Let’s analyze each piece of given information step by step and determine which statements must be true based on the geometric properties.
### Step 1: Understanding Perpendicular and Parallel Lines
1. Perpendicular Lines: Two lines are perpendicular if they meet at a right angle (90 degrees).
2. Parallel Lines: Two lines are parallel if they are always the same distance apart and never meet.
### Step 2: Relationship Between Lines
- Line [tex]$b$[/tex] is perpendicular to line [tex]$f$[/tex].
- Line [tex]$f$[/tex] is perpendicular to line [tex]$g$[/tex].
- Line [tex]$h$[/tex] is parallel to line [tex]$f$[/tex].
### Step 3: Analyze Each Relationship
1. [tex]$f \perp g$[/tex]:
- Given that line [tex]$f$[/tex] is perpendicular to line [tex]$g$[/tex]. This is a direct statement.
2. [tex]$b \perp f$[/tex]:
- Given that line [tex]$b$[/tex] is perpendicular to line [tex]$f$[/tex]. This is also a direct statement.
3. [tex]$h \parallel f$[/tex]:
- Given that line [tex]$h$[/tex] is parallel to line [tex]$f$[/tex]. This is another direct statement.
### Step 4: Derive New Relationships
1. [tex]$b \parallel g$[/tex]:
- Since line [tex]$b$[/tex] is perpendicular to line [tex]$f$[/tex] and line [tex]$f$[/tex] is perpendicular to line [tex]$g$[/tex], line [tex]$b$[/tex] must be parallel to line [tex]$g$[/tex]. This follows because if two lines are both perpendicular to the same line, they must be parallel to each other.
2. [tex]$h \perp g$[/tex]:
- Since line [tex]$h$[/tex] is parallel to line [tex]$f$[/tex] and line [tex]$f$[/tex] is perpendicular to line [tex]$g$[/tex], line [tex]$h$[/tex] must also be perpendicular to line [tex]$g$[/tex]. Parallel lines to a perpendicular line remain perpendicular.
### Step 5: Evaluate Given Statements
- [tex]$f \perp h$[/tex]:
- This is not true. Since [tex]$h$[/tex] is parallel to [tex]$f$[/tex], they are not perpendicular.
- [tex]$h \perp g$[/tex]:
- This is true. From our derived relationship, line [tex]$h$[/tex] is perpendicular to line [tex]$g$[/tex].
- [tex]$b \parallel h$[/tex]:
- We do not have enough information to determine this relationship directly, so we cannot conclude that [tex]$b$[/tex] is parallel to [tex]$h$[/tex].
### Conclusion
Based on the analysis, the following must be true:
- Line [tex]$h$[/tex] is perpendicular to line [tex]$g$[/tex] [tex]$(h \perp g)$[/tex].
So, the only correct answer given the relationships is:
[tex]$h \perp g$[/tex]
