Answer:
[tex]x=50\\ \\m\angle a=m\angle (x+75)^{\circ}=m\angle (2x+25)^{\circ}=m\angle e=125^{\circ}\\ \\m\angle b=m\angle c=m\angle d=m\angle f=55^{\circ}[/tex]
Step-by-step explanation:
Part A.
Angles with measures [tex](2x+25)^{\circ}[/tex] and [tex](x+75)^{\circ}[/tex] are alternate interior angles when two parallel lines are cut by a transversal. By Alternate Interior Angles theorem,
[tex]2x+25=x+75\\ \\2x-x=75-25\\ \\x=50[/tex]
Part B.
Angles a and [tex](x+75)^{\circ}[/tex] are congruent as vertical angles, so
[tex]m\angle a=(50+75)^{\circ}=125^{\circ}[/tex]
Angles [tex](x+75)^{\circ}[/tex] and d are the same side interior angles, so the add up to 180°, thus
[tex]m\angle d=180^{\circ}-125^{\circ}=55^{\circ}[/tex]
Angles d and f are congruent as vertical angles.
Angles e and [tex](2x+25)^{\circ}[/tex] are congruent as vertical angles.
Angles c and d are congruent as alternate interior angles.
Angles c and b are congruent as vertical angles.
Therefore,
[tex]m\angle a=m\angle (x+75)^{\circ}=m\angle (2x+25)^{\circ}=m\angle e=125^{\circ}\\ \\m\angle b=m\angle c=m\angle d=m\angle f=55^{\circ}[/tex]
