Construction Techniques: Perpendicular Bisectors

Based on the construction below, which statement must be true?

• m<ABD = 1/2 m<CBD
• m<ABD = m<CBD
• m<ABD = m<ABC
• m<CBD = 1/2 m<ABD

As shown in the diagram below of triangle ABC, a compass is used to find points D and E, equidistant from point A. Next, the compass is used to find point F, equidistant from points D and E. Finally, a straight-edge is used to draw ray AF. Then, point G, the intersection of ray AF and side BC of triangle ABC, is labeled. Which statement must be true?

• Ray AF bisects side BC
• Ray AF bisects <BAC
• Ray AF is perpendicular to BC
• Triangle ABG ~ Triangle ACG

Which diagram shows the construction of a perpendicular bisector of line segment AB?

• A
• B
• C
• D

In the accompanying figure, triangle ABC is scalene. The construction shows that CD is the 

• Median to side AB
• Bisector of <C
• Altitude of side AB
• Perpendicular bisector to side AB

A straightedge and compass were use to create the construction below. Arc EF was drawn from point B, and arcs with equal radii were drawn from E and F. Which statement is false?

• m<ABD=m<DBC
• 1/2(m<ABC)=m<ABD
• 2(m<DBC)=m<ABC
• 2(M<ABC)=M<CBD

In triangle DEF, EG bisects DF. Which of the following is a valid conclusion?

• DG is congruent to DF
• EG is congruent to GF
• <1 is congruent to <2
• DG = 1/2 DF

In constructing a perpendicular from an outside point P to line l as shown below, the main reason that PD is perpendicular to line l is that

• Two points equidistant from the endpoints of a line segment determine the perpendicular bisector of the segment
• Perpendicular bisector of the segment 
• The shortest distance from a point to a line can never be determined
• Intersecting lines are either parallel or perpendicular

The diagram below shows the construction of line AB through point P parallel to line CD. Which theorem justifies this method of construction?

• If two lines in a plane are perpendicular to a transversal at different points, then the lines are parallel
• If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel
• If two lines in a plane are cut by a transversal to form congruent alternate interior angles, then the lines are parallel

Which geometric principle is used to justify the construction below?

• A line perpendicular to one of two parallel lines is perpendicular to the other
• Two lines are perpendicular if they intersect to form congruent adjacent angles
• When two lines are intersected by a transversal and alternate interior angles are congruent, the lines are parallel
• When two lines are intersected by a traversal and the coorespondingangles are congruent, the lines are parallel

In the accompanying diagram of a construction, what does PC represent?

• An altitude drawn to AB
• A median drawn to AB
• The bisector of <APB
• The perpendicular bisector of AB