Chapter Closure
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ii.What is the sum of m∠ b and m∠ e?
iii. What positions with respect to the transversal do the angles have?
iv. What shape is formed by three angles?
ii. What kind of angle does ∠ c, ∠ d, and ∠ e form together?
iii. What kind of angles do ∠ a and ∠ e form?
iv. Consider the position of ∠ g and ∠ c.
ii. Straight angle pair, supplementary
iii. Alternate interior angles, congruent
iv. Triangle angle sum is 180^(∘)
ii. 93^(∘), straight angle with ∠ e and ∠ c.
iii. 55^(∘), vertical to ∠ e
iv. 32^(∘), alternate interior angle to ∠ c
i. These angles have corresponding positions with respect to the transversal. Therefore, these are corresponding angles. Since the lines cut by the transversal are parallel, we can use the Corresponding Angles Theorem and say that they are congruent. This means that the measures are equal.
ii. ∠ b and ∠ e are adjacent angles with a noncommon side that form opposite rays. This fits the description of a straight angle pair which means they are supplementary angles.
iii. The angles both lie between the two lines but on opposite sides of the transversal which makes them alternate interior angles. Because the lines cut by the transversal are parallel, we can use the Alternate Interior Angles Theorem and say that they are congruent. This means that the measures are equal.
iv. Each of the three angles occupy a vertex of a triangle. Therefore, by the Triangle Angle Sum Theorem, we know they sum to 180^(∘).
iii. From the diagram, we see that ∠ a and ∠ e are vertical angles. Therefore, by the Vertical Angles Congruence Theorem, we know that they are congruent making the angle measures equal. ∠ a ≅ ∠ e ⇒ m∠ a = m∠ e iv. From the diagram, we see that m∠ g and m∠ c are alternate interior angles. Because the lines cut by the transversal are parallel, we can by the Alternative Interior Angles Theorem say that they are congruent. This means that the measures are equal. ∠ g ≅ ∠ c ⇒ m∠ g = m∠ c