Question

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

Answer 1

Proved

Explanation:

Required;

Prove that the above triangle is a triangle

An isosceles triangle has 2 equal sides and 2 equal angles;

But in this case, we're dealing with only angles.

Once, we can prove that triangle EDF has 2 angles, then it has been proven that it is an isosceles triangle.

The first step is to calculate
`<DFE`

First, it should be noted that angle on a straight line equals 180;

This implies that

`<DFE + 90 + (y)/(2) = 180`

Subtract
`90 +(y)/(2)` from both sides

`<DFE + 90 + (y)/(2) - ( 90 + (y)/(2))= 180 - ( 90 + (y)/(2))`

`<DFE + 90 + (y)/(2) - 90 - (y)/(2)= 180 - 90 - (y)/(2)`

`<DFE = 90 - (y)/(2)`

The next step is to calculate the measure of
`<DE F`

It should be noted that angles in a triangle add up to 180;

This implies that

`<DE F + <DFE + <EDF = 180`

Where

`<EDF = y --- (Given)\\<DFE = 90 - (y)/(2) --- (Calculated)`

`<DE F + <DFE + <EDF = 180` becomes

`<DE F + 90 - (y)/(2) + y = 180`

Subtract 90 from both sides

`<DE F + 90 - (y)/(2) + y - 90 = 180 - 90`

`<DE F - (y)/(2) + y = 90`

Perform arithmetic operation on y and -y/2

`<DE F + (-y +2y)/(2) = 90`

`<DE F + (y)/(2) = 90`

Subtract y/2 from both sides

`<DE F + (y)/(2) - (y)/(2) = 90 - (y)/(2)`

`<DE F = 90 - (y)/(2)`

At this point, it has been proven that triangle EDF is isosceles because
`<DE F = <DFE = 90 - (y)/(2)`