Pearson Geometry Common Core, 2011
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Pearson Geometry Common Core, 2011 View details
2. Perpendicular and Angle Bisectors
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Exercise 25 Page 297

Practice makes perfect
a Given the segment a compass and straightedge can be used to draw its perpendicular bisector.

Place the compass' sharp end at one of the segment's endpoints. Draw an arc with a radius larger than half the distance between and

Keeping the compass length the same, draw a corresponding arc on the opposite side. The two arcs should now intersect at two distinct points.

The line that contains these two intersections can now be drawn using a straightedge.

This line is perpendicular to and their intersection is at the midpoint of

Finally, we will draw a segment from and to an arbitrary point on the perpendicular bisector, to create

b According to the Perpendicular Bisector Theorem, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If we can show that the distance from and to our arbitrary point are congruent, we know that is an isosceles triangle.

We can prove that if we show that Notice that and form a linear pair, and since it must be that as well. We also notice that the triangles share as a side, which means it is congruent by the Reflexive Property of Congruence.

Now we can claim that by the SAS (Side-Angle-Side) Congruence Theorem. Since and are corresponding sides we know that they are congruent, which means is equidistant from the endpoints of Therefore, which means is an isosceles triangle.