When two segments have the same length, they are said to be congruent segments. If AB = AC Measure of segments Congruent Segments then AB = AC A BC Is read: “ Segment AB is congruent to segment AC” Is read: “ The measure of segment AB is equal to the measure of segment AC” Do modeling activity on page 36: Locating the midpoint of a segment You can also compare the measure of segments. For example you can say: AB AC Congruent Segments

The midpoint of a segment is the point equidistant from the endpoints of the segment. Definition of Midpoint: The midpoint M of PQ is the point between P and Q such that PM=MQ P Q M MIDPOINTS AND SEGMENT CONGRUENCE...

 On the number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is a+b 2  In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x 1, y 1 ) and (x 2, y 2 ) are (x 1 +x 2, y 1 +y 2 ) 2 2 Midpoint Formulas

Example 1 (p.37) a) Use the number line to find the coordinates of the midpoint of FG b) Find the coordinates of Q, the midpoint of RS, if the endpoints of RS are R (-3,-4) and S (5,7). F G -3 –2 – Examples

Example 2: p. 38 (Students do then I explain)  Find the coordinates of point Q if L(4,-6) is the midpoint of NQ and the coordinates of N are (8,-9)  If Y is the midpoint of XZ, XY = 2a + 11, and YZ = 4a –5, find the value of a and the the measure of XZ More examples

Segment bisector: is a segment line or plane that intersects a segment at its midpoint. P Q M T..... R N M M, TM, RM and plane N are all bisectors of PQ Do construction p. 39 Segment Bisector..

 Proof of theorem 1-1: Given that M is the midpoint of AB, write a paragraph proof to show that AM = MB. (Theorem 1-1) Midpoint theorem: If M is the midpoint of AB then AM = MB The Midpoint Theorem

A proof: is a logical argument in which each statement you make is backed up by a statement that is accepted as true. Paragraph proof or informal proof: a paragraph that explains why a conjecture for a given situation is true. Conjecture: is an educated guess. What are proofs?

From the definition of midpoint of a segment, we know that AM = MB. A. B. M. That means that AM and MB have the same measures By the definition of congruence, if AM and MB have the same measure, they are congruent segments. Thus, AM = MB. Paragraph proof of theorem 1-1