Let's begin by analyzing the given information. We are given that l is perpendicular to AB, and bisects it at C. We also know that point P is on l. This is how we will begin our proof!
Statement1)& l ⊥ AB, l bisects AB at C & P is on l Reason1)& Given
Notice that C is also on l, as l bisects AB at C. Therefore, PC ⊥ AB. By the definition of perpendicular segments, ∠ ACP and ∠ BCP are right angles, and hence they are congruent.
Statement2)& ∠ ACP ≅ ∠ BCP Reason2)& Definition of perpendicular & segments
We are also given that l bisects AB. Therefore, by the definition of a segment bisector, it divides AB into two congruent sides AC and BC.
Statement3)& AC ≅ BC Reason3)& Definition of a segment bisector
From the diagram we can tell that △ ACP and △ BCP share the side PC. By the Reflexive Property of Congruence we know that PC ≅ CP.
Statement 4)& PC ≅ CP Reason 4)& Reflexive Property & of Congruence
Now, we know that two sides and the included angle of △ ACP are congruent to two sides and the included angle of △ BCP. Therefore, by the Side-Angle-Side (SAS) Congruence Theorem, △ ACP ≅ △ BCP.
Statement5)& △ ACP ≅ △ BCP Reason5)& SAS Theorem
Corresponding parts of congruent triangles are congruent. Thus, the corresponding parts of △ ACP and △ BCP are congruent, and hence PA ≅ PB.
Statement6)& PA ≅ PB Reason6)& Corresponding parts of & congruent triangles are & congruent.
By the definition of congruent segments, we can conclude that PA and PB have equal lengths.
Statement7)& PA=PB Reason7)& Definition of congruent segments
Completed Proof
Statements
Reasons
1.
l ⊥ AB, l bisects AB at C, P is on l
1.
Given
2.
∠ ABP ≅ ∠ BCP
2.
Definition of perpendicular segments
3.
AC ≅ BC
3.
Definition of a segment bisector
4.
PC ≅ CP
4.
Reflexive Property of Congruence
5.
△ ACP ≅ △ BCP
5.
SAS Theorem
6.
PA ≅PB
6.
Corresponding parts of congruent triangles are congruent.
