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Altitude

activity
Cut out a triangular shaped cardboard ABC.
Place it upright on a table.
Take a ruler and place it upright over the triangle from vertex A to the opposite A
side BC perpendicularly.
Now draw a line segment starting from Vertex A that comes straight down L
to BC, and is perpendicular to BC. Mark the point L at which the line segment B C
intersects BC.
The length of this line segment AL represents the height of the triangle ABC
from vertex A. Thus, the line segment AL is the altitude of the triangle ABC.

Hence, the altitude of a triangle is a line segment that is Remember
perpendicular to the side opposite to a vertex and passes through An altitude has one end point at a
that vertex. It is also known as the height of the triangle. vertex of the triangle and the other on
Or, if a perpendicular line is drawn from the vertex of a triangle the line containing the opposite side.
to the opposite side, it is called the altitude of the triangle.
A
Through each vertex, an altitude can be drawn. Thus, a triangle has three altitudes.

In the adjoining figure in DABC, AD, BE, and CF are the altitudes drawn from the F E
vertex A, B, and C, respectively.

B D C
Quick Check

Draw altitudes from A to BC for the following triangles.
A
A A

B C B C B C

Exterior Angle of a Triangle

We know that a triangle has three angles which is also called interior angles. A
If any side of a triangle is extended, then the angle between this extended
line and the adjacent side is known as an exterior angle.

In the adjacent figure, side BC is extended to point D. Thus, ∠ACD is the B C D
exterior angle between the side AC and the extended line CD.
The measure of an exterior angle is always equal to the sum of the measurement of the interior
opposite angles.
i.e., ∠ACD = ∠CAB + ∠CBA

Justification: In DABC, let us draw a line CE parallel to the side AB and name the angles formed.
Since, AB || CE and AC is the transversal.

173 The Triangle and Its Properties