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Math SAT - Triangles

Triangles
Triangles pop up all over the Math section. There are questions specifically about triangles, questions that ask about triangles inscribed in polygons and circles, and questions about triangles in coordinate geometry.
Three Sides, Four Fundamental Properties
Every triangle, no matter how special, follows four main rules.
1. Sum of the Interior Angles
If you were trapped on a deserted island with tons of SAT questions about triangles, this is the one rule you’d need to know:
The sum of the interior angles of a triangle is 180°.
If you know the measures of two of a triangle’s angles, you’ll always be able to find the third by subtracting the sum of the first two from 180.
2. Measure of an Exterior Angle
The exterior angle of a triangle is always supplementary to the interior angle with which it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex. In the image below, d is the exterior angle.
Since d and c together form a straight angle, they are supplementary: . According to the first rule of triangles, the three angles of a triangle always add up to , so . Since  and , d must equal a + b.
3. Triangle Inequality Rule
This rule states that:
The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
Take a look at the figure below:
The triangle inequality rule says that c – b < a and a < c + b. The exact length of side a depends on the measure of the angle created by sides b and c. 
4. Proportionality of Triangles
Here’s the final fundamental triangle property. This one explains the relationships between the angles of a triangle and the lengths of the triangle’s sides.
In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.

In this figure, side a is clearly the longest side and  is the largest angle. Meanwhile, side c is the shortest side and  is the smallest angle. So c < b < a and C < B < A. This proportionality of side lengths and angle measures holds true for all triangles.
Special Triangles
Isosceles Triangles

An isosceles triangle has two sides of equal length, and those two sides are opposite congruent angles. These equal angles are usually called as base angles. In the isosceles triangle below, side a = b and :
Equilateral Triangles
An equilateral triangle has three equal sides and three congruent 60º angles.
Right Triangles

A triangle that contains a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. The angles opposite the legs of a right triangle are complementary .
The Pythagorean Theorem

If triangles are considered a SAT favorite then right triangles and the Pythagorean Theorem are SAT darlings, so you jolly well know a lot about these two. Here’s the Pythagorean Theorem: In a right triangle, a2 + b2 = hypotenuse2 :

PYTHAGOREAN TRIPLES

There are few predefined sets of integers that always end up being what we call "Pythagorean triples".  Here are some common ones:

{3, 4, 5}
{5, 12, 13}
{7, 24, 25}
{8, 15, 17}
In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}. The SAT is full of these, so you must learn a few, to cut down the time you waste on calculations.  
Extra-Special Right Triangles
Right triangles are pretty special in their own way. But there are two extra-special right triangles. They are 30-60-90 triangles and 45-45-90 triangles, and they appear all the time on the SAT.
30-60-90 Triangles
The guy who name the 30-60-90 triangle didn't have much of an imagination. As the name clearly suggests, these triangles have angles of , , and . Before you say "So what!" here's what's special: The sides of every 30-60-90 triangle will follow this ratio of 1: : 2 . the diagram explains this better:
This constant ratio means that if you know the length of just one side in the triangle, you’ll immediately be able to calculate the lengths of all the sides.
And there’s another amazing thing about 30-60-90 triangles. Two of these triangles joined at the side opposite the 60º angle will form an equilateral triangle.
Here’s why you need to pay attention to this extra-special feature of 30-60-90 triangles. If you know the side length of an equilateral triangle, you can figure out the triangle’s height: Divide the side length by two and multiply it by . 
45-45-90 Triangles

A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x:

Know this 1: 1:  ratio for 45-45-90 triangles. It will save you time. Also, just as two 30-60-90 triangles form an equilateral triangles, two 45-45-90 triangles form a square. 
Similar Triangles
Two triangles are “similar” if the ratio of the lengths of their corresponding sides is constant (which you now know means that their corresponding angles must be congruent). Take a look at a few similar triangles and you will learn that the symbol for “is similar to” is ~  :

There are two crucial facts about similar triangles.
  • Corresponding sides of similar triangles are proportional.
  • Corresponding angles of similar triangles are identical.
Congruent Triangles
Congruent triangles are identical. 
Two triangles are congruent if they meet any of the following criteria:
  1. All the corresponding sides of the two triangles are equal. This is known as the Side-Side-Side (SSS) method of determining congruency.
  2. The corresponding sides of each triangle are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency
    .
  3. The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency
Perimeter of a Triangle
The perimeter of a triangle is equal to the sum of the lengths of the triangle’s three sides. 
Area of a Triangle
The formula for the area of a triangle is
where b is the length of a base of the triangle, and h is height (also called the altitude). The height of the triangle will sometimes be shown as a dotted line, as in the figures below:
But you probably won’t get such an easy question. It’s more likely that you’ll have to find the altitude, using other tools and techniques. For example, try to find the area of the triangle below:

To find the area of this triangle, draw in the altitude from the base (of length 9) to the opposite vertex. Notice that now you have two triangles, and one of them (the smaller one on the right) is a 30-60-90 triangle.
The hypotenuse of this 30-60-90 triangle is 4, so according to the ratio 1: : 2, the short side must be 2 and the medium side, which is also the altitude of the original triangle, is 2. Now you can plug the base and altitude into the formula to find the area of the original triangle: 1/ 2bh = 1/2(9)(2) = 9.
Trig or Treat?
“Oh my gosh! The new SAT includes trigonometry!” If you’ve heard people talking this, don’t listen to it. Here’s what the actual SAT people say about trig questions on the new SAT: “These questions can be answered by using trigonometric methods, but may also be answered using other methods.” You will never have to use trig to solve a problem, and I’ll come right out and say it: You never should use trig. That’s right. At least not until you are a genius at it!
The questions on which you could (but shouldn’t) use trig on the new SAT will cover 30-60-90 and 45-45-90 triangles. 
acrosstheuniverse
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acrosstheuniverse said:

WOW! Great lesson Astha! Very excited to see more. :)

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aarushibahl
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aarushibahl said:

very nice lesson but dont u think in the asa method of determining congruency, the congruent sides should be in between the 2 congruent angles?im not quite sure about this , anyways nice work. :)

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Astha
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Astha said in response to:
acrosstheuniverse
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WOW! Great lesson Astha! Very excited to see more. :)

Thanks, I hope i can contribute more!

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Astha
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Astha said in response to:
aarushibahl
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very nice lesson but dont u think in the asa method of determining congruency, the congruent sides should be in between the 2 congruent angles?im not quite sure about this , anyways nice work. :)

Aarushi you are right… actually they are two separate rules but for convenience i merged both! The property is as follows: The triangle can have 2 corresponding angles equal and the side in between them equal or any other side equal. For further clarification look up these links: http://www.mathopenref.com/congruentaas.html.

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oLahav
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oLahav said in response to:
aarushibahl
aarushibahl’s post:
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very nice lesson but dont u think in the asa method of determining congruency, the congruent sides should be in between the 2 congruent angles?im not quite sure about this , anyways nice work. :)

Technically, if you know 2 angles you can figure out the third in one second, so it wouldn’t really matter where the side lies, in between or not. You do have to make sure the side makes sense though, but this can be done easily by noting its direction relative to the angles given.

Incredible lesson Astha! Keep up the fantastic work!

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