Question

select all the true statements. a. ∠3≅∠2 because they are alternate interior angles. b. m∠1 + m∠3 = 180 because they form a straight angle. c. ∠3≅∠6 because they are alternate interior angles. d. ∠1 and ∠6 are supplementary because ∠3≅∠6 and m∠1 + m∠3 = 180. e. ∠1≅∠3 because they are vertical angles.

Explanation:

Step1: Recall angle - pair relationships

Alternate interior angles are non - adjacent angles between two parallel lines and on opposite sides of a transversal. Vertical angles are opposite each other when two lines intersect. Supplementary angles add up to 180 degrees.

Step2: Analyze option A

∠3 and ∠2 are not alternate interior angles. Alternate interior angles for the given parallel lines and transversal would be pairs like ∠3 and ∠6 or ∠4 and ∠5. So, option A is false.

Step3: Analyze option B

∠1 and ∠3 form a straight - angle. A straight - angle measures 180 degrees. So, \(m\angle1 + m\angle3=180\). Option B is true.

Step4: Analyze option C

∠3 and ∠6 are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are congruent. So, ∠3≅∠6. Option C is true.

Step5: Analyze option D

Since ∠3≅∠6 (alternate interior angles) and \(m\angle1 + m\angle3 = 180\) (linear pair), we can substitute ∠3 with ∠6. Then \(m\angle1 + m\angle6=180\), which means ∠1 and ∠6 are supplementary. Option D is true.

Step6: Analyze option E

∠1 and ∠3 are not vertical angles. Vertical angles are formed when two lines intersect, and ∠1 and ∠3 are adjacent angles that form a straight - angle. So, option E is false.

Answer:

B. \(m\angle1 + m\angle3 = 180\) because they form a straight angle.
C. ∠3≅∠6 because they are alternate interior angles.
D. ∠1 and ∠6 are supplementary because ∠3≅∠6 and \(m\angle1 + m\angle3 = 180\).