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 (?).
Ohhhh boy...
To prove that angles ABC and RPQ are congruent, we need to show that triangle ABC is congruent to triangle RPQ by using the side-side-side (SSS) congruence criterion.
Let's start by finding the length of segment BC. To find BC, we can use the distance formula. The formula to find the distance between two points (x1, y1) and (x2, y2) is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Applying this formula to points B(5,2) and C(-2,0):
BC = √[(-2 - 5)^2 + (0 - 2)^2] = √[49 + 4] = √53
So we have BC = √53.
Next, we need to find the length of segment PQ. Applying the distance formula to points P(-3, 4) and Q(-5, -3):
PQ = √[(-5 - (-3))^2 + (-3 - 4)^2] = √[4 + 49] = √53
So we have PQ = √53.
Now, let's find the length of segment CA. Applying the distance formula to points C(-2, 0) and A(3, -1):
CA = √[(3 - (-2))^2 + (-1 - 0)^2] = √[25 + 1] = √26
So we have CA = √26.
Now, we can state the congruence of the triangles: triangle ABC is congruent to triangle RPQ.
By using the SSS congruence criterion, we have AB = RP = 13, BC = PQ = √53, and CA = QR = √26.
Therefore, triangle ABC is congruent to triangle RPQ.
Finally, according to the definition of congruence, if two triangles are congruent, then their corresponding angles are congruent.
Hence, angles ABC and RPQ are congruent.