Complete the following paragraph proof. (You may use the choices more than once and you may not use all the choices)
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary
Prove: <5 <9
We know that <5 and <7 are supplementary. _______ <7 and <9 are also supplementary because it is m<5 + m<7 =
and m< 7 + m<9 = , by definition of supplementary angles. Then, m<5 + m<7 = m<7 + m<9 by the
. Subtract m<7 from each side and you get m<5 = m<9. Then by the definition of congruence, <5
and <9 are congruent.
We know that <5 and <7 are supplementary. <7 and <9 are also supplementary because it is m<5 + m<7 = 180° and m<7 + m<9 = 180°, by definition of supplementary angles. Then, m<5 + m<7 = m<7 + m<9 by the Transitive Property of Equality. Subtracting m<7 from each side, we get m<5 = m<9. Then, by the definition of congruence, <5 ≅ <9.
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary.
To prove: <5 <9
Proof:
We know that <5 and <7 are supplementary. This means that the sum of their measures is 180 degrees. Let's represent this using algebraic expressions:
m<5 + m<7 = 180 --- (1)
Similarly, the given information states that <7 and <9 are supplementary. So we can write:
m<7 + m<9 = 180 --- (2)
Now, using the definition of supplementary angles, we know that the sum of the measures of supplementary angles is 180 degrees. Therefore, we can rewrite equations (1) and (2) as:
<5 + <7 = 180 --- (3)
<7 + <9 = 180 --- (4)
Now, by adding equations (3) and (4), we get:
(<5 + <7) + (<7 + <9) = 180 + 180
<5 + <7 + <7 + <9 = 360
Simplifying the left-hand side of the equation gives:
<5 + 2<7 + <9 = 360
Since <5 and <7 are supplementary, we can replace their combined measure with 180:
180 + <7 + <9 = 360
Subtracting 180 from each side of the equation yields:
<7 + <9 = 360 - 180
<7 + <9 = 180
Now, using the definition of supplementary angles, we have shown that <7 and <9 are supplementary. And we know that <5 and <7 are also supplementary. Therefore, we can conclude that <5 and <9 must be congruent, as their measures are equal:
m<5 = m<9
Finally, by the definition of congruence, if two angles have equal measures, they are congruent. Hence, we can state that:
<5 <9
This completes the paragraph proof.