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The Triangle and its PropertiesTriangle 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.
Based on the sidesScalene Triangles
No equal sidesNo equal angles
Isosceles TrianglesTwo equal sides Two equal angles
Equilateral TrianglesThree equal sidesThree equal angles, always 60°
Classification of triangles
Scalene
Isosceles
Equilateral
Classification of triangles Based on AnglesAcute-angled Triangle
All angles are less than 90°
Obtuse-angled TriangleHas an angle more than 90°
Right-angled triangles Has a right angle (90°)
Acute Triangl
e
Right Triangl
e
Obtuse Triangle
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.
ALTITUDES OF A TRIANGLE• Altitude – line segment from a vertex
that intersects the opposite side at a right angle.
Any triangle has three altitudes.
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
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
Proof: ÐC + ÐD + ÐE = 1800
……..Straight line
ÐA = ÐD and ÐB = ÐE….Alternate angles
Þ ÐC + ÐB + ÐA = 1800
ÐA + ÐB + ÐC = 1800
D E
Given: Triangle
A B
CConstruction: Draw line ‘l’ through ÐC parallel
to the base AB
The measure of the three angles of a triangle sum to 1800 .
To Prove : ÐA + ÐB + ÐC = 1800
l
ANGLE SUM PROPERTY OF A TRIANGLE
An exterior angle of a triangle equals the sum of the two interior opposite angles in measure.
To Prove: ÐACD = ÐABC + ÐBAC
Proof: ÐACB + ÐACD = 1800 …………………. Straight line
ÐABC + ÐACB + ÐBAC = 1800 …………………sum of the triangle
Þ ÐACB + ÐACD = ÐABC + ÐACB + ÐBAC
Þ ÐACD = ÐABC + ÐBAC
A
B C D
Given: In Δ ABC extend BC to D
EXTERIOR ANGLE OF A TRIANGLE AND ITS PROPERTY
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