Question

Proving the Congruent Supplements Theorem

Answer 1

Answer:

From

m∠1 = m∠4  

m∠1 + m∠2 = m∠3 + m∠4

m∠2 = m∠3 ${}$                                                Identity property

∠2 ≅ ∠3 ${}$                                      ${}$               Equal angles are congruent

Step-by-step explanation:

Given ${}$                                                                            Reason

∠1 and ∠2 are supplementary    ${}$             Given

Therefore;

m∠1 + m∠2 = 180°                              ${}$       Supplementary ∠s sum up to 180°

∠3 and ∠4 are supplementary    ${}$             Given

Therefore;

m∠3 + m∠4 = 180°                              ${}$       Supplementary ∠s sum up to 180°

From which we have;

m∠1 + m∠2 = 180° = m∠3 + m∠4       ${}$   ${}$    Transitive property

m∠1 + m∠2 = m∠3 + m∠4

∠1 ≅ ∠4                          ${}$                             Given

m∠1 = m∠4                                    ${}$              Congruent ∠s have equal measure

Therefore;

m∠1 + m∠2 = m∠3 + m∠1              ${}$         ${}$   Transitive property

Therefore;

m∠1 + m∠2 - m∠1= m∠3 + m∠1 - m∠1       ${}$Subtraction property

m∠1 - m∠1 + m∠2 = m∠3 + m∠1 - m∠1  ${}$    

0 + m∠2 = m∠3 + 0                             ${}$       Inverse property

Therefore;

m∠2 = m∠3 ${}$                                                Identity property

∠2 ≅ ∠3 ${}$                                      ${}$               Equal angles are congruent.

Answer 2

Answer:

1. <1 and <2 are supp. given
2. m <1+ m<2 =180. def. of supplementary angles
3. <1≅<4. given
4. <3 and <4 are supp. given
5. m< 3 + m<4 = 180. def. of supplementary angles
6. m<1+m<2 = m< 3 + m<4. substitution property
7. m<1 = m<4. definition of ≅ angles
8. m<1+m<2 = m< 3 + m<1. Substitution property
9. m<2 = m< 3. subtraction property
10. <2 ≅ < 3. definition of ≅ angle