Question

part 6 of 7

write a two - column proof for the following.

given: ( mangle e = 30 )
( mangle a = 2mangle e )

prove: ( mangle 3 + mangle a = 90 )

(do not include the degrees symbol in your answers.)

statements | reasons
--- | ---

1. ( mangle e = 30 ) | 1) given
2. ( mangle a = 2mangle e ) | 2) given
3. ( mangle a = 2(30) ) | 3) substitution
4. ( mangle a = 60 ) | 4) simplification
5. ( mangle 3 = 30 ) | 5) blank
6. ( mangle 3 + mangle a = 30 + mangle a ) | 6) blank
7. ( mangle a = 60 ) | 7) blank

(diagram of intersecting lines with angles labeled 1,2,3,4, and others.)

Explanation:

Step1: Identify the goal

We need to find the reason for statement 7: \( m\angle A = 60 \). Let's track the previous statements.

Step2: Analyze previous steps

- Statement 2: \( m\angle A = 2m\angle E \) (Given)
- Statement 1: \( m\angle E = 30 \) (Given)
- Statement 3: Substitute \( m\angle E = 30 \) into \( m\angle A = 2m\angle E \), so \( m\angle A = 2(30) \) (Substitution)
- Statement 4: Calculate \( 2(30) = 60 \), so \( m\angle A = 60 \) (Simplification/Calculation)

Wait, but looking at the table, statement 7 is \( m\angle A = 60 \), and we need its reason. From the steps, after statement 4 (which is \( m\angle A = 60 \) by simplifying \( 2(30) \)), but maybe there's a relation with other angles. Wait, the original proof has \( m\angle 3 + m\angle A = 90 \)? Wait, no, the initial problem says "Prove: \( m\angle 3 + m\angle A = 90 \)"? Wait, maybe I misread. Wait, the given is \( m\angle E = 30 \), \( m\angle A = 2m\angle E \), and we need to prove something, but the table has statements. Wait, statement 7 is \( m\angle A = 60 \), and we need its reason. From statement 2 (\( m\angle A = 2m\angle E \)) and statement 1 (\( m\angle E = 30 \)), then statement 3 is substitution (replacing \( m\angle E \) with 30 in statement 2), statement 4 is simplifying \( 2\times30 = 60 \), so the reason for statement 7 (if statement 4 is \( m\angle A = 60 \)) would be "Simplification" or "Calculation" from statement 3. But maybe the table has a typo, and statement 7 is actually using the value from statement 4. Alternatively, if we look at the initial proof: Given \( m\angle E = 30 \), \( m\angle A = 2m\angle E \), so \( m\angle A = 2\times30 = 60 \) (by substitution and then simplification). So the reason for \( m\angle A = 60 \) (statement 7) is that it's the result of simplifying \( 2\times30 \) from statement 3, so the reason is "Simplification" or "Calculation". But in the table, statement 4 is \( m\angle 1 = 60 \)? Wait, maybe the angle labels are different. Wait, the diagram has angles 1,2,3,4 and another angle. Maybe \( \angle E \) is \( \angle 3 \)? Wait, statement 1: \( m\angle E = 30 \) (maybe \( \angle 3 \)), statement 2: \( m\angle A = 2m\angle E \), statement 3: \( m\angle 1 = 2(30) \)? No, the table has statement 3: \( m\angle 1 = 2(30) \), statement 4: \( m\angle 1 = 60 \), statement 5: \( m\angle 3 = 30 \), statement 6: \( m\angle 3 + m\angle A = 30 + m\angle A \), statement 7: \( m\angle A = 60 \). Wait, maybe \( \angle A \) is \( \angle 1 \)? No, this is confusing. Wait, the key is that statement 7 is \( m\angle A = 60 \), and we need its reason. From the given \( m\angle A = 2m\angle E \) and \( m\angle E = 30 \), so substituting \( m\angle E = 30 \) into \( m\angle A = 2m\angle E \) gives \( m\angle A = 2\times30 = 60 \). So the reason is "Substitution and Simplification" or just "Simplification" after substitution. But in the table, statement 3 is substitution (using \( m\angle E = 30 \) in \( m\angle 1 = 2m\angle E \)), statement 4 is simplification (calculating \( 2\times30 \)). Then statement 7: \( m\angle A = 60 \) – maybe \( \angle A \) is \( \angle 1 \), so the reason is the same as statement 4, but maybe the table has a different flow. Alternatively, if we look at the initial proof's "Prove: \( m\angle 3 + m\angle A = 90 \)", then if \( m\angle 3 = 30 \) and \( m\angle A = 60 \), then \( 30 + 60 = 90 \). So to get \( m\angle A = 60 \), we use the given \( m\angle A = 2m\angle E \) and \( m\angle E = 30 \), so substitution (statement 2 and 1) gives \( m\angle A = 2\times30 \), then simplification gives \( 60 \). So the reason for statement 7 is "Simplification" (or "Calculation") because we calculated \( 2\times30 = 60 \).

Answer:

The reason for statement 7 (\( m\angle A = 60 \)) is Simplification (or Calculation) because we simplify \( 2 \times 30 \) (from substituting \( m\angle E = 30 \) into \( m\angle A = 2m\angle E \)) to get \( 60 \).