Exterior Angle of a Triangle and its Property
In a triangle ABC, and extend one of its sides say, BC to point D.
∠ACD is formed at the vertex C. This angle lies in the exterior of ∆ABC. We call it an exterior angle of the ∆ABC formed at vertex C.
We can see that, ∠BCA is an adjacent angle to ∠ACD. The remaining two angles of the triangle namely ∠A and ∠B are called the two interior opposite angles or the two remote interior angles of ∠ACD.
Draw a line from C to E.
Now we have created a line segment CE which is parallel to AB. Move the vertex A and observe how the angles are changing accordingly.
We see that ∠BAC =
We see that ∠ABC =
We observe m ∠ACD = m ∠ACE + m ∠ECD = m ∠BAC + m ∠ABC.
Let us prove this statement.
The exterior angle of a triangle is equal to the sum of its two interior opposite angles
As we saw in the above experiment, we extended BC to D where ∠ACD is an
We drew CE through C , parallel to BA.
Need to be proved:
∠ACD(exterior angle)= ∠BAC + ∠ABC where ∠BAC, ∠ABC are two
Proof
AB || CE (by construction) AC is a
Therefore, alternate angles should be
∠BAC=∠ACE AB || CE (by construction) BC is a
Therefore, alternate angles should be
∠ABC = ∠ECD
∠ACD = ∠ACE + ∠ECD = ∠
Hence, ∠ACD = ∠BAC + ∠ABC.
The above relation between an exterior angle and its two interior opposite angles is referred to as the Exterior Angle Property of a triangle.
Exterior angles can be formed for a triangle in many ways.
Click on the BC ray line after C.
Click on the CA ray line after A.
Click on the AB ray line after B.
There are three more ways of getting exterior angles. Try to produce those rough sketches.
Are the exterior angles formed at each vertex of a triangle always equal?
The exterior angles will be equal if the triangle is equilateral, else they will be different.
The sum of an exterior angle of a triangle and its adjacent interior angle is
Find angle x in below figure:
Fig
Solution:
Sum of interior opposite angles = Exterior angle
(or) 50° + x =
(or) x =
1. An exterior angle of a triangle is of measure 70º and one of its interior opposite angles is of measure 25º. Find the measure of the other interior opposite angle.
Solution:
Given:
An exterior angle of a triangle is
One of the interior opposite angles is
To find:
The measure of the other
Let ∠ACD = 70° (exterior angle).
Let ∠A = 25° (one of the interior
Use the Exterior Angle Theorem:
The exterior angle of a triangle is
Therefore: ∠ACD = ∠A + ∠
Substitute the Known Values:
We have ∠ACD = 70° and ∠A = 25°:
70° =
Solve for ∠B:
Subtract 25° from both sides of the equation:
∠B = 70° - 25° =
The measure of the other interior opposite angle is 45°.
2. The two interior opposite angles of an exterior angle of a triangle are 60° and 80°. Find the measure of the exterior angle.
Solution:
interior opposite angles
Given:
The two interior opposite angles of an exterior angle of a triangle are
Let ∠A = 60° and ∠B = 80° be the two
Use the Exterior Angle Theorem:
The exterior angle of a triangle is
Therefore: ∠ACD = ∠A + ∠
Substitute the Known Values:
We have ∠A =
∠ACD = 60° +
∠ACD =
Therefore,the measure of the exterior angle is 140°.
3. Is something wrong in this diagram? Comment.
Fig
Solution:
In the given diagram, the measure of the exterior angle is
However, according to the properties of triangles, the exterior angle should be
If we assume the triangle is △ABC and the exterior angle is formed by extending side
Exterior Angle = Sum of the two opposite interior angles
Given the two interior angles are 50° each, the exterior angle should be:
50° + 50° =
Thus, the correct measure of the exterior angle should be 100°, not 50°. The given exterior angle in the diagram is incorrect.