Postulate Addition Segnent 2 Postulate POSTULATE: A C, Is A Rule This .

ulate POSTULATE: A rule this is accepted without proof because it’ s something obvious. PARALLEL LINE POSTULATE: Through a point not a given line, there is a line parallel to the given line. THEOREM: A rule that is proven to be true. BC x 6, and PARALLEL LINE POSTULATE: x*6 6 Through a point not a given line, there is a line parallel to the given line RULER POSTULATE 1. Any two points can have coordinates 0 and 1 2. Distance a – b SEGMENT ADDITION POSTULE: If B is in between A and C, then AB BC AC CONGRUENT SEGMENTS: Segments that have same shape and equal lengths. CONGRUENT Same Shape and Equal in Size qual you Z"). y be the ure, DE FG MIDPOINT DEFINTION: .2a2. IF M is the midpoint of AB , DE FG DE 3 FG 3 A point that divides the segment into two congruent segments. then !###" !##" AM MB and AM MB lane is a MIDPOINT FORMULA: (A B) / 2 If B is midpoint in between A and C, and A -1, C -5 Then B [-1 (-5)] / 2 (-6)/2 -3 Points, Lines, Planes, and Angles MIDOINT THEOREM: /7 If M is the midpoint of AB then AM MB 1 AB 2

exactly one line through ,4 and edge B. We agree that we can use a straight- to draw iE or parts of the line, such as lS ana AB. Using a Compass in Constructions CONSTRUCTIONS www.mathopenref.com Given a point O and a length r, we know from the definition of a circle one circle with center O and radius r. We agree that there is exactly CIRCLE: Set compass to a specific radius length. Construct a Circle. to draw this circle or arcs of the circle. compass that we can use aThe distance from the center of the circle to the circle is the . All points on a circle are from the center Construction 7 CONGRUENT SEGMENTS Given a segment, construct a segment congruent to the given segment. Given: AB Construct: A segment congruent to ,,q.A Procedure: L Use a straightedge to draw a line. Call it /. 2. Choose any point on / and label it X. 3. Set your compass for radius ,48. Using X as center, draw an arc intersecting line l. Label the point of intersection X XY is congruent to l,B. Justification: Since AB was used for the radius of OX, X)'is congruentto AB. Written Exercises CONSTRUCTING SEGMENTS WITH SPECIFIC LENGHTS Constructions and Loci On your paper, draw two segments roughly like those shown. Use these lengths in Exercises l-4 to construct a segment having the indicated length. A l.a b 2.b-a 3.3a-b 4.a 2b 5. Using any convenient length for a side, construct an equilateral triangle. 6. a. Construct a 30' angle. b. Construct a 15' angle. 7. Draw any acute LACU. Use a method based on the SSS Postulate to construct a triangle congruent to LACU. 8. Draw any obtuse LOBT. Use the SSS method to construct a triangle congruent to LOBT. 9. Repeat Exercise 7, but use the SAS method. 10. Repeat Exercise 8, but use the ASA method. On your paper, draw two angles roughly like those shown. Then for Exercises / 335

8-2 Perpendiculars and Parallels The next three constructions are based on the following theorem and postulate. (l) If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. (2) Through any two points there is exactly one line. PERPENDICULAR BISECTOR Construction 4 Given a segment, construct the perpendicular bisector of the segment Given: AB Construct: The perpendicular bisector of AB Procedure: l. Using any radius greater thanlAB, draw four arcs of equal radii, two with center I and two with center B. Label the points of intersection of these arcs X and y. 2. Draw 7i. A XI is the perpendicular bisector of AB. Justification: Points X and Y are equidistant from,4 and B. Thus i7 is ttre perpendicular bisector of AB. Construction 5 Constructions and Loci Given a point on a line, construct the perpendicular to the line at the given point. Given: C Point C on line k Construct: The perpendicular to k at C Procedure: 1. Using C as center and any radius, draw arcs intersecting k at X and Y. 2. Using X as center and a radius greater than CX, draw an arc. Using Y as center and the same radius, draw at arc intersecting the first arc at Z. 3. Draw tZ. E i, p"rp"ndicular to k at C. Justification: Points X and Y were constructed so that C is equidistant from X and X Then point Z was constructed so that Z is equidistant from X and Y. Sirce tZ is the perpendicular bisector of {y, tZ is perpendicular to k at C. Construction 6 Given a point outside a line, construct the perpendicular to the line from the given point. P. / 339

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PARALLEL LINE POSTULATE: Through a point not a given line, there is a line parallel to the given line RULER POSTULATE BC 1. Any two points can have coordinates 0 and 1 2. Distance a - b SEGMENT ADDITION POSTULE: If B is in between A and C, then AB BC AC CONGRUENT SEGMENTS: Segments that have same shape and equal lengths.