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Name: Date: Page 1 of 2 Activity 5.2.2b Proof of the Perpendicular Bisector Theorem The Perpendicular Bisector Theorems says that the locus of points that are equidistant from the endpoints of a segment is the perpendicular bisector of the segment. To prove this theorem we need to prove two things: (1) If a point lies on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the segment, and (2) If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Prove part (1): Given: β‘π«πͺ is the perpendicular bisector of Μ Μ Μ Μ π¨π© β‘ P lies on π«πͺ Prove: PA = PB Fill in the blanks to complete the proof below Statements Reasons β‘π«πͺ is the perpendicular bisector of Μ Μ Μ Μ π¨π© β‘ and P lies on π«πͺ Given AC = CB (a) Definition of _________________________ β π΄πΆπ and β π΅πΆπ are both right angles (b) Definition of _________________________ (c) _______________________________ All right angles are congruent PC = PC (d) ___________________________________ (e) _______________________________ (f) ___________________________________ (g) _______________________________ Corresponding parts of congruent triangles are congruent Activity 5.2.2b Connecticut Core Geometry Curriculum Version 3.0 Name: Date: Page 2 of 2 Prove part (2) Given: PA = PB C is the midpoint of Μ Μ Μ Μ π΄π΅ Prove: β‘ππΆ β₯ Μ Μ Μ Μ π΄π΅ Statements Reasons PA = PB Given C is the midpoint of Μ Μ Μ Μ π¨π© Given AC = CB (a) Definition of _________________________ (b) _______________________________ Reflexive Property (c) _______________________________ (d) ___________________________________ πβ π΄πΆπ = πβ π΅πΆπ (e) ___________________________________ (f) πβ ____________ + πβ __________ = 180° (g) ___________________________________ πβ π΄πΆπ + πβ π΄πΆπ = 180° Substitution (substitute πβ π΄πΆπ for πβ π΅πΆπ) 2πβ π΄πΆπ = 180° (h) __________________________________ (i) ______________________________ Division property of equality (j) ______________________________ Definition of perpendicular lines Activity 5.2.2b Connecticut Core Geometry Curriculum Version 3.0