Question

lesson 8 practice problems
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 the 2 lines 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: if a triangle is isosceles, 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: Vertical angles are congruent, but congruent angles don't have to be vertical (e.g., two 30° angles in different triangles). So the converse is false.
• Option B: By definition, perpendicular lines form 4 right angles, and if 2 lines form 4 right angles, they must be perpendicular. The converse is true.
• Option C: The perpendicular bisector theorem states a point equidistant from endpoints lies on the perpendicular bisector, and its converse (a point on the perpendicular bisector is equidistant from endpoints) is also a theorem, so the converse is true.
• Option D: In a triangle, if base angles are congruent, the triangle is isosceles (converse of the isosceles triangle theorem), so the converse is true.
• Option E: Supplementary angles sum to 180°, but they don't have to form a straight angle (e.g., two angles in different places each 90°). So the converse is false.

Answer:

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 the 2 lines 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: If a triangle is isosceles, the base angles are congruent. Converse: If the base angles of a triangle are congruent, then the triangle is isosceles.