Exterior and Interior Angles of Triangle

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In this section we will discuss about exterior and interior angles of triangle and the relation between them.


Exterior Angle : If the side BC of a triangle ABC is produced to form ray BX, then ∠ACX is called an Exterior angle of ΔABC at C.

Interior Opposite Angle : According to exterior angle ACX in triangle ABC, ∠BAC and ∠ABC are Interior Opposite Angles.

Interior Adjacent Angle : According to exterior angle ACX the interior adjacent angle in triangle ABC is ∠ACB.

Relation between exterior and interior angles of triangle.
Theorem : The measure of exterior angle is equal to the sum of two interior opposite angles.


Given : ABC is a triangle.

Prove that : ∠ACX = ∠A + ∠B

Statements	Reasons
1) ABC is a triangle.	1) Given
2) ∠A + ∠B + ∠ACB = 180	2) By angle sum property.
3) ∠ACB + ∠ACX = 180	3) Linear pair angles and they are supplementary.
4) ∠A + ∠B + ∠ACB = ∠ACB + ∠ACX	4) From (2) and (3)
5) ∠A + ∠B = ∠ACX	5) Subtraction property.

Examples :

1)
From the above figure, find 1) ∠ABC 2) ∠BAC.
Solution : ∠ABC and 110⁰ form a linear pair angles.
∠ABC + 110 = 180 (Linear pair angles are supplementary)
∠ABC = 180 – 110
∴ ∠ABC = 70⁰
By exterior and interior angles of triangle theorem
∠ACB + ∠BAC = 110⁰
60 + ∠BAC = 110
∴ ∠BAC = 110 – 60
∴ ∠BAC = 50⁰
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2)
From the above figure, find 1) ∠b 2) ∠C 3) ∠DAE
Solution : AB = DB (given)
So ΔABD is an isosceles triangle.
∠BDA = ∠DAB
∠ DAB = 35⁰
∠b = ∠DAB + ∠ADB ( Exterior angle theorem)
∠b = 35 + 35 = 70⁰
Similarly in ΔACE
AC = CE (given)
∴ ΔACE is an isosceles triangle.
∠AEC = ∠CAE
∠CAE= 46⁰ ∠c = ∠CAE + ∠AEC
∠c = 46 + 46 = 92⁰
∠a = 180 – (∠b + ∠c )
∠a = 18⁰
∴ ∠DAE = ∠DAB + ∠EAC + ∠a
∴ ∠DAE = 35 + 46 + 18
∴ ∠DAE = 99⁰

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Triangles

• Introduction to Triangles
• Types of Triangles on the basis of Sides
• Types of Triangles on the basis of Angles
• Angle Sum Property of Triangles
• Exterior and Interior angles of Triangle
• Triangle Inequality Property
• Congruent Triangles
• Postulates of Congruent Triangle
• Inequality in Triangle

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