5-2 Bisectors in Triangles What is a perpendicular bisector? a line or line segment that cuts another line or line segment into two congruent pieces and makes a right angle perpendicular bisectors can also cut triangles into two congruent pieces Theorem 5-2

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

Theorem 5-3 Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Converse? What is a converse statement? a converse switches the hypothesis and the conclusion of a conditional Conditional? What is a conditional!? a conditional is an "if-then" statement, like "if it rains today, then we stay inside." Equidistant? equidistant means that each point is the exact same distance from another point. Like, the 400 wing and the 100 wing are equidistant from the chapel. Use the information given in the diagram. Line CD is the perpendicular bisector of line segment AB. Find CA and DB. Explain your reasoning. Example 1 What do we know? CB = 5 and AD = 6 What do we need to find? The measure of CA and DB what term did we use when we talked about the chapel? Which Theorem proves that point A and point B are EQUIDISTANT from point C? So, if the distance from point C to point B is 5, then what is the distance from point C to point A? BECAUSE the the perpendicular bisector cuts line segment AB into two congruent pieces, we know that point A and point B are equidistant from point C CB = 5, AD = 6 Given CD is a perpendicular bisector of line segment AB Given CA = CB and DA = DB Perpendicular Bisector Theorem CA = 5 and DB = 6 Theorem 5-4 Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point in the interior of an angle is equidistant from the sides of the angle, the the point if on the angle bisector. Theorem 5-5 Converse of the Angle Bisector Theorem Example 2 Find the value of x, then find FD and FB. From the diagram you can see that F is on the bisector of angle ACE. Therefore, FB = FD. Subtract 2x. 5x = 2x +24 divide by 3. 3x = 24 Substitute. x = 8 FB = 5x = 5(8) = 40 -2x -2x FD = 40 FB =FD Substitute. Substitute. Homework: pg. 251

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# 5-2 Bisectors of Triangles

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