FIND ANGLES OF ISOCELES TRAINGLE

An isosceles triangle is a triangle in which two sides are equal in length.

The two sides which are having equal measures will have equal angles.

Example 1 :

Solution :

Since AB  =  BC, <C  =  <A

The Sum of interior angles of a triangle is 180˚

So, <A + <B + <C  =  180˚

72˚+x˚+72˚  =  180˚

144˚+x˚  =  180˚

x  =  36˚

<B  =  36˚

So, the missing angle x is 36˚.

Example 2 :

Solution :

Since PQ  =  PR, <PQR  =  <PRQ  =  x

So, <P + <Q + <R  =  180˚

70˚ + x˚ + x˚   =  180˚

70˚ + 2x˚  =  180˚

2x˚  =  180˚ - 70˚

x  =  110/2

x  =  55˚

<Q  =  <R  =  55˚

<Q  =  55˚ and <R  =  55˚

So, the missing angle x is 55˚

Example 3 :

Solution :

Since AB  =  AC, <B  =  <C

<B  =  <C  =  (2x)˚

The Sum of interior angles of a triangle is 180˚

So, <A + <B + <C  =  180˚

x˚ + 2x˚ + 2x˚  =  180˚

x˚ + 4x˚  =  180˚

5x˚  =  180˚

x˚  =  180/5

x  =  36˚

So, the missing angle x is 36˚

Example 4 :

Since RQ  =  RP

<P  =  <Q

x˚  =  (146 – x)˚

x  =  146 – x

x + x  =  146

2x  =  146

x  =  146/2

x  =  73˚

So, the missing angle x is 73˚.

Example 5 :

Solution :

<B  =  <D

<DBC + <DBA  =  180˚ (linear pair of angles)

<DBC + 120˚  =  180˚

<DBC  =  180˚-120˚

<DBC  =  60˚

<B  =  60˚

<B  =  <D  =  60˚

The Sum of interior angles of a triangle is 180˚

<C + <B + <D  =  180˚

x˚ + 60˚ + 60˚  =  180˚

x˚ + 120˚  =  180˚

x  =  180˚ - 120˚

x  =  60˚

So, the missing angle x is 60˚.

Example 6 :

Solution :

Since AD  =  DB, ∆ABC is an isosceles triangle.

DB  =  BC, ∆DBC is an isosceles triangle.

<DAB  =  <DBA  =  65

<DBA + <DBC  =  180

65 + <DBC  =  180

<DBC  =  115

In triangle DBC.

Let x be <BCD and <CDB

<DBC + <BCD + <CDB  =  180

115 + x + x  =  180

2x  =  180-115

2x  =  65

x  =  32.5

Example 7 :

Solution :

Since <B  =  <C  =  75˚

AB  =  x cm, AC  =  16 cm

AB  =  AC

x  =  16 cm

Example 8 :

Solution :

DA  =  DB

BD  =  9 cm

So,

x  =  BD

x  =  9 cm and DA  =  9 cm

Example 9 :

Solution :

∆ABC is an isosceles triangle (AB  =  AC).

 base <B  =  <C are equal.

M is the midpoint of the angle bisects the base at right angles.

<AMC  =  <AMB  =  90˚

x  =  90˚

Example 10 :

The figure alongside has not been drawn accurately:

a) Find x.

b) What can be deduced about the triangle?

Solution :

(a)  <A + <B + <C  =  180

x + x + 24 + 52  =  180

2x + 76  =  180

2x  =  104

x  =  52

(b)  In the above triangle, two angles are equal. So it is isosceles triangle.

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