2. 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.
3. 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°
Classification of triangles
Scalene
Isosceles
Equilateral
4. 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
Right
Triangle
Obtuse
Triangle
5. 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
.
6. ALTITUDES OF A TRIANGLE
• Altitude – line segment from a vertex
that intersects the opposite side at a
right angle.
Any triangle has three altitudes.
7. 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.
ACUTE OBTUSE
B
A
C
ALTITUDES OF A TRIANGLE
8. RIGHT
A
B C
If ABC is a right triangle, identify its altitudes.
BG, AB and BC are its altitudes.
G
Can a side of a triangle be its altitude? YES!
ALTITUDES OF A TRIANGLE
9. 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
A B
C
Construction: Draw line ‘l’ through C parallel
to the base AB
The measure of the three angles of a triangle sum
to 1800 .
To Prove : A + B + C = 1800
l
ANGLE SUM PROPERTY OF A TRIANGLE
10. An exterior angle of a triangle equals the sum of the
two interior opposite angles in measure.
To Prove: ACD = ABC + BAC
Proof: CB + ACD = 1800 …………………. Straight line
ABC + ACB + BAC = 1800 …………………sum of the triangle
ACB + ACD = ABC + ACB + BAC
ACD = ABC + BAC
A
B C D
Given: In Δ ABC extend BC
to D
EXTERIOR ANGLE OF A TRIANGLE AND
ITS PROPERTY
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