Question

1. each statement is always true. select all statements for which the converse is also always true.

a. statement: if 2 angles are vertical, then they are congruent. converse: if 2 angles are congruent, then they are vertical.
b. statement: if 2 lines are perpendicular, then they intersect to form 4 right angles. converse: if 2 lines intersect to form 4 right angles, then they are perpendicular.
c. statement: if a point is equidistant from the 2 endpoints of a segment, then it lies on the perpendicular bisector of the segment. converse: if a point lies on the perpendicular bisector of a segment, then it is equidistant from the 2 endpoints of the segment.
d. statement: in an isosceles triangle, the base angles are congruent. converse: if the base angles of a triangle are congruent, then the triangle is isosceles.
e. statement: if 2 angles form a straight angle, then they are supplementary. converse: if 2 angles are supplementary, then they form a straight angle.

Explanation:

• Option A: Congruent angles are not necessarily vertical. For example, two angles in a parallelogram (that are not vertical angles) can be congruent. So the converse is false.
• Option B: By the definition of perpendicular lines, if two lines intersect to form 4 right angles, they are perpendicular. The converse is true.
• Option C: By the perpendicular bisector theorem, if a point lies on the perpendicular bisector of a segment, it is equidistant from the two endpoints of the segment. The converse is true.
• Option D: By the isosceles triangle theorem (converse), if the base angles of a triangle are congruent, then the triangle is isosceles. The converse is true.
• Option E: Supplementary angles (sum to \(180^{\circ}\)) do not have to form a straight - angle. They can be two non - adjacent angles. So the converse is false.

Answer:

B. If 2 lines intersect to form 4 right angles, then they are perpendicular.
C. If a point lies on the perpendicular bisector of a segment, then it is equidistant from the 2 endpoints of the segment.
D. If the base angles of a triangle are congruent, then the triangle is isosceles.