Question

given: ∠1 and ∠2 are complementary ∠3 and ∠4 are complementary prove: ∠1≅∠4 statements reasons 1. ∠1 and ∠2 are complementary 1. 2. ∠3 and ∠4 are complementary 2. 3. m∠1 + m∠2 = 90° 3. 4. m∠3 + m∠4 = 90° 4. 5. ∠2≅∠3 5. 6. m∠2 = m∠3 6. 7. m∠1 + m∠2 = m∠3 + m∠4 7. 8. m∠1 + m∠3 = m∠3 + m∠4 8. 9. m∠1 = m∠4 9. 10. ∠1≅∠4 10.

Explanation:

Step1: Given information

1. Given
2. Given

Step2: Definition of complementary angles

1. Definition of complementary angles (If two angles are complementary, the sum of their measures is 90°)
2. Definition of complementary angles

Step3: Vertical - angle theorem

1. Vertical angles are congruent
2. Definition of congruent angles (Congruent angles have equal measures)

Step4: Substitution property

1. Substitution property of equality (Since \(m\angle1 + m\angle2=90^{\circ}\) and \(m\angle3 + m\angle4 = 90^{\circ}\), then \(m\angle1 + m\angle2=m\angle3 + m\angle4\))

Step5: Substitution (replace \(\angle2\) with \(\angle3\))

1. Substitution property of equality (Substitute \(m\angle2\) with \(m\angle3\) in \(m\angle1 + m\angle2=m\angle3 + m\angle4\))

Step6: Subtraction property of equality

1. Subtraction property of equality (Subtract \(m\angle3\) from both sides of \(m\angle1 + m\angle3=m\angle3 + m\angle4\))

Step7: Definition of congruent angles

1. Definition of congruent angles (If \(m\angle1 = m\angle4\), then \(\angle1\cong\angle4\))

Answer:

1. Given
2. Given
3. Definition of complementary angles
4. Definition of complementary angles
5. Vertical angles are congruent
6. Definition of congruent angles
7. Substitution property of equality
8. Substitution property of equality
9. Subtraction property of equality
10. Definition of congruent angles