Answer:
a = 30
b = 15
c = 3
d = 30
e = 10
f = 20
Explanation:
According to the Alternate Interior Angles Theorem, when two parallel lines are intersected by a transversal, the angles that are interior to the parallel lines and on the alternate sides of the transversal are congruent. Therefore:
(a + 30)° = 60°
a + 30 = 60
a = 30
According to the Corresponding Angles Postulate, when a straight line intersects two parallel straight lines, the resulting corresponding angles are congruent. Therefore:
(a + 30)° = (a + 2b)°
a + 30 = a + 2b
30 = 2b
b = 15
According to the Vertical Angles Theorem, when two straight lines intersect, the opposite vertical angles are congruent. Therefore:
(a + 2b)° = (5b - 5c)°
(30 + 2(15))° = (5(15) - 5c)°
60° = (75 - 5c)°
60 = 75 - 5c
5c = 75 - 60
5c = 15
c = 3
According to the Corresponding Angles Postulate, when a straight line intersects two parallel straight lines, the resulting corresponding angles are congruent. Therefore:
(a + 2b)° = (10c + d)°
(30 + 2(15))° = (10(3) + d)°
60° = (30 + d)°
60 = 30 + d
d = 30
Angles on a straight line sum to 180°. Therefore:
(a + 2b)° + (2d + 6e)° = 180°
(30 + 2(15))° + (2(30) + 6e)° = 180°
60° + (60 + 6e)° = 180°
60 + 60 + 6e = 180
120 + 6e = 180
6e = 60
e = 10
According to the Same-side Interior Angles Theorem, when two parallel lines are intersected by a transversal, the angles that are interior to the parallel lines and on the same side of the transversal line are supplementary (sum to 180°). Therefore:
(a + 2b)° + (4f + 4e)° = 180°
(30 + 2(15))° + (4f + 4(10))° = 180°
60° + (4f + 40)° = 180°
60 + 4f + 40 = 180
100 + 4f = 180
4f = 80
f = 20