Question

complete the proof in paragraph form.
\\( \angle 1 \\) is complementary to \\( \angle 2 \\) given
\\( m \angle 1 + m \angle 2 = 90 ^ { \circ } \\) definition of complementary
\\( \angle 4 \cong \angle 1 \\) vertical angles eorem
\\( m \angle 4 = m \angle 1 \\) definition of congruence
\\( 40 ^ { \circ } = m \angle 1 \\) substitution property of equality
\\( m \angle 4 = 40 ^ { \circ } \\) given
complete the proof in paragraph form.
by the definition of complementary angles, since \\( \angle 1 \\) is complementary to \\( \angle 2, m \angle 1 + m \angle 2 = 90 ^ { \circ } \\). by the theorem, \\( \angle 4 \cong \angle 1 \\), and \\( m \angle 4 = m \angle 1 \\) by the definition of congruence. combined with the given equation, \\( m \angle 4 = 40 ^ { \circ } \\), the substitution quality means that \\( 40 ^ { \circ } = m \angle 1 \\). using the, \\( 40 ^ { \circ } + m \angle 2 = \\). finally, using the
vertical angles theorem
linear pair theorem
substitution property of equality
subtraction property of equality

Explanation:

Step1: Substitute \(m\angle1\) with \(40^{\circ}\)

Since \(m\angle1 + m\angle2=90^{\circ}\) and \(m\angle1 = 40^{\circ}\) (from substitution property of equality as \(m\angle4 = 40^{\circ}\) and \(m\angle4=m\angle1\)), we substitute \(m\angle1\) in the equation \(m\angle1 + m\angle2=90^{\circ}\). So, \(40^{\circ}+m\angle2 = 90^{\circ}\).

Step2: Solve for \(m\angle2\)

Using the subtraction property of equality (\(a + b=c\) implies \(b=c - a\)), we subtract \(40^{\circ}\) from both sides of the equation \(40^{\circ}+m\angle2 = 90^{\circ}\). So, \(m\angle2=90^{\circ}- 40^{\circ}\).

Answer:

First blank: substitution property of equality; Second blank: \(90^{\circ}\); Third blank: subtraction property of equality.