Question

Prove that the triangle EDF is isosceles. Give reasons for your answer.

Answer 1

`\triangle EDF` is isosceles.

Explanation:

Please have a look at the attached figure.

We are given the following things:

`\angle EDF = y`

`\text{External }\angle DFG = 90 +(y)/(2)`

Let us try to find out
`\angle E` and
`\angle DFE`. After that we will compare them.

Finding
`\angle DFE`:

Side EG is a straight line so
`\angle GFE = 180`

`\angle GFE` is sum of internal
`\angle DFE` and external
`\angle DFG`

`\angle GFE = 180 = \angle DFE + \angle DFG\\\Rightarrow 180 = \angle DFE + (90+(y)/(2))\\\Rightarrow \angle DFE = 180 - 90 - (y)/(2)\\\Rightarrow \angle DFE = 90 - (y)/(2) ....... (1)`

Finding
`\angle E`:

Property of external angle: External angle in a triangle is equal to the sum of two opposite internal angles of a triangle.

i.e. external
`\angle DFG` =
`\angle E + \angle EDF`

`\Rightarrow 90+(y)/(2) = \angle E + y\\\Rightarrow \angle E = 90+(y)/(2) -y\\\Rightarrow \angle E = 90-(y)/(2) ....... (2)`

Comparing equations (1) and (2):

It can be clearly seen that:

`\angle DFE = \angle E =90-(y)/(2)`

The two angles of
`\triangle EDF` are equal hence
`\triangle EDF` is isosceles.