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
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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