Question

statements: 1) ( mangle 2 = 30 ) 2) ( mangle 5 = mangle 2 ) 3) ( mangle 5 = \boxed{30} ) 4) ( mangle 5 = 30 ) 5) ( mangle 3 = 180 ) 6) ( mangle 5 + mangle 4 = 180 - mangle 4 ) 7) ( mangle 4 = 30 ) 8) ( mangle 5 + mangle 4 = 30 + square ) reasons: 1) given 2) given 3) substitution 4) simplification 5) vertical angles theorem 6) subtraction property of equality 7) vertical angles theorem 8) substitution (diagram of intersecting lines included)

Explanation:

Step1: Identify \( m\angle4 \)

From statement 7, we know \( m\angle4 = 90^\circ \) (by Vertical Angles Theorem, as vertical angles are equal and if one is \( 90^\circ \), the other is too).

Step2: Substitute \( m\angle4 \) into the equation

In statement 8, the equation is \( m\angle5 - m\angle4 = 30 - \square \)? Wait, actually, looking at the structure, statement 6 is \( m\angle5 - m\angle4 = 30 - m\angle4 \)? Wait, no, let's re - examine. Wait, the last statement (8) is \( m\angle5 - m\angle4 = 30 - \square \), and we know from statement 7 that \( m\angle4 = 90^\circ \)? Wait, no, maybe I misread. Wait, statement 7: \( m\angle4 = 90^\circ \) (assuming from vertical angles, if \( \angle2 \) is \( 90^\circ \), then \( \angle4 \) as vertical angle is \( 90^\circ \)). Then in statement 8, we substitute \( m\angle4 = 90^\circ \) into the right - hand side. Wait, the equation in statement 8 is \( m\angle5 - m\angle4 = 30 - m\angle4 \)? No, maybe the original problem is about angle measures. Wait, let's start over.

From statement 1: \( m\angle2 = 90^\circ \) (Given).

Statement 2: \( m\angle5 = 30 + m\angle2 \) (Given). Wait, no, the original statement 2 is \( m\angle5 = 30 + m\angle2 \)? Wait, the user's image has statement 2 as \( m\angle5 = 30 + m\angle2 \)? Wait, no, the first part: "Given: \( m\angle2 = 90 \), \( m\angle5 = 30 + m\angle2 \)". Then "Prove: \( m\angle5 - m\angle4 = 30 \)".

Wait, let's do the proof steps:

1. \( m\angle2 = 90^\circ \) (Given)
2. \( m\angle5 = 30^\circ + m\angle2 \) (Given)
3. \( m\angle4 = m\angle2 \) (Vertical Angles Theorem, since \( \angle2 \) and \( \angle4 \) are vertical angles)
4. Substitute \( m\angle2 = 90^\circ \) into \( m\angle5 = 30^\circ + m\angle2 \), we get \( m\angle5 = 30^\circ+90^\circ = 120^\circ \) (Simplification)
5. \( m\angle4 = 90^\circ \) (From step 3, since \( m\angle2 = 90^\circ \))
6. \( m\angle5 - m\angle4 = 120^\circ - 90^\circ \) (Subtraction Property of Equality, subtracting \( m\angle4 \) from both sides of \( m\angle5 = 30^\circ + m\angle2 \) and since \( m\angle4 = m\angle2 \))
7. \( 120^\circ - 90^\circ = 30^\circ \) (Simplification)

But in the last step of the user's problem (statement 8: \( m\angle5 - m\angle4 = 30 - \square \)? No, maybe it's \( m\angle5 - m\angle4 = 30 \) and we need to find the value to substitute. Wait, from step 3, \( m\angle4 = m\angle2 = 90^\circ \). So in the equation \( m\angle5 - m\angle4 = 30 + (m\angle2 - m\angle4) \)? No, let's look at the table:

Statements:
1. \( m\angle2 = 90 \)
2. \( m\angle5 = 30 + m\angle2 \)
3. \( m\angle4 = \square \) (maybe \( m\angle2 \))
4. \( m\angle5 = \square \) (30 + 90 = 120)
5. \( m\angle4 = 90 \)
6. \( m\angle5 - m\angle4 = 30 + m\angle2 - m\angle4 \) (Addition Property of Equality? Wait, the reason for 6 is Addition Property of Equality? No, it's substituting \( m\angle5 = 30 + m\angle2 \) into \( m\angle5 - m\angle4 \))
7. \( m\angle4 = 90 \) (Vertical Angles Theorem)
8. \( m\angle5 - m\angle4 = 30 + \square - \square \) (Substitution, substituting \( m\angle2 = 90 \) and \( m\angle4 = 90 \))

Wait, the key is that \( m\angle4 = m\angle2 = 90^\circ \) (Vertical Angles Theorem). So when we have \( m\angle5 - m\angle4 \), and \( m\angle5 = 30 + m\angle2 \), then \( m\angle5 - m\angle4 = 30 + m\angle2 - m\angle4 \). Since \( m\angle4 = m\angle2 \), this becomes \( 30+(m\angle2 - m\angle4)=30 + 0=30 \). But in the last step of the table, the equation is \( m\angle5 - m\angle4 = 30 - \square \)? No, maybe it's a typo and should be \( m\angle5 - m\angle4 = 30 + (m\angle2 - m\angle4) \), and since \( m\angle2 = m\angle4 = 90 \), then \( m\angle2 - m\angle4 = 0 \), so \( m\angle5 - m\angle4 = 30+0 = 30 \). But if we look at the last statement in the table: "8) \( m\angle5 - m\angle4 = 30 - \square \)" and the reason is "Substitution", we need to find the value in the square. Since \( m\angle4 = 90 \) and \( m\angle2 = 90 \), and from \( m\angle5 = 30 + m\angle2 \), then \( m\angle5 - m\angle4 = 30 + m\angle2 - m\angle4 \). Substituting \( m\angle2 = 90 \) and \( m\angle4 = 90 \), we get \( 30 + 90 - 90=30 \). So the square should be \( 90 \) (the value of \( m\angle4 \) or \( m\angle2 \)).

Step1: Recall \( m\angle4 \) value

From vertical angles theorem, \( m\angle4=m\angle2 \). Given \( m\angle2 = 90^\circ \), so \( m\angle4 = 90^\circ \).

Step2: Substitute into the equation

The equation is \( m\angle5 - m\angle4 = 30 + m\angle2 - m\angle4 \). Substituting \( m\angle2 = 90^\circ \) and \( m\angle4 = 90^\circ \), we have \( m\angle5 - m\angle4=30 + 90 - 90 \). The value in the square (the second square, since it's \( 30 - \square \)? Wait, no, maybe the equation is \( m\angle5 - m\angle4 = 30+(m\angle2 - m\angle4) \), and \( m\angle2 - m\angle4 = 0 \), but if we consider the form \( 30 - \square \), it's a mistake, and the correct substitution is using \( m\angle4 = 90 \), so the square should be \( 90 \) (because \( 30+90 - 90 = 30 \), so \( 30 - 0=30 \), but maybe the intended equation is \( m\angle5 - m\angle4 = 30+(m\angle2 - m\angle4) \), and since \( m\angle2 = m\angle4 \), \( m\angle2 - m\angle4 = 0 \), but the table has \( 30 - \square \), so perhaps it's a typo and the square is \( 0 \), but that doesn't make sense. Wait, going back to the given:

Given \( m\angle2 = 90 \), \( m\angle5 = 30 + m\angle2 \)

Prove \( m\angle5 - m\angle4 = 30 \)

1. \( m\angle2 = 90 \) (Given)
2. \( m\angle5 = 30 + m\angle2 \) (Given)
3. \( m\angle4 = m\angle2 \) (Vertical Angles Theorem)
4. Substitute \( m\angle2 = 90 \) into \( m\angle5 = 30 + m\angle2 \): \( m\angle5 = 30 + 90 = 120 \) (Simplification)
5. \( m\angle4 = 90 \) (From step 3)
6. \( m\angle5 - m\angle4 = 120 - 90 \) (Subtraction)
7. \( 120 - 90 = 30 \) (Simplification)

In the table, statement 8 is \( m\angle5 - m\angle4 = 30 - \square \), and the reason is Substitution. From step 3, \( m\angle4 = m\angle2 \), so \( m\angle5 - m\angle4 = 30 + m\angle2 - m\angle4 = 30+(m\angle2 - m\angle4) \). Since \( m\angle2 = m\angle4 \), \( m\angle2 - m\angle4 = 0 \), so \( m\angle5 - m\angle4 = 30+0 = 30 \). If we write it as \( 30 - \square \), then \( \square = 0 \), but that's because \( 30 - 0 = 30 \). But more likely, the intended substitution is using \( m\angle4 = 90 \) in the equation \( m\angle5 - m\angle4 = 30+(m\angle2 - m\angle4) \), and since \( m\angle2 = 90 \), we substitute \( m\angle2 = 90 \) and \( m\angle4 = 90 \), so the square (the value of \( m\angle4 \)) is \( 90 \).

Answer:

If we consider the substitution of \( m\angle4 = 90^\circ \) (or \( m\angle2 = 90^\circ \)) into the equation, the value in the square is \( 90 \) (or if we consider \( m\angle2 - m\angle4 = 0 \), the square is \( 0 \)). But based on the vertical angles and the given \( m\angle2 = 90^\circ \), the most probable value to substitute is \( 90 \) (the measure of \( m\angle4 \)). So the answer is \( \boldsymbol{90} \) (assuming the equation is \( m\angle5 - m\angle4 = 30+(m\angle2 - m\angle4) \) and we substitute \( m\angle4 = 90 \)).