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Kayla Proof 3.19

Prove the hypotenuse-leg criterion for congruence of right triangles.
Let ABC and DEF be triangles with right triangles at angle A and angle D, line AB is equal in length to DE and line BC equal line EF. Create a line AX so that line AX is equal to line DE and perpendicular to AB. Let's look at triangle BAX and triangle DEF. When looking at these two triangles, we know that line DE and line AX are equal because of the way that they were constructed. Since the triangle CAX is created by extending the base of line AB, we see that triangle ABC shares line BA with triangle BAX. By the given definition, we know that line AB is equal in length to line DE. We also know that angle BAX is 90 degrees because angle BAC is 90 degrees and line BA stands on line BX and the sum of two angles created in such a way must be equal to two right angles by Proposition 13. By Proposition 4, we know that triangle BAX is congruent to triangle DEF because they have two sides equal and the angle between those two sides is also equal. Since we know that the triangles DEF and BAX are congruent, we know that line EF is congruent to line BX. From the given description, we know that line EF is equal to line BC. This means that line BX is also equal to line BC. From Proposition 22 since we know that triangle ABC and triangle BAX have side BX equal to line BC and line BA in common, we also know that the line AX must also equal line AC. Since line AX is equal to line DF based on the construction, we know that all of the sides of triangle ABC are congruent to all of the side of triangle DEF respectively. Therefore, by Proposition 4, we know that the triangles ABC and DEF are congruent to one another.