Let's begin by analyzing the desired outcome of our proof. We want to show that KQ bisects JP. Notice that JM and PM will be congruent if we prove that △ KJM and △ QPM are congruent. Then, point M would be the midpoint, and hence KQ would bisect JP.
We are given that JK ≅ PQ. This is how we will begin our proof!
Statement1)& JK ≅ PQ Reason1)& Given
We are also given that JK ∥ QP. Therefore, if we draw the lines passing through these sides, then we will get two parallel lines a and b. If we also draw a line passing through the side KQ, then we will get a transversal t that intersects these parallel lines.
Therefore, by the Vertical Angles Theorem, these angles are congruent.
Statement 3)& ∠ KMJ ≅ ∠ QMP Reason 3)& Vertical Angles Theorem
Now, we know that two angles and a non-included side of △ KJM are congruent to two angles and a non-included side of △ QPM. Therefore, by the Angle-Angle-Side (AAS) Congruence Theorem, △ KJM ≅ △ QPM.
Statement4)& △ KJM ≅ △ QPM Reason4)& AAS Theorem
Corresponding parts of congruent triangles are congruent. Therefore, JM ≅ PM.
Statement5)& JM ≅ PM Reason5)& Corresponding parts of & congruent triangles are & congruent.
Since JM ≅ PM, we can conclude that the point M is the midpoint of JP. Hence, KQ bisects JP.
Statement6)& KQ bisects JP Reason6)& Definition of a segment bisector
Completed Proof
Statements
Reasons
1.
JK ≅ PQ
1.
Given
2.
∠ JKM ≅ ∠ PQM
2.
Alternate Interior Angles Theorem
3.
∠ KMJ ≅ ∠ QMP
3.
Vertical Angles Theorem
4.
△ KJM ≅ △ QPM
4.
AAS Theorem
5.
JM ≅ CPM
5.
Corresponding parts of congruent triangles are congruent.
