Question

complete the proof in paragraph form.
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 vertical angles 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 property of equality means that $40^\circ = m\angle 1$. using the \\(\boldsymbol{?}\\), $40^\circ + m\angle 2 = \boldsymbol{?}$. finally, using
the
linear pair theorem
vertical angles theorem
substitution property of equality
subtraction property of equality

Explanation:

Step1: Substitute $\boldsymbol{m\angle1=40^\circ}$

Using substitution property of equality, substitute $m\angle1=40^\circ$ into $m\angle1+m\angle2=90^\circ$:
$40^\circ + m\angle2 = 90^\circ$

Step2: Solve for $\boldsymbol{m\angle2}$

Apply subtraction property of equality, subtract $40^\circ$ from both sides:
$m\angle2 = 90^\circ - 40^\circ = 50^\circ$

Answer:

First blank (justification): substitution property of equality
Second blank (equation): $90^\circ$
Final measure of $\angle2$: $50^\circ$

Paragraph proof completion:
By the definition of complementary angles, since $\angle1$ is complementary to $\angle2$, $m\angle1+m\angle2 = 90^\circ$. By the vertical angles theorem, $\angle4 \cong \angle1$, and $m\angle4 = m\angle1$ by the definition of congruence. Combined with the given equation, $m\angle4 = 40^\circ$, the substitution property of equality means that $40^\circ = m\angle1$. Using the substitution property of equality, $40^\circ + m\angle2 = \boldsymbol{90^\circ}$. Finally, using the subtraction property of equality, $m\angle2=50^\circ$.