Question

1. which construction could be used to construct an isosceles triangle abc given line segment ab? a. mark a third point c not on segment ab. draw segments ac and bc. b. label a point c on segment ab and construct a line perpendicular to ab through point c. draw segments ac and bc. c. construct the perpendicular bisector of segment ab. mark the intersection of this line and ab and label it c. draw segments ac and bc. d. construct the perpendicular bisector of segment ab. mark any point c on the perpendicular bisector except where it intersects ab. draw segments ac and bc. 5. select all true statements about regular polygons. a. all angles are right angles. b. all angles are congruent. c. all side lengths are equal. d. there are exactly 4 sides. e. there are at least 3 sides. (from unit 1, lesson 7.)

Explanation:

Step1: Recall isosceles - triangle property

An isosceles triangle has two equal - length sides. If we construct the perpendicular bisector of segment \(AB\), any point \(C\) on the perpendicular bisector (except the mid - point of \(AB\)) will be equidistant from \(A\) and \(B\) (by the property of the perpendicular bisector: any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment). So, \(AC = BC\), and \(\triangle ABC\) is isosceles.

Step2: Recall regular - polygon properties

A regular polygon is a polygon that is equilateral (all side lengths are equal) and equiangular (all angles are congruent). Also, a polygon must have at least 3 sides.

• Option A: Not all regular polygons have right angles. For example, a regular hexagon does not have right angles.
• Option B: All angles are congruent in a regular polygon.
• Option C: All side lengths are equal in a regular polygon.
• Option D: A regular polygon does not have exactly 4 sides. There are regular polygons with 3 sides (equilateral triangle), 5 sides (regular pentagon), etc.
• Option E: A polygon is defined as a closed - figure with at least 3 sides, and a regular polygon is a type of polygon, so it has at least 3 sides.

Answer:

1. D. Construct the perpendicular bisector of segment \(AB\). Mark any point \(C\) on the perpendicular bisector except where it intersects \(AB\). Draw segments \(AC\) and \(BC\).
2. B. All angles are congruent., C. All side lengths are equal., E. There are at least 3 sides.