# Exterior and Interior Angles of Triangle

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 110

^{0}form a linear pair angles.

∠ABC + 110 = 180 (Linear pair angles are supplementary)

∠ABC = 180 – 110

∴ ∠ABC = 70

^{0}

By exterior and interior angles of triangle theorem

∠ACB + ∠BAC = 110

^{0}

60 + ∠BAC = 110

∴ ∠BAC = 110 – 60

∴ ∠BAC = 50

^{0}

-------------------------------------------------------------

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 = 35

^{0}

∠b = ∠DAB + ∠ADB ( Exterior angle theorem)

∠b = 35 + 35 = 70

^{0}

Similarly in ΔACE

AC = CE (given)

∴ ΔACE is an isosceles triangle.

∠AEC = ∠CAE

∠CAE= 46

^{0}∠c = ∠CAE + ∠AEC

∠c = 46 + 46 = 92

^{0}

∠a = 180 – (∠b + ∠c )

∠a = 18

^{0}

∴ ∠DAE = ∠DAB + ∠EAC + ∠a

∴ ∠DAE = 35 + 46 + 18

∴ ∠DAE = 99

^{0}

**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

• 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