In this guide, we are going to tackle one of the biggest items on the ACT Math portion: Geometry. As I am sure you know, geometry is the study of various types of shapes within mathematics. With geometry comes a plethora of rules, theorems, and postulates that are all important to the world around us and discovering how things function. In this article, we are going to break down some of the most critical aspects of the ACT Math test: triangles, parallel lines, trigonometry, area, perimeter, and volume.
Angles of Triangles
The most common shape you will see on the ACT is triangles. It is the most basic shape studied in mathematics and, therefore, serves as the baseline for many of our rules moving forward into more complex shapes like quadrilaterals, pentagons, etc.
One of the major themes with triangles that you will encounter is the angles inside of a triangle. As the name says, a triangle has three angles. There are many rules you can use to solve for various angles inside a triangle that are detailed below.
- The interior angles of a triangle add up to 180°. This is always relevant and leads into the pattern that a quadrilateral adds to 360°, pentagon 540°, etc. In other words, the formula is (n-2)180°, where n is the number of sides.
- The sum of the exterior angles is 360° for ANY shape, regardless of the number of sides.
- The sum of a linear pair (any two angles that combine to make a straight line) is 180°. This will play into our next section as well.
- An excellent rule of thumb is if an angle looks like it’s equal to another one in the diagram, it probably is. If it doesn’t look equal, it is likely going to be complementary or add to 180°.
In the diagram below, what is the value of x?
In this diagram, we are given two angles and are searching for the third angle. The biggest mistake someone might make is just subtracting the other two from 180°. This would provide us with an incorrect answer because 135° is not an interior angle, so we have to use the linear pair-rule to find the interior angle, which comes out to be 45°. Therefore, to find x, we need to do 180-45-60=75. Therefore, x=75.
Angles of Parallel Lines
Another popular topic is using different rules about parallel lines to find missing angles. There are several vital pairs of angles regarding parallel lines that are important to remember, but the most important thing if you ever get bogged down by rules is this: If they look equal, they probably are. If they don’t look equal, they probably add to 180°.
Different Pairs of Angles
- The first pair of angles you are bound to encounter is the linear pair we mentioned earlier. It is also worth noting here since it is likely to show up multiple times on the ACT.
- The next pair is vertical angles. These are angles that are directly across from each other when two lines intersect. Vertical angles are congruent, or in other words, have equal measures. When any two lines intersect, you will have equal vertical angles and a linear pair that add up to 180°.
- When we are dealing with parallel lines, we also have the following:
- Alternate interior angles: These are opposite sides of a transversal and the inside of the parallel lines. They are congruent to each other.
- Alternate exterior angles: These are on opposite sides of a transversal and the exterior of the parallel lines. They are also congruent to each other.
- Same side (or consecutive) interior angles: These are on the same side of the transversal but the interior of the parallel lines. These angles complement each other, which can also be seen by piecing together vertical angles, alternate interior angles, and linear pairs.
This section is so crucial to the ACT that is actually designated its own section among the breakdown. While some of the trigonometry falls in the functions section, some of it does fall within the geometry section, particularly dealing with right triangles.
Regardless of how you learned the trig functions and what they mean, there are plenty of ways to remember what sine, cosine, and tangent mean. Just hit a quick Google search. Whether it is outhouses, hippies, or cats, you’re going to be good on this section.
- Sine is opposite over hypotenuse. SOH
- Cosine is adjacent over hypotenuse. CAH
- Tangent is opposite over adjacent. TOA
The most important thing when you encounter trigonometry within a right triangle is labeling each side. The hypotenuse is the largest side and across from the right angle. The opposite side is across from your reference angle, or the angle you are interested in dealing with. The adjacent side is the one attached to your reference angle that is not the hypotenuse.
If sin𝜃 = 7/25, what is the tan𝜃?
First off, we know that since sin is 7/25, our opposite side is seven, and our hypotenuse is 25. Next, we can use the Pythagorean theorem to find the third side of our right triangle, which will be 24. By process of elimination, we know this is our adjacent side to 𝜃.
Put the pieces together to find tan, which is opposite over adjacent, so 7/24.
Area, Perimeter and Volume
The last section we are going to focus on is the area, perimeter, and volume. Area and perimeter are relevant for two-dimensional shapes like triangles and rectangles, while volume applies to three-dimensional shapes like cones, spheres, and cubes.
The most important formulas to know is area; for rectangles, squares, and parallelograms, they are all base*height or length*width. For a triangle, it is 1/2bh.
You can calculate the perimeter by adding up all the sides regardless of the shape.
For 3D shapes, most of the time, the volume or surface area formulas are provided in the question prompt, so you will simply be plugging in numbers from a diagram. The only formula you might need to memorize is for a cube: V=lwh.
The length of a rectangle is three units larger than its width. The perimeter of the rectangle is 30 units. Find the area of the rectangle.
The first thing we need to note is that the perimeter is all sides added up. Since a rectangle has four sides, we end up with w+w+(w+3)+(w+3)=30. Solving this equation, we can easily find that the width is six. Therefore, since we know that the length is three times larger than the width, the length is 9. So we have a rectangle with sides 6-6-9-9, which check out on the perimeter being 30.
Now that we know the length and width, the area is very simple: 6*9=54. Therefore, the area is 54.
Give yourself a pat on the back for making it this far. 🎉 We went through some tips on geometry questions, as well as thoroughly walked through some example questions 🚶♀️. If you need extra ACT help, check out this list of awesome resources, or the ultimate ACT study guides listed below to help you strive for that 36!
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