Question

1. use a compass and a straight edge to construct an angle bisector of angle ∠cde shown below. leave all construction marks 11) based on the construction below, which conclusion is not always true? 1) $overline{ab}perpoverline{cd}$ 2) $ab = cd$ 3) $ae = eb$ 4) $ce = de$ 11a) explain your answer choice. provide evidence to support your explanation.

Explanation:

Step1: Construction of angle - bisector

Place the compass at vertex D of ∠CDE. Draw an arc that intersects the two rays of the angle (DC and DE) at two distinct points. Let these points be F and G. Then, from points F and G, draw arcs of equal radius inside the angle. The point of intersection of these two arcs is on the angle - bisector. Draw a ray from D through this intersection point to construct the angle - bisector.

Step2: Analyze statements for question 11

1. For $\overline{AB}\perp\overline{CD}$: In a general angle - bisector construction, there is no reason for a line related to the construction (like AB) to be perpendicular to CD. This is not always true.
2. For $AB = CD$: There is no geometric relationship in angle - bisector construction that implies $AB = CD$. But this is not the key property related to angle - bisector construction.
3. For $AE=EB$: In a proper angle - bisector construction, if AB is related to the construction (for example, if it is part of the process to find equidistant points), it is possible to have $AE = EB$ when constructing the angle - bisector.
4. For $CE = DE$: When constructing an angle - bisector, if we consider the points of intersection of the initial arc with the sides of the angle, we can have $CE = DE$ (when the initial arc is drawn from the vertex of the angle with a fixed radius intersecting the sides).

Answer:

1. $\overline{AB}\perp\overline{CD}$