<p>Hey guys I hope everyone is having a great 4th of July! I have a quick question from the math section of the 2012-2013 Preparing for the ACT math section. Its #47 on Page 31 in the booklet.
<a href="http://media.act.org/documents/preparing.pdf%5B/url%5D">http://media.act.org/documents/preparing.pdf</a></p>
<p>Could somebody please explain to me how we know that the angle that is being asked for is 90 degrees? Its been a long time since I took Geometry in 8th grade online, and I don't have a fresh memory on the general principles. Any help and explanations would be appreciated! :D</p>
<p>The two parallel lines are cut by a transversal. Using the properties of parallel lines, you can determine that angles BAC and ACD are supplementary(if you have trouble with this part, extend all three lines and use either alternate interior angles or corresponding angles). Now we know that, since supplementary angles add to 180 degrees, angle ACD measures 98 degrees. Next, using the definition of a bisector, we know that EAC measures 41 degrees and ACE measures 49 degrees(a bisector cuts an angle in half). Finally, since the three angles of a triangle add to 180 degrees, we can calculate that angle AEC measures 90 degrees(180-41-49=90).</p>
<p>So you know the value of angle BAC is 82. Extend line AC to include a point F, so you end up with line ACF. Since line ACF intersects two parallel lines (AB and CD), angle DCF will be the same value as angle BAC (you should know this property, angle relationships in parallel lines cut by a transversal, for the ACT). From this you find that angles BAC and ACD are supplementary, they will add to 180 degrees. Since both angles BAC and ACD are being bisected, or halved, to form angles EAC and ACE, respectively, the sum of EAC and ACE will be 90 degrees. Now you can conclude that angle AEC is 90 degrees.</p>
<p>ACE + EAC + AEC = 180
ACE + EAC = 90
90 + AEC = 180
AEC = 90</p>
<p>That’s if you don’t want to deal with all the values. It might seem a little confusing with all the letters though, haha, but these geometry problems get ridiculously easy and become very intuitive once you familiarize yourself with the big concepts. boogapotamus offered a good solution, too.</p>
<p>Thank you guys so much! Both ways of solving it helped me out tremendously to understand it! 
By any chance are there any threads that show all of the geometric principles that we need to know for the ACT? Because I think that is the only thing keeping me from a high score on the math. The problem is that I’m currently taking Calculus II and my past memory of geometry is pretty much fading away haha. :)</p>
<p>I’d say buy the ACT official guide and go through all the practice tests. It would be easier than going back and re-learning geometry since you will only need to cover material that will be on the test.</p>
<p>Should I just look over them or should I take them seriously as if I was taking a real test?</p>
<p>Definitely treat the questions as if you were taking the real thing. Then go back and make sure you understand every question and its answer (you can even post here to help with that!).</p>
<p>Ok great I will do that! By the way could you help me out with #60 on the math section as well? I don’t understand particularly what they are asking.</p>
<p>There’s probably a short, intuitive way to do this, but I can’t remember it haha. For the sake of realism, I will show my way of solving it as if I were taking the test right now with no outside info.</p>
<p>I agree this one is kinda confusing so I’ll turn it into a statement:
The solution set of answer choice F/G/H/J/K is the set of real numbers that are 5 units from -3. </p>
<p>Now you need to find the answer choice which proves the statement correct. You know the only real numbers that are 5 units from -3 are -8 and 2, so the set of solutions for x will be {-8,2}. Start plugging in -8 and 2 for x and see which choice works.</p>
<p>F. |-8+3| = 5
|2+3| = 5
G. |-8-3| = 11
|2-3| = 1
H. |-8+5| = 3
|2+5| = 7
J. |-8-5| = 13
|2-5| = 3
K. Will not work because absolute values cannot be negative.</p>
<p>The only equation that holds true for the solution set {-8,2} is equation (F).</p>