Proposition 3.9
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Proposition 3.9: Supplements of congruent angles are congruent. Let ABC DEF .
Axiom C-1 tells us that we can locate our point F on the ray EF so that BC EF . We can also locate our point D on the ray ED so that BA ED . Axiom B-2 assures us of the existence of points X and Y such that X*B*C and Y*E*F, and again, Axiom C-1 tells us that we can locate Y on the ray EY so that BX EY . Definition 2.16 tells us that BC and BY are opposite rays, and that EF and EY are opposite rays, while Definition 3.8 tells us that ABC and ABX are supplementary angles, and DEF and DEY are supplementary angles. We want to show that ABX DEY .
Axiom C-6 tells us that ABC DEF by side-angle-side (SAS). As a consequence of the congruency of ABC and DEF , we have ACB DFE , and AC DF . SAS now gives us ACX DFY .
As a consequence of the congruency of ACX and DFY , we have AXC DYF , and AX DY . SAS delivers again with AXB DYE .
As a consequence of the congruency of AXB and DYE , we have ABX DEY , which is what we sought to prove.
Axiom C-1 tells us that we can locate our point F on the ray EF so that BC EF . We can also locate our point D on the ray ED so that BA ED . Axiom B-2 assures us of the existence of points X and Y such that X*B*C and Y*E*F, and again, Axiom C-1 tells us that we can locate Y on the ray EY so that BX EY . Definition 2.16 tells us that BC and BY are opposite rays, and that EF and EY are opposite rays, while Definition 3.8 tells us that ABC and ABX are supplementary angles, and DEF and DEY are supplementary angles. We want to show that ABX DEY .
Axiom C-6 tells us that ABC DEF by side-angle-side (SAS). As a consequence of the congruency of ABC and DEF , we have ACB DFE , and AC DF . SAS now gives us ACX DFY .
As a consequence of the congruency of ACX and DFY , we have AXC DYF , and AX DY . SAS delivers again with AXB DYE .
As a consequence of the congruency of AXB and DYE , we have ABX DEY , which is what we sought to prove.
