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
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
(5 points)
≅ ≅
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary
To Prove: <5 ≅ <9
Proof:
1. <5 and <7 are supplementary (Given)
2. <7 and <9 are supplementary (Given)
3. <5 + <7 = 180° (Definition of supplementary angles)
4. <7 + <9 = 180° (Definition of supplementary angles)
5. <5 + <7 = <7 + <9 (Transitive property of equality)
6. <5 = <9 (Subtraction property of equality)
7. <5 ≅ <9 (Definition of congruence)
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary.
Prove: <5 ≅ <9.
Proof:
1. By the definition of supplementary angles, we know that the sum of the measures of two supplementary angles is 180 degrees.
2. Let m<5 represent the measure of angle <5 and m<7 represent the measure of angle <7. Similarly, let m<9 represent the measure of angle <9.
3. Since <5 and <7 are supplementary, we have m<5 + m<7 = 180 (definition of supplementary angles).
4. Similarly, since <7 and <9 are supplementary, we have m<7 + m<9 = 180 (definition of supplementary angles).
5. Combining equations (3) and (4), we have m<5 + m<7 = m<7 + m<9.
6. Subtracting m<7 from both sides of equation (5), we get m<5 = m<9 (subtraction property of equality).
7. By the definition of congruence, m<5 ≅ m<9, which implies that <5 ≅ <9.
Therefore, <5 ≅ <9.
Given: <5 and <7 are supplementary. <7 and <9 are also supplementary.
Prove: <5 <9
Proof:
1. <5 and <7 are supplementary. (Given)
2. <7 and <9 are supplementary. (Given)
3. By definition of supplementary angles, m<5 + m<7 = 180°. (Definition of supplementary angles)
4. By substituting the value of m<7 from equation (2) into equation (3), we get m<5 + m<9 = 180°. (Substitution property)
5. By addition property of equality, we have m<9 + m<5 = 180°. (Communitive property of addition)
6. Subtracting m<5 from both sides of the equation in (5), we obtain m<9 = 180° - m<5. (Subtraction property of equality)
7. By the definition of congruence, if two angles have equal measures, they are congruent. Therefore, if m<5 = m<9, then <5 ≅ <9. (Definition of congruence)
Hence, we have proved that <5 ≅ <9.