3. The Triangle and its Properties
Triangle is a simple closed curve made of three line
segments.
Triangle has three vertices, three sides and three angles.
In Δ ABC
Sides: AB, BC and CA
Angles: ∠BAC, ∠ABC and ∠BCA
Vertices: A, B and C
The side opposite to the vertex A is BC.
4. Classification of triangles
Based on the sides
Scalene Triangles
No equal sides
No equal angles
Isosceles Triangles
Two equal sides
Two equal angles
Equilateral Triangles
Three equal sides
Three equal angles,
always 60°
Scalene
Isosceles
Equilateral
5. Classification of triangles
Based on Angles
Acute-angled Triangle
All angles are less than 90°
Obtuse-angled Triangle
Has an angle more than 90°
Right-angled triangles
Has a right angle (90°)
Acute
Triangle
Obtuse
Triangle
Right
Triangle
6. MEDIANS OF A TRIANGLE
A median of a triangle is a line segment joining
a vertex to the midpoint of the opposite side
A triangle has three medians.
• The three medians always meet at a single point.
• Each median divides the triangle into two smaller
triangles which have the same area
• The centroid (point where they meet) is the center of gravity of
the triangle
.
7. ALTITUDES OF A TRIANGLE
• Altitude – line segment from a vertex
that intersects the opposite side at a
right angle.
Any triangle has three altitudes.
8. Definition of an Altitude of a Triangle
A segment is an altitude of a triangle if and only if it
has one endpoint at a vertex of a triangle and the
other on the line that contains the side opposite that
vertex so that the segment is perpendicular to this line.
B
A
ACUTE OBTUSE
C
ALTITUDES OF A TRIANGLE
9. ALTITUDES OF A TRIANGLE
Can a side of a triangle be its altitude? YES!
G
RIGHT
A
B C
If ABC is a right triangle, identify its altitudes.
BG, AB and BC are its altitudes.
10. The measure of the three angles of a triangle sum
to 1800 .
To Prove : A + B + C = 1800
Proof: C + D + E = 1800 ……..Straight line
A = D and B = E….Alternate angles
C + B + A = 1800
A + B + C = 1800
D E
Given: Triangle
C
A B
Construction: Draw line ‘l’ through C parallel
to the base AB
l
ANGLE SUM PROPERTY OF A
TRIANGLE
11. EXTERIOR ANGLE OF A TRIANGLE
AND ITS PROPERTY
An exterior angle of a triangle equals the sum of the
two interior opposite angles in measure.
Given: In Δ ABC extend BC
to D
To Prove: ACD = ABC + BAC
A
B C D
Proof: CB + ACD = 1800 …………………. Straight line
ABC + ACB + BAC = 1800 …………………sum of the triangle
ACB + ACD = ABC + ACB + BAC
ACD = ABC + BAC
12. PYTHAGORAS THEOREM
In a right angled triangle the square of the hypotenuse is
equal to the sum of the squares of the other two sides.
In ABC :
• AC is the hypotenuse
• AB and BC are the 2 sides
A
B C
Then according to Pythagoras theorem ,
AC² = AB² + BC²
