Posted by Wendell on Tuesday, July 26, 2011 at 1:00am.
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 (?).

Geometry  Reiny, Tuesday, July 26, 2011 at 8:29am
AB = √[(53)^2 + (2+1)^2] = √(4+9) = √13
RP = √([3+6)^2 +(42)^2] = √(9+4) = √13
BC = √[(5+2)^2 + (20)^2] = √(49+4) = √53
PQ = √[3+5)^2 + (4+3)^2] = √(4+49) = √53
AC = √[3+2)^2 + (10)^2] = √(25+1) = √26
RQ = √[(6+5)^2 + (2+3)^2] = √(1+25) = √26
clearly we have corresponding pairs of sides equal, so
by SSS, ∆ABC≅∆RPQ
(Your length of 13, 53, and 26 should have been √13, √53, and √26)

Geometry  Wendell, Tuesday, July 26, 2011 at 2:59pm
I guess I don't understand how to write the answer.

Geometry  Wendell, Tuesday, July 26, 2011 at 3:06pm
Angle ABC is congruent to RPQ by____? and angles ABC and RPQ are congruent by _____?
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