Question

6 given: ∠1 and ∠2 are complementary, ∠3 and ∠4 are complementary
prove: ∠1 ≅ ∠4

statements\treasons

1. ∠1 and ∠2 are complementary\t1.
2. ∠3 and ∠4 are complementary\t2.
3. m∠1 + m∠2 = 90°\t3.
4. m∠3 + m∠4 = 90°\t4.
5. ∠2 ≅ ∠3\t5.
6. m∠2 = m∠3\t6.
7. m∠1 + m∠2 = m∠3 + m∠4\t7.
8. m∠1 + m∠3 = m∠3 + m∠4\t8.
9. m∠1 = m∠4\t9.
10. ∠1 ≅ ∠4\t10.

Explanation:

Step1: Reason for Statement 1

The first statement is given, so the reason is "Given".

Step2: Reason for Statement 2

The second statement is also given, so the reason is "Given".

Step3: Reason for Statement 3

By the definition of complementary angles (if two angles are complementary, the sum of their measures is \(90^\circ\)), so reason is "Definition of complementary angles".

Step4: Reason for Statement 4

Similarly, by the definition of complementary angles, reason is "Definition of complementary angles".

Step5: Reason for Statement 5

\(\angle 2\) and \(\angle 3\) are vertical angles, and vertical angles are congruent, so reason is "Vertical angles are congruent".

Step6: Reason for Statement 6

If two angles are congruent, their measures are equal (definition of congruent angles), so reason is "Definition of congruent angles".

Step7: Reason for Statement 7

From steps 3 and 4, both sums equal \(90^\circ\), so by the Transitive Property of Equality (\(a = c\) and \(b = c\) implies \(a = b\)), reason is "Transitive Property of Equality".

Step8: Reason for Statement 8

Substitute \(m\angle 2\) with \(m\angle 3\) (from step 6) in the equation from step 7, so reason is "Substitution Property".

Step9: Reason for Statement 9

Subtract \(m\angle 3\) from both sides of the equation in step 8, by the Subtraction Property of Equality, reason is "Subtraction Property of Equality".

Step10: Reason for Statement 10

If two angles have equal measures, they are congruent (definition of congruent angles), so reason is "Definition of congruent angles".

Filling in the reasons:

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. Transitive Property of Equality
8. Substitution Property
9. Subtraction Property of Equality
10. Definition of congruent angles

Answer:

The reasons for each statement are filled as above.