**The Triangle and Its Properties**

**Exercise 6.3**

1). Find the value of the unknown *x *in the following diagrams:

i).

by angle sum property of a triangle

mÐA + mÐB + mÐC = 180^{0}

*x* + 50^{0} + 60^{0} = 180^{0 }

*x* + 110^{0} = 180^{0}

* x* = 180^{0 }– 110^{0}

*x* = 70^{0}

ii).

by angle sum property of a triangle

mÐP + mÐQ + mÐR = 180^{0}

90^{0} + 30^{0 }+ *x* = 180^{0 }

*x* + 120^{0} = 180^{0}

* x* = 180^{0 }– 120^{0}

*x* = 60^{0}

iii).

by angle sum property of a triangle

mÐX + mÐY + mÐZ = 180^{0}

30^{0} + 110^{0 }+ *x* = 180^{0 }

*x* + 140^{0} = 180^{0}

* x* = 180^{0 }– 140^{0}

*x* = 40^{0}

iv).

by angle sum property of a triangle

*x* + *x* + 50^{0} = 180^{0 }

2*x* + 50^{0} = 180^{0}

* *2*x* = 180^{0 }– 50^{0}

2*x* = 130^{0}

*x* = 130/2

*x* = 65^{0}

^{ }v).

by angle sum property of a triangle

*x* + *x* + *x* = 180^{0 }

3*x* = 180^{0}

*x* = 180^{0 }/3

*x* = 60^{0}

^{ }vi).

by angle sum property of a triangle

*x* + 2*x* + 90^{0} = 180^{0 }

3*x* = 180^{0 }– 90^{0}

3*x* = 90^{0 }

*x* = 90^{0}/3

*x* = 30^{0}

Ans: x = 30^{0}

2x = 60^{0}

2). Find the value of the unknown *x *and* y *in the following figures:

i).

the sum of interior opposite angles = exterior angle

50^{0} + *x* = 120^{0}

*x* = 120^{0 }– 50^{0}

*x* = 70^{0}

by angle sum property of a triangle

50^{0} + *x* + *y* = 180^{0 }

50^{0} + 70^{0 }+ *y* = 180^{0 }

*y* + 120^{0} = 180^{0}

* y* = 180^{0 }– 120^{0}

*y* = 60^{0}

Ans: *x* = 70^{0}

*y* = 60^{0}

ii).

vertically opposite angles are congruent

*y* = 80^{0}

by angle sum property of a triangle

50^{0} + *x* + *y* = 180^{0 }

50^{0} + *x* + 80^{0} = 180^{0 }

*x* + 130^{0} = 180^{0}

* x* = 180^{0 }– 130^{0}

*x* = 50^{0}

Ans: *x* = 50^{0}

*y* = 80^{0}

iii).

the sum of interior opposite angles = exterior angle

50^{0} + 60^{0} = *x*

*x* = 50^{0} + 60^{0}

*x* = 110^{0}

by angle sum property of a triangle

50^{0} + 60^{0 }+ *y* = 180^{0 }

110^{0} + *y* = 180^{0 }

*y* = 180^{0} – 110^{0}

* y* = 70^{0 }

Ans: *x* = 110^{0}

*y* = 70^{0}

iv).

vertically opposite angles are congruent

*x* = 60^{0}

by angle sum property of a triangle

*x* + *y* + 30^{0} = 180^{0 }

60^{0} + *y* + 30^{0} = 180^{0 }

*y* + 90^{0} = 180^{0}

* y* = 180^{0 }– 90^{0}

*y* = 90^{0}

Ans: *x* = 60^{0}

*y* = 90^{0}

v).

vertically opposite angles are congruent

*x* = 90^{0}

by angle sum property of a triangle

*x* + *x* + 90^{0} = 180^{0 }

2*x* + 90^{0}= 180^{0 }

2*x* = 180^{0 }– 90^{0}

2*x* = 90^{0}

*x *= 90^{0 }/ 2

*x*= 45^{0}

Ans: *x* = 45^{0}

*y* = 90^{0}

vi).

vertically opposite angles are congruent

*x* = *y*

by angle sum property of a triangle

*x* + *x* + *x* = 180^{0 }

3*x* = 180^{0 }

*x* = 180^{0 }/3

*x* = 60^{0}

*x *=* y*

*y *= 60^{0}

Ans: *x* = 60^{0}

*y* = 60^{0}

Click here for the solutions of