle
A triangle is a geometrical figure. Tri means three. So Triangle is a geometrical figure having 3
angles.
A triangle is consisting of three line segments linked end to end. As the figure linked end to end,
triangle is a closed figure. The line segments are sides of that triangle. The starting common
point of two line segments is the vertex of triangle. The angle between two adjacent sides is the
angle of that triangle.
In any triangle the angle opposite to smallest side is smallest and angle opposite to largest side is
largest.
Conventionally; in a triangle side opposite to any angle is written by the same small letter. As
shown in the following figure. Side opposite to angle A is ‘a’.
Sum of three angles of any triangle is always
.
We can write name of any triangle by writing vertices of that triangle in order.
Activity: Write the name of above triangle in 6 different ways.
For
ABC
Pair of sides
Angle
Vertex
AB, BC
BC, AC
AB, AC
ABC
BCA
BAC
B
C
A
The closed region of plane of a triangle bounded by the three sides of a triangle is known as the
interior of that triangle.
The interior and three sides of a triangle together form a triangular region.
Exterior of a triangle is the region outside the sides of that triangle.
Sides of a triangle are boundaries of interior and exterior of a triangle.
Triangle, interior of a triangle and exterior of a triangle are disjoint parts.
For
ABC
Points
Interior points
P, Q, S, R
Points on Triangle
A, B, C, E, F, H
Exterior points
X, Y, Z
Triangle and its exterior are non-convex sets but interior is a convex set.
The triangle has exterior angles also. The exterior angle is an angle between one side of a
triangle and the extension of an adjacent side.



The exterior angle of a triangle forms a linear pair with the adjacent interior angle.
Every angle of a triangle has 2 exterior angles.
The measure of exterior angle is equal to the sum of measure of remote interior angles of
that triangle.
For
ABC
Exterior angle
ACF
BAD
CBE
Adjacent interior
angle
ACB
BAC
CBA
Remote interior
angles
CAB, CBA
ABC, ACB
BCA, BAC
Activity: Draw and write the same table for other exterior angles of given
ABC.
For the same refer the following link.
https://www.youtube.com/watch?v=s4KPzc6rSyE
Depending upon the angles of a triangle, there are 3 types.



Acute angle triangle
Right angle triangle
Obtuse angle triangle
Acute angle triangle: A triangle whose all
three interior angles are acute (less than
.)
Right angle triangle: A triangle in which one of the interior
angle is of measure
.
Obtuse angle triangle: A triangle whose one of the
interior angles is obtuse (greater than
.)
Depending upon the sides of a triangle, there are 3 types.



Equilateral triangle
Isosceles triangle
Scalene triangle
Equilateral triangle: A triangle having all three sides of
equal length is called an equilateral triangle.
1. All three angles are equal in measure.
2. Each angle is of
.
3. This triangle is always an acute angle triangle.
4. Also known as Equiangular triangle.
Isosceles triangle: A triangle which has two of its sides
equal in length is called a isosceles triangle.
1. Angles opposite to equal sides are equal.
2. May be acute angled or right angle or obtuse angled triangle.
3. All equilateral triangles are isosceles triangle.
Scalene Triangle: A triangle having all three sides
of different lengths is called scalene triangle.
1. May be acute angled or right angle or obtuse angled triangle.
http://www.youtube.com/watch?v=diS6TJLpqBk
 The next part is theorems based on Triangles.
1st Theorem
Statement: The sum of the measures of the angles of a triangle is
Given:
ABC is any triangle.
To prove:
ABC +
ACB +
BAC =
Previous knowledge:
1) a + b + c =
- straight angles
.
.
2) Angles form by 2 parallel lines and a transversal –Alternate angles
Activity:
Proof: Observing the following figure and above two points write the
proof in statement reason form.
Statement
Reason
1
2
3
4
5
Using the above theorem and properties try to prove the following corollary’s in minimum steps.
Corollary:
1. If measures of all angles of triangles are equal then measure of each is
2. If one angle of triangle is right angle then other two are supplementary.
3. If one of the angles of a triangle is obtuse then other two are acute.
.
2nd Theorem – Remote interior angles’ theorem
Statement: The measure of an exterior angle of a triangle is equal to the sum of the measures of
its remote interior angles.
Given: In
ABC,
1. P is the point on ray BC such that B – C – P.
2.
ACP is an exterior angle of the triangle.
To prove:
ACP =
BAC +
ABC
Previous knowledge:
1. a + b =
– angles in a linear pair.
2. The 1st theorem - The sum of the measures of the angles of a triangle is
.
Activity:
Proof: Observing the given figure and above two points write the
proof in statement reason form.
Statement
Reason
1
2
3
4
5
Using the above theorem and properties try to prove the following corollary’s in minimum steps.
Corollary:
1. The measure of any one of exterior angles of a triangle is
then the triangle is right
angled triangle.
2. Sum of measures of exterior angles of a triangle in same sense is always
.
3rd Theorem:
Statement: Exterior angle of a triangle is always greater than each of its remote interior angle.
Given: In
ABC,
1. P is the point on ray BC such that B – C – P.
2.
ACP is an exterior angle of the triangle.
To prove:
ACP
ABC and
ACP
ACB
Previous knowledge:
1. If a + b = c then c > a and c > b; where a, b , c each is non zero.
2. The measure of an exterior angle of a triangle is equal to the sum of the measures of its
remote interior angles.
Proof: Observing the given figure and above two points write the
proof in statement reason form.
Statement
Reason
1
2
3
4
5
 Refer S.S.C textbooks for writing proofs of theorems.