Final answer:
To prove the given statements, we use the fact that ∠2 and ∠5 are supplementary to show that ∠4 = ∠7 and ∠4 + ∠7 = 180°. We also prove that ∠4 ≠ 180° ∠7 and that ∠4 and ∠7 are not complementary.
Step-by-step explanation:
To prove:
a. ∠4 = ∠7
b. ∠4 + ∠7 = 180°
c. ∠4 ≠ 180° ∠7
d. ∠4 and ∠7 are complementary
Proof:
Given: ∠2 and ∠5 are supplementary, which means ∠2 + ∠5 = 180°
To prove ∠4 = ∠7, we need to show that ∠4 equals the same measure as ∠7. Since ∠2 and ∠5 are supplementary, ∠2 + ∠5 = 180°. Then, we can deduce that ∠4 = ∠2 and ∠7 = ∠5. Therefore, ∠4 = ∠7.
To prove that ∠4 + ∠7 = 180°, we can substitute the measures of ∠2 and ∠5. From the given information, ∠2 + ∠5 = 180°, which means ∠4 + ∠7 = 180°.
To prove that ∠4 ≠ 180° ∠7, we can use the fact that ∠4 = ∠2 and ∠7 = ∠5. If ∠2 + ∠5 = 180°, it does not mean ∠4 + ∠7 = 180° ∠7. Therefore, ∠4 ≠ 180° ∠7.
To prove that ∠4 and ∠7 are complementary, we need to show that the sum of their measures is 90 degrees. Since ∠4 = ∠2 and ∠7 = ∠5, if ∠2 + ∠5 = 180° then ∠4 + ∠7 = 180°. Therefore, ∠4 and ∠7 are not complementary.