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 ≅
The completed paragraph proof would be:
We know that <5 and <7 are supplementary. <7 and <9 are also supplementary because it is given. By the definition of supplementary angles, we have m<5 + m<7 = 180 and m<7 + m<9 = 180. Therefore, m<5 + m<7 = m<7 + m<9. Subtracting m<7 from each side, we get m<5 = m<9. Then, by the definition of congruence, we have <5 ≅ <9.
To complete the paragraph proof, we need to use the transitive property of equality to show that <5 is congruent to <9.
Let's review the steps we have so far:
1. Given: <5 and <7 are supplementary. <7 and <9 are also supplementary.
2. We know that m<5 + m<7 = 180° by definition of supplementary angles.
3. We also know that m<7 + m<9 = 180° by definition of supplementary angles.
Now, we can continue with the proof:
4. Since m<5 + m<7 = 180° and m<7 + m<9 = 180°, we can equate the expressions on both sides:
m<5 + m<7 = m<7 + m<9. This is due to the transitive property of equality.
5. To isolate the m<5 term on one side of the equation, we subtract m<7 from both sides:
m<5 + m<7 - m<7 = m<7 + m<9 - m<7.
This simplifies to: m<5 = m<9.
6. Now, we can use the definition of congruence: If two angles have the same measure, they are congruent. Since m<5 = m<9, we can conclude that <5 is congruent to <9.
Therefore, the complete proof is:
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary.
Prove: <5 ≅ <9
Proof:
1. <5 and <7 are supplementary. <7 and <9 are also supplementary. (Given)
2. m<5 + m<7 = 180° (Definition of supplementary angles)
3. m<7 + m<9 = 180° (Definition of supplementary angles)
4. m<5 + m<7 = m<7 + m<9 (Transitive property of equality)
5. m<5 = m<9 (Subtraction property of equality)
6. <5 ≅ <9 (Definition of congruence)