Answer:
(a) ∠4 = 115°
(b) ∠7 = 65°
(c) ∠3 = 65°
Explanation:
There are several applicable relationships in this figure. Perhaps the simplest is to take advantage of the relationships wherever any lines cross:
- vertical angles are congruent
- angles forming a linear pair are supplementary
When two parallel lines are crossed by a transversal, the Corresponding Angle Theorem comes into play. It tells you ...
- corresponding angles are congruent.
The rule for vertical angles gives you ...
- ∠1 ≅ ∠4
- ∠2 ≅ ∠3
- ∠5 ≅ ∠8
- ∠6 ≅ ∠7
The rule for corresponding angles tells you ...
Angles 2 and 4 are a linear pair, so are supplementary.
There are many other relationships that can be derived from these, but these are sufficient to provide a basis for answering the questions.
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(a)
Angle 4 is supplementary to angle 2 so measures ...
∠4 = 180° -65° = 115°
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(b)
Angle 7 is a vertical angle to angle 6, which is a corresponding angle with angle 2. Vertical and corresponding angles are congruent, so ...
∠7 ≅ ∠6 ≅ ∠2 = 65°
(See also the comment below.)
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(c)
Angle 3 is a vertical angle to angle 2, so is congruent to it.
∠3 ≅ ∠2 = 65°
(Angle 3 is also "corresponding" to angle 7, which we just showed was 65°.)
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Additional comment
When the transversal crosses parallel lines at other than 90°, all obtuse angles are congruent, and all acute angles are congruent. The obtuse and acute angles are supplementary. Once you know one acute angle is 65°, you know all acute angles are 65°, and all obtuse angles are 115°.
Various theorems relate different angles pairs. For example, angles 2 and 7 are called "alternate exterior angles" (as are angles 1 and 8). There is a theorem that says alternate exterior angles are congruent. We arrived at the same fact using the congruence of corresponding and vertical angles.