Answer:
27. 48°, 132°
28. The expressions resolve to both angles being 0°, a contradiction to the lines being distinct.
Explanation:
Given a figure showing parallel lines crossed by a transversal with marked angles, you want to know (27) the measures of consecutive interior angles that have a ratio of 4:11, and (28) the contradiction in corresponding angles being marked (60-2x)° and (2x-60)°.
27. Algebra.
The two marked angles are consecutive interior angles, so are supplementary. Using x for the smaller angle value, we can write the equation ...
x/(180-x) = 4/11 . . . . . the given ratio of the two angles
Multiplying by 11(180-x) gives ...
11x = 4(180-x)
15x = 4(180) . . . . add 4x and simplify
x = 48 . . . . . . . . divide by 15
Then the obtuse angle is ...
180 -48 = 132
The measures of the angles are ∠1 = 48°, ∠2 = 132°.
2. Error Analysis.
The two marked corresponding angles are congruent, so ...
(60 -2x)° = (2x -60)°
4x = 120 . . . . . . . . . . . divide by °, add 2x+60
x = 30 . . . . . . . . . . divide by 4
60 -2x = 60 -2(30) = 0
2x -60 = 2(30) -60 = 0
The marked values require the angles to be 0°, which means the transversal is coincident with each of the "parallel" lines.
The given angle values require the lines not to be distinct, as the diagram shows they are.