Question

For questions 1 through 4, use the diagram below.#1. If a l b, then which of the following conjectures is true?a. If a l b, then angles 3 and 10 are congruent by alternate interior angles.b. If a I b, then angles 3 and 4 are congruent by alternate interior angles theorem.c. If all b, then angle 1 is congruent to angle 9 because corresponding angles are congruent.If a ll b, then corresponding angles 3 and 8 are congruent.d.2. Which of the following conjectures cannot be made?a. If c ll d, then 4 8 and 4 13 are supplementary.b. Ifc | d, then & 9 is congruent to 4 7.c. If c ll d, then 47 is congruent to $13.d. If c ll d, then 41 is congruent to 493. If 84 249, then what conclusion can be made?a. If 44 249, then c ll d by corresponding angles converse.b. If 44 % 49, then cl d by same side interior angle converse.C. If 44 % 49, then cll d by vertical angle converse.d. If 44 249, then cll d by alternate interior angle converse.4. If 41 % 413, then what conclusion can be made?

Answer 1

Using the concept of transversal line passing through parallel lines, we can say that ∠3 is congruent to ∠8 since the two angles are corresponding angles.

Note that since ∠1 and ∠9 are on the same line, we cannot say that the two angles are congruent. On the otherhand, ∠3 and ∠10 are also on the same line and ∠3 and ∠4 are also on line a. This means that the angles cannot be alternate interior angles.

Thus, the correct answer for question #1 is option D.

For question #2, since ∠8 and ∠13 are interior angles on the same side of the transversal, the two angles must be supplementary.

On the other hand, ∠7 and ∠13 are alternate interior angles. Thus, the angles must be congruent.

Similarly, ∠1 and ∠9 are corresponding angles. Thus, the angles are also congruent.

However, ∠9 and ∠7 are not on the same transversal line. Therefore, we cannot obtain a conjecture on these angles. Thus, the correct answer must be option B.

For question #3, if ∠4 and ∠9 are congruent, then line c and line d must be parallel. Thus, ∠4 and ∠9 must be alternate interior angles. This means the correct answer is option D.

For question 4, if ∠1 and ∠13 are congruent, then ∠1 are ∠13 must be alternate exterior angles. Thus, line c and line d must be parallel. This means the correct answer must be option A.