Question

answer

1. \\( \angle 2 \cong \angle 4 \\)
2. given
3. \\( m \angle 2 = m \angle 4 \\)
4. angle congruence postulate
5. \\( \angle 2 \\) and \\( \angle 3 \\) are supplementary
6. given
7. \\( m \angle 2 + m \angle 3 = 180 ^ { \circ } \\)
8. definition of supplementary angles
9. \\( \angle 1 \\) and \\( \angle 4 \\) are supplementary

5.

1. \\( m \angle 1 + m \angle 4 = 180 ^ { \circ } \\)
2. definition of supplementary angles
3. \\( m \angle 1 + m \angle 4 = m \angle 2 + m \angle 3 \\)

7.

1. \\( m \angle 1 + m \angle 4 = m \angle 4 + m \angle 3 \\)

8.

1. \\( m \angle 1 = m \angle 3 \\)

9.

1. \\( \angle 1 \cong \angle 3 \\)

10.

Explanation:

Step1: Determine the reason for statement 5

Since \(m\angle2 + m\angle3=180^{\circ}\) (from statement 4) and \(m\angle2 = m\angle4\) (from statement 2), and \(\angle1\) and \(\angle4\) form a linear - pair (assuming a geometric figure where this is visually or contextually implied). By the definition of supplementary angles (two angles that form a linear - pair are supplementary), the reason for statement 5 is "Definition of supplementary angles".

Step2: Determine the reason for statement 7

Since \(m\angle1 + m\angle4=180^{\circ}\) (statement 6) and \(m\angle2 + m\angle3 = 180^{\circ}\) (statement 4), by the transitive property of equality (if \(a = c\) and \(b = c\), then \(a = b\)), the reason for statement 7 is "Transitive property of equality".

Step3: Determine the reason for statement 8

Since \(m\angle2=m\angle4\) (statement 2), we substitute \(m\angle2\) with \(m\angle4\) in the equation \(m\angle1 + m\angle4=m\angle2 + m\angle3\) (statement 7). By the substitution property of equality (if \(a = b\), then \(a\) can be substituted for \(b\) in an equation), the reason for statement 8 is "Substitution property of equality".

Step4: Determine the reason for statement 9

Starting from \(m\angle1 + m\angle4=m\angle4 + m\angle3\) (statement 8), we subtract \(m\angle4\) from both sides of the equation. Using the subtraction property of equality (\(a + c=b + c\) implies \(a = b\)), we get \(m\angle1=m\angle3\). So the reason for statement 9 is "Subtraction property of equality".

Step5: Determine the reason for statement 10

Since \(m\angle1=m\angle3\) (statement 9), by the angle - congruence postulate (if the measures of two angles are equal, then the angles are congruent), the reason for statement 10 is "Angle congruence postulate".

Answer:

1. Definition of supplementary angles
2. Transitive property of equality
3. Substitution property of equality
4. Subtraction property of equality
5. Angle congruence postulate