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: a 2 + b 2 = c 2, 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 triangle  Measured 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.)