Question

given: \\( \angle 1 \\) is supplementary to \\( \angle 2 ; m \angle 3 + m \angle 4 = 180 ^ { \circ } ; m \angle 2 = m \angle 4 \\) prove: \\( \angle 1 \cong \angle 3 \\) statements reasons 1. \\( \angle 1 \\) is supplementary to \\( \angle 2 ; m \angle 3 + m \angle 4 = 180 ^ { \circ } ; m \angle 2 = m \angle 4 \\) 1. given 2. \\( \angle 3 \\) is supplementary to \\( \angle 4 \\) 2. 3. 3. def. of congruent angles 4. \\( \angle 1 \cong \angle 3 \\) 4. a. def. of supplementary angles b. \\( m \angle 1 + m \angle 2 = 180 ^ { \circ } \\) c. \\( \cong \\) supplements theorem d. \\( m \angle 2 + m \angle 3 = 180 ^ { \circ } \\) e. \\( \angle 2 \cong \angle 4 \\)

Explanation:

Step1: Determine the reason for statement 2

Since \(m\angle3 + m\angle4=180^{\circ}\), by the definition of supplementary angles (if the sum of the measures of two angles is \(180^{\circ}\), then the angles are supplementary), the reason for statement 2 is "Def. of supplementary angles".

Step2: Fill in statement 3

Since \(m\angle2 = m\angle4\), by the definition of congruent angles (if the measures of two angles are equal, then the angles are congruent), \(\angle2\cong\angle4\). So statement 3 is \(\angle2\cong\angle4\).

Step3: Determine the reason for statement 4

We know that \(\angle1\) is supplementary to \(\angle2\) (\(m\angle1 + m\angle2=180^{\circ}\)) and \(\angle3\) is supplementary to \(\angle4\) (\(m\angle3 + m\angle4 = 180^{\circ}\)), and \(\angle2\cong\angle4\). By the Congruent Supplements Theorem (if two angles are supplementary to congruent angles, then the two angles are congruent), the reason for statement 4 is "\(\cong\) Supplements Theorem".

Answer:

1. a
2. e
3. c