Given: A(3,-1), B(5,2), C(-2,0), P(-3,4), Q(-5,-3), R(-6,2).
Prove: angles ABC and RPQ are congruent by completing the paragraph proof.
AB=RP=13, BC=(?)=53, and CA=QR=26. So segment AB is congruent to (?), segments BC and PQ are congruent and segment CA is congruent to segment QR. Therefore triangle ABC is congruent to (?) by (?), and angles ABC and RPQ are congruent by (?).
((AB)^2 = (5-3)^2 + (2-(-1))^2 = 13,
AB = sqrt(13).
AB = RP = sqrt(13).
BC = PQ = sqrt(53).
CA = QR = sqrt(26).
AB Congruent to RP.
Triangle ABC is Congruent to Triangle
RPQ by S.S.S.
Angle ABC is Congruent to angle RPQ by Correspondant angles of Congruent
triangles.
To prove that angles ABC and RPQ are congruent, we need to show that triangle ABC is congruent to triangle RPQ.
First, let's find the length of segment BC. To find the length of a line segment, we can use the distance formula:
For segment BC:
BC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
BC = sqrt((-2 - 5)^2 + (0 - 2)^2)
BC = sqrt((-7)^2 + (-2)^2)
BC = sqrt(49 + 4)
BC = sqrt(53)
So we can update the given lengths as follows:
AB = RP = 13, BC = sqrt(53), CA = QR = 26.
Now let's analyze the updated information. We have:
AB = RP = 13 (Given)
BC = sqrt(53) (Calculated)
CA = QR = 26 (Given)
Using the side-lengths, we can conclude that segment AB is congruent to RP (both have a length of 13), segments BC and PQ are congruent (both have a length of sqrt(53)), and segment CA is congruent to segment QR (both have a length of 26).
By the Side-Side-Side (SSS) congruence criterion, we can state that triangle ABC is congruent to triangle RPQ.
Therefore, we can conclude that angles ABC and RPQ are congruent by the corresponding parts of congruent triangles (CPCTC) theorem.