$I$ is a point of the circle of diameter $JK$. The perpendicular bisector of $JK$ cut the semi-circle not containing $I$ at $M$. Let $N$ and $P$ be the orthogonal projections of $M$ on $IJ$ and $IP$. Prove that $\angle JMN=\angle PMK$. The next question asks to prove that $JMN$ and $KMP$ are congruent.

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How I did it:

$\angle JMN=90-\angle MJN=90-\left(\angle MJK+\angle KJI\right)=90-(45+\angle KJI)=90-\angle KJI-45=\angle IKJ -45=180-45-\angle MKP -45=90-\angle MKP=\angle PMK$

Maybe there's a shorter way in this case but how can I simplify angle chasing for students?

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    $\begingroup$ Do you mean "How can I ask simpler angle-chasing questions?" or "How can I provide a simpler argument for this particular angle-chasing question?" or, from the title, "How can I simply explain to my students what angle chasing is?" $\endgroup$ – Nick C Nov 7 '18 at 16:13
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    $\begingroup$ @NickC Why math.SE? I'm not asking how to solve it. I'm asking how do experienced teachers handle this type of questions. $\endgroup$ – Any Nov 7 '18 at 16:45
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    $\begingroup$ Side note: I am amazed by the ability of American educators to invent names for the most mundane and basic things, "angle hunting" or "angle chasing" being one of them. I presume that such a name makes an important task by itself, instead of just tinkering with known information and applying axioms and known theorems. It seems that in American education everything should have a name, everything is bullet-pointed, everything is going according to a plan. Smells like Soviet central planning. Then everyone is surprised why kids cannot solve a problem a little different to one "solved" in class. $\endgroup$ – Rusty Core Nov 7 '18 at 20:28
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    $\begingroup$ @RustyCore The perpendicular bisector is given in the first line. I'm not an american educator, nor do I teach in English. I just read the term angle hunting/ angle chasing somewhere. $\endgroup$ – Any Nov 8 '18 at 12:20
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    $\begingroup$ A meta-comment: the issue raised in this question strikes me as fairly inappropriate or pointless. Namely, the literal question cited is pointlessly extreme for most students... and even for the best students, surely there are better things they could put their attention to. "Angle hunting", forsooth. Let's do real math instead... I'm sorry, I can't take this seriously, even though I did once do well at such (silly) stuff. $\endgroup$ – paul garrett Nov 8 '18 at 23:42

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