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ACT Plane Geometry – Working Angle Problems

Since there are a number of angle problems on the ACT test, it’s worth thinking about some strategies for solving them.  While I can’t identify every possibility, there are some things that you can do to help you approach an angle problem.

The Secrets to Working Angle Problems

First, start with what you are given.  That should be pretty obvious, since it should be either stated in the text of the problem, or on the diagram.  Make sure you understand what you are told.

Next, figure out what you need to end up with.  This also may be obvious, but it’s important to identify anything that might be a little different or unusual, so you don’t get tripped up on them (like needing the answer in units that are different from what you are given).

Now the trick is to use what you know about angles to get you from what you are given to what you need to end up with.  There may be multiple steps.  The key is to go from one fact to another, using what you know.   Eventually, that should lead you to the correct answer.

An Example of Working Angle Problems

This problem is similar to one that appeared in the 2011-2012 Preparing for the ACT Guide.

Working Angle Problems

In this problem, you are asked to find what x, y, and z add up to.  In reality, there is a very simple answer to this that doesn’t require any calculations.  But if you don’t know that answer, you’ll have to go with what you know.

You are given the size of two of the angles, and you know that you need to find the values of x, y, and z before you can sum them.

First, you remember that supplementary angles add up to 180°.  And since ∠x and the angle with the measure of 69 are supplementary angles (since they are on a straight line and a straight angle is 180°), you can find ∠x by subtracting 69 from 180 (getting 111° for ∠x).

Do the same thing with ∠y (180° – 85°  = 95°).  Now you know what the measure of ∠y is.

You also know that all angles in triangle add up to 180° and you already know two of them, so you can calculate the measure of the angle that is supplementary to ∠z (180° – (69° + 85°) or 26°).

Now that you know the measure of the angle next to ∠z, you can easily find ∠z using the supplementary rule we used earlier (∠z = 180° – 26° or 154° ).  Now you know the size of angle ∠z.

Knowing the size of all the angles, add them together:  111° + 95° + 154° = 360°, which is the answer.

See how we went from one thing to another until we got the answer?  Even if you don’t know the exact direction to go, start adding information that you don’t have by applying the rules that you’ve learned to the things that you do know.  Soon enough, the path will be made clear and you’ll know what you need to do to get the answer.

Hope this was helpful!


P.S., the shortcut path that didn’t rely on all the calculations we did–All that is required is to memorize the fact that the exterior angles of any p0lygon all add up to 360°.  If you remember that, you’re able to do this problem in a matter of seconds, and not risk making a mistake on calculations.

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