You know: all squares are rectangles, but not all rectangles are squares.
(That’s about the extent of what I remember from higher mathematics classes.)
Similarly, all dangling participles (danglers) are misplaced modifiers, but not all misplaced modifiers are danglers. I’ll provide some examples and links. As I’ve said before, I’m very bad at creating poor writing on purpose; when it goes from my brain to my fingers through the keyboard onto the screen, it’s grammatically correct but not necessarily the cleanest copy on the planet. I have a very difficult time purposely making mistakes like these. (Perhaps I should work on that …) Lucky for me (and you), they’ve been corralled elsewhere. I’ll write a couple of my own, and link to more.
First, though, some definitions. A misplaced modifier is just what it says: it modifies the wrong word, because the sentence is poorly constructed. Here are five examples, none of which are danglers. For my money, these are far more insidious and difficult for the untrained eye to spot. (Feel free to disagree with the first one; it’s an ongoing discussion. I happen to concur with the writer, and keep a sharp eye out for “onlies” that aren’t where they would be best used. Placement does make a semantic difference.)
She was the first to arrive dressed in a sharp blue suit.
Taken at face value, that sentence implies that others arrived before her, but none of them wore a sharp blue suit. If we move the modifying phrase to the front, it’s much clearer (and probably what was intended in the first place):
Dressed in a sharp blue suit, she was the first to arrive.
If you’re curious, the answer is yes; I would make the change and then add a comment, querying the author if this is what was intended. Never assume. (Even when you’re sure you’re right.)
Now, on to danglers. Quickly: a participle (for this purpose) is a verb form that ends in -ing. Like “dangling.” Heh. The reason we say they’re “danglers” is they hang at the beginning of the sentence with nothing appropriate to modify. Like this:
Running down the street after the car, it swerved into an alley and I lost sight of it.
Now, we can tell that the intention is to say “I was running down the street after the car, it swerved into an alley, and I lost sight of it.” But grammatically, that isn’t what this sentence says. “It” can only refer, grammatically, to “the car” at the end of that opening pair of phrases. “Running down the street after the car” is dangling, without a clear noun to modify.
We can rewrite the sentence, of course, to fix the problem.
Running down the street after the car, I lost sight of it when it swerved into an alley.
(Is that the best sentence ever? Nah. But it serves the purpose of illustration.)
Grammar Girl (Mignon Fogarty) has a wonderful article about this, right here.
As an editor, I see errors of both kinds pretty frequently. As a writing coach, I do my best to teach my clients how to spot them and correct them on their end, so that their next project can be cleaner. It’s a win-win as far as I’m concerned.