Answer :
Let's go through the steps to identify the missing reason in this proof:
1. You've been given that [tex]$\angle CXB$[/tex] is supplementary to [tex]$\angle AXY$[/tex].
2. You've also been given that [tex]$\angle CXB$[/tex] is supplementary to [tex]$\angle CXY$[/tex].
By definition, supplementary angles are two angles that add up to 180 degrees. Consequently, this translates into the following mathematical relations:
[tex]\[ m\angle CXB + m\angle AXY = 180^\circ \][/tex]
[tex]\[ m\angle CXB + m\angle CXY = 180^\circ \][/tex]
3. Now, according to the Congruent Supplements Theorem, if two angles are each supplementary to the same angle or to congruent angles, then those two angles are congruent. Therefore, the theorem directly tells us that:
[tex]\[ \angle AXY \cong \angle CXY \][/tex]
4. The missing reason for establishing that [tex]$\angle AXY \cong \angle CXY$[/tex] is indeed the Congruent Supplements Theorem.
So, the completed proof with the missing reason is:
[tex]\[ \angle CXB \text{ is supplementary to } \angle AXY. \][/tex]
[tex]\[ \angle CXB \text{ is supplementary to } \angle CXY. \quad \text{(definition of supplementary)} \][/tex]
[tex]\[ \angle AXY \cong \angle CXY \quad \text{(Congruent Supplements Theorem)} \][/tex]
Thus, the missing reason is: Congruent Supplements Theorem.
1. You've been given that [tex]$\angle CXB$[/tex] is supplementary to [tex]$\angle AXY$[/tex].
2. You've also been given that [tex]$\angle CXB$[/tex] is supplementary to [tex]$\angle CXY$[/tex].
By definition, supplementary angles are two angles that add up to 180 degrees. Consequently, this translates into the following mathematical relations:
[tex]\[ m\angle CXB + m\angle AXY = 180^\circ \][/tex]
[tex]\[ m\angle CXB + m\angle CXY = 180^\circ \][/tex]
3. Now, according to the Congruent Supplements Theorem, if two angles are each supplementary to the same angle or to congruent angles, then those two angles are congruent. Therefore, the theorem directly tells us that:
[tex]\[ \angle AXY \cong \angle CXY \][/tex]
4. The missing reason for establishing that [tex]$\angle AXY \cong \angle CXY$[/tex] is indeed the Congruent Supplements Theorem.
So, the completed proof with the missing reason is:
[tex]\[ \angle CXB \text{ is supplementary to } \angle AXY. \][/tex]
[tex]\[ \angle CXB \text{ is supplementary to } \angle CXY. \quad \text{(definition of supplementary)} \][/tex]
[tex]\[ \angle AXY \cong \angle CXY \quad \text{(Congruent Supplements Theorem)} \][/tex]
Thus, the missing reason is: Congruent Supplements Theorem.
