Answer:
Proved
Step-by-step 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 [tex]<DFE[/tex]
First, it should be noted that angle on a straight line equals 180;
This implies that
[tex]<DFE + 90 + \frac{y}{2} = 180[/tex]
Subtract [tex]90 +\frac{y}{2}[/tex] from both sides
[tex]<DFE + 90 + \frac{y}{2} - ( 90 + \frac{y}{2})= 180 - ( 90 + \frac{y}{2})[/tex]
[tex]<DFE + 90 + \frac{y}{2} - 90 - \frac{y}{2}= 180 - 90 - \frac{y}{2}[/tex]
[tex]<DFE = 90 - \frac{y}{2}[/tex]
The next step is to calculate the measure of [tex]<DE F[/tex]
It should be noted that angles in a triangle add up to 180;
This implies that
[tex]<DE F + <DFE + <EDF = 180[/tex]
Where
[tex]<EDF = y --- (Given)\\<DFE = 90 - \frac{y}{2} --- (Calculated)[/tex]
[tex]<DE F + <DFE + <EDF = 180[/tex] becomes
[tex]<DE F + 90 - \frac{y}{2} + y = 180[/tex]
Subtract 90 from both sides
[tex]<DE F + 90 - \frac{y}{2} + y - 90 = 180 - 90[/tex]
[tex]<DE F - \frac{y}{2} + y = 90[/tex]
Perform arithmetic operation on y and -y/2
[tex]<DE F + \frac{-y +2y}{2} = 90[/tex]
[tex]<DE F + \frac{y}{2} = 90[/tex]
Subtract y/2 from both sides
[tex]<DE F + \frac{y}{2} - \frac{y}{2} = 90 - \frac{y}{2}[/tex]
[tex]<DE F = 90 - \frac{y}{2}[/tex]
At this point, it has been proven that triangle EDF is isosceles because [tex]<DE F = <DFE = 90 - \frac{y}{2}[/tex]
