Complete the following proof. You may use an answer more than once. You will not use all the possible answers.
(3 points)
Statements Reasons
1. <1 and <2 are complementary 1. Given
2. m<1 +m<2 = 90° 2. Response area
3. <1≅<4 3. Given
4. m<1 = m<4 4. Definition of congruence
5. <2 ≅<3 5. Response area
6. m<2 = m<3 6. Definition of
7. m<4 + m<3 = 90° 7. Response area
8. <3 and <4 are complementary 8. Definition of Complementary Angles
9. m<1 + m<4 = 90° 9. Substitution property (using statements 3 and 4)
10. m<1 + m<2 = 90° 10. Transitive property (using statements 2 and 9)
11. <1 and <2 are also complementary 11. Definition of Complementary Angles
Statements Reasons
1. <1 and <2 are complementary 1. Given
2. m<1 + m<2 = 90° 2. Definition of complementary angles
3. <1≅<4 3. Given
4. m<1 = m<4 4. Definition of congruence
5. <2 ≅ <3 5. Response area
6. m<2 = m<3 6. Definition of congruence
7. m<4 + m<3 = 90° 7. Substitution of congruent angles
8. <3 and <4 are complementary 8. Definition of complementary angles
To complete the proof, we need to provide reasons for statements 2, 5, and 7.
2. m<1 + m<2 = 90° - Since <1 and <2 are complementary (given in statement 1), their measures sum up to 90°.
5. <2 ≅ <3 - This statement does not have a specific reason given, but it seems that it can be inferred from the congruence of <1 and <4 (statement 3) and the fact that <1 and <2 are complementary (statement 1). If <1 and <4 are congruent, and <1 and <2 are complementary, then it follows that <4 and <2 are also complementary. By definition, if two angles are complementary to the same angle, then they are congruent. Therefore, <2 and <3 must be congruent.
7. m<4 + m<3 = 90° - This statement can be justified by the fact that <4 and <3 are complementary. The definition of complementary angles states that the sum of the measures of two complementary angles is equal to 90°.
Therefore, the completed proof with all reasons would look like this:
Statements Reasons
1. <1 and <2 are complementary Given
2. m<1 + m<2 = 90° Sum of complementary angles
3. <1 ≅ <4 Given
4. m<1 = m<4 Definition of congruence
5. <2 ≅ <3 Inference from statements 1 and 3
6. m<2 = m<3 Definition of congruence
7. m<4 + m<3 = 90° Sum of complementary angles
8. <3 and <4 are complementary Definition of complementary angles