Chapter 5

Congruent Triangles

5.1 Perpendiculars and Bisectors Perpendicular Bisector: segment, line, or

ray that is perpendicular and cuts a figure into two parts

Equidistant: same distance away Distance from Point to Line: the space

between a point and a line

Perpendicular Bisector Theorem:If a point is on a perpendicular bisector, then it is equidistant from the segment endpoints.

Converse of Perpendicular Bisector Theorem:If a point is equidistant from the segment endpoints, then the point is on the perpendicular bisector.

Angle Bisector Theorem: If a point is on an angle bisector, then it is equidistant from the sides of the angle

Converse of Angle Bisector Theorem:If a point is equidistant from the sides of an angle, then it lies on the angle bisector.

Example 1: Using Perpendicular Bisectors

a. AB = BC

b. P is on the perpendicular bisector

Example 2: Using Angle Bisectors

It will be half of 90°, or 45°

Checkpoint

Point Q will be on the perpendicular bisector QS

No, you don’t know any equal lengths of sides PS and RS

Checkpoint

DC = 6, because it must be equidistant to the sides of the angle

5.2 Perpendiculars and Bisectors Perpendicular Bisector of a Triangle: segment

from the midpoint of each side extended perpendicular to the interior of the triangle

Concurrent Lines: lines that intersect in one point

Circumcenter: point where all perpendicular bisectors intersect

Angle Bisector: ray that divides the angle in half

Pythagorean Theorem: a2 + b2 = c2, where “c” is the hypotenuse

5.2 Continued Incenter: point where all angle bisectors

intersect Perpendicular Bisector Theorem: the circumcenter is equal distance away from the three vertices

5.2 Continued Angle Bisector Theorem: the incenter is

equal distance from the sides of the triangleMeasured perpendicular to the sides

Example 1: Using Perpendicular Bisectors

Find the perpendicular bisectors of each side of the triangle. The concurrent point (circumcenter) is the equidistant buoy

Checkpoint

PS = QS = RS because S is the circumcenter. So, RS = 10 which means PS = 10 and QS = 10

Example 2: Using Angle Bisectors

HM = HN = HP because of incenter

Pythagorean Theorem now has to be used to find missing lengths

Checkpoint

So W is the incenter and XW = WY = WZ.

2d + 7 = 4d – 1 (Solve.)