Question: Write a two-column proof for the following.
Given: $m\angle 2 = 30$
$m\angle 1 = 2m\angle 2$
Prove: $m\angle 3 + m\angle 4 = 90$
(Do not include the degree symbol in your answers.)
Statements Reasons
1) $m\angle 2 = 30$ 1) Given
2) $m\angle 1 = 2m\angle 2$ 2) Given
3) $m\angle 1 = 2 (30)$ 3) Substitution
4) $m\angle 1 = 60$ 4) Simplification
5) $m\angle 3 = 30$ 5) Vertical Angles Theorem
6) $m\angle 3 + m\angle 4 = 30 + m\angle 4$ 6) Addition Property of Equality
7) $m\angle 4 = 60$ 7) Vertical Angles Theorem
8) $m\angle 3 + m\angle 4 = 30 + 60$ 8) Substitution
1. **State the problem:** Given $m\angle 2 = 30$ and $m\angle 1 = 2m\angle 2$, prove that $m\angle 3 + m\angle 4 = 90$.
2. **Given information:**
- $m\angle 2 = 30$
- $m\angle 1 = 2m\angle 2$
3. **Substitute the value of $m\angle 2$ into $m\angle 1$:**
$$m\angle 1 = 2 \times 30$$
4. **Simplify:**
$$m\angle 1 = 60$$
5. **Use the Vertical Angles Theorem:**
Vertical angles are equal, so
$$m\angle 3 = m\angle 2 = 30$$
6. **Express $m\angle 3 + m\angle 4$ using addition property:**
$$m\angle 3 + m\angle 4 = 30 + m\angle 4$$
7. **Use Vertical Angles Theorem again:**
$$m\angle 4 = m\angle 1 = 60$$
8. **Substitute $m\angle 4$ into the sum:**
$$m\angle 3 + m\angle 4 = 30 + 60$$
9. **Add to find the sum:**
$$m\angle 3 + m\angle 4 = 90$$
**Final conclusion:** We have proven that $m\angle 3 + m\angle 4 = 90$ as required.