Answer:
Part a: m ∠ F = 72°
Part b: DF is the shortest side.
Step-by-step explanation:
Part a:
Given from the diagram:
m ∠ E = 36°
and
DE = FE
Since, DE = FE
So, m ∠ D = m ∠ F
Note: Angle opposite to the equal sides of an isosceles triangle are also equal.
In Δ DEF,
m ∠ D + m ∠ E + m ∠ F = 180° {Sum of interior angles of a triangle is 180°}
Substitute 36° for E in the equation
m ∠ D + m ∠ 36° + m ∠ F = 180°
Subtract 36 from both sides
m ∠ D + m ∠ 36° - 36° + m ∠ F = 180° - 36°
Simplify
m ∠ D + m ∠ F = 144° {Since m ∠ D = m ∠ F, we can put m ∠ F + m ∠ F}
Change Equation
m ∠ F + m ∠ F = 144°
Add m ∠ F + m ∠ F and get 2 m ∠ F
2 m ∠ F = 144°
Divide by 2 on both sides
[tex]\mathrm{\dfrac{ 2\:m\angle\:F }{ 2 } = \dfrac{ 144\° }{ 2 }}[/tex]
Simplify
m ∠ F = 72°
Part b:
In a triangle, the shortest side is always opposite the smallest interior angle. Here in Δ DEF, the smallest angle is ∠ E. The side opposite to ∠ E is DF, so DF is the shortest side.
