Question

1 multiple choice 15 points based on the construction below, which conclusion is not always true? 1) \\( \overline{ab} \perp \overline{cd} \\) 2) \\( ab = cd \\) 3) \\( ae = eb \\) 4) \\( ce = de \\)

Explanation:

Step1: Analyze Construction Type

The construction shows two intersecting lines (AB and CD) with arcs, indicating perpendicular bisector construction. So, AB is the perpendicular bisector of CD, and CD is the perpendicular bisector of AB.

Step2: Evaluate Each Option

• Option 1: $\overline{AB} \perp \overline{CD}$ – True, as it's a perpendicular bisector construction.
• Option 2: $AB = CD$ – Not always true. The perpendicular bisector construction ensures bisecting (so $AE = EB$, $CE = DE$) and perpendicularity, but lengths of AB and CD can differ.
• Option 3: $AE = EB$ – True, since AB is bisected by CD.
• Option 4: $CE = DE$ – True, since CD is bisected by AB.

Answer:

1. $AB = CD$