Question

complete the proof in paragraph form.

\\( \angle 1 \\) is complementary to \\( \angle 2 \\)
\\( m \angle 1 + m \angle 2 = 90 ^ { \circ } \\)
given
definition of
complementary

\\( \angle 4 \cong \angle 1 \\)
\\( m \angle 4 = m \angle 1 \\)
vertical angles
definition of
theorem
congruence

\\( 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 = \\)
and \\( m \angle 4 = m \angle 1 \\) by the definition of congruence. combined with the given equation, \\( m \angle 4 = \\)
that \\( 40 ^ { \circ } = m \angle 1 \\). using the subtraction property of equality, \\( 40 ^ { \circ } + m \angle 2 = 90 \\) degrees

the

substitution property of equality
subtraction property of equality
vertical angles theorem
linear pair theorem

Explanation:

Step1: Substitute \(m\angle1\) with \(40^{\circ}\) in \(m\angle1 + m\angle2=90^{\circ}\)

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

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:

\(m\angle2 = 50^{\circ}\)