# Prove the second case of the Congruent Supplements Theorem where two angles are supplementary to congruent angles. Given: <1 and <2 are supplements <3 and <4 are supplements <1 congruent <4 Prove: <2 congruent <3 We are given that angle 1 and angle 2 are supplements, angle 3 and angle 4 are supplements, and angle 1 is congruent to angle 4.

We are asked to prove that angle 2 is congruent to angle 3.

`/_1 "supplement"/_2`              Given

`m/_1+m/_2=180`                 Definition of supp. angles

`/_3"supplement" /_4`             ...

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We are given that angle 1 and angle 2 are supplements, angle 3 and angle 4 are supplements, and angle 1 is congruent to angle 4.

We are asked to prove that angle 2 is congruent to angle 3.

`/_1 "supplement"/_2`              Given

`m/_1+m/_2=180`                 Definition of supp. angles

`/_3"supplement" /_4`              Given

`m/_3+m/_4=180`                  Definition of supp angles

`m/_1+m/_2=m/_3+m/_4` Substitution

`/_1 cong /_4`                                     Given

`m/_1=m/_4`                             Definition of congruent

`m/_2=m/_3`                      Subtraction property of equality

`/_2 cong /_3`                            Definition of congruent

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