$I$ 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?

  • 3
    $\begingroup$ 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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    $\begingroup$ It's pointless to exactly the same degree that physical exercise is "pointless". Nobody who doesn't work in a sports equipment store will ever be asked to lift a dumbbell in "the real world", so what's the point? Obviously, it is to train the mind and muscles in an abstract environment so that one is able to lift heavy objects efficiently in the real world. In the same way, students practice with artificially difficult geometry problems so that they can recognize the same patterns when they are faced with carpentry or orienteering or whatever angle problems face them outside class. $\endgroup$ Nov 25 '19 at 14:45

There can be no standard algorithm to solve angle hunting problems. You have to build this skill in students by exposing them to simpler problems and gradually increasing the complexity. I suppose the difficulty of these problems can be guaged by the number of properties/constructions required to solve them completely.

The key thing is to begin by working backwards i.e., identify the triangle containing the missing angle. From there keep working your way backwards trying to find the other missing angles or their sums. Some of the properties that could be used in doing this are (not necessarily in the order of difficulty):

  • Sum of angles in a triangle is 180

  • Sum of supplementary angles is 180

  • Sum of complimentary angles is 90

  • Vertically opposite angles are equal

  • Corresponding angles between parallel lines are equal

  • Drop an orthogonal to form a right angled triangle

  • Angle subtended by a diameter on a semicircle is 90

  • Angle subtended by a chord at centre is twice the angle subtended in the major arc

  • Properties of congruent triangles and similar triangles

  • ...

The list can't be exhaustive and the properties really depend on the level of students. As I said in the beginning these are open ended problems which can be mastered only through practice. Start with easy and move on to complicated problems.


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