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.
See if you can use this rule to solve the question below:
|
|
|
What is one possible value of x if angle C < A < B? |
|
(A) |
1 |
|
(B) |
6 |
|
(C) |
7 |
|
(D) |
10 |
|
(E) |
15 | |
According to the proportionality of triangles rule, the longest side of a triangle is opposite the largest angle. Likewise, the shortest side of a triangle is opposite the smallest angle. The largest angle in triangle ABC is 
, which is opposite the side of length 8. The smallest angle in triangle ABC is 
, which is opposite the side of length 6. This means that the third side, of length x, measures between 6 and 8 units in length. The only choice that fits the criteria is 7. C is the correct answer.
Special Triangles
Special triangles are “special” not because they get to follow fewer rules than other triangles but because they get to follow more. Each type of special triangle has its own special name: isosceles, equilateral, and right. Knowing the properties of each will help you tremendously, humongously, a lot, on the SAT.
But first we have to take a second to explain the markings we use to describe the properties of special triangles. The little arcs and ticks drawn in the figure below show that this triangle has two sides of equal length and three equal angle pairs. The sides that each have one tick through them are equal, as are the sides that each have two ticks through them. The angles with one little arc are equal to each other, the angles with two little arcs are equal to each other, and the angles with three little arcs are all equal to each other.
Isosceles Triangles
In ancient Greece, Isosceles was the god of triangles. His legs were of perfectly equal length and formed two opposing congruent angles when he stood up straight. Isosceles triangles share many of the same properties, naturally. 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 
:
If you know the value of one of the base angles in an isosceles triangle, you can figure out all the angles. Let’s say you’ve got an isosceles triangle with a base angle of 35º. Since you know isosceles triangles have two congruent base angles by definition, you know that the other base angle is also 35º. All three angles in a triangle must always add up to 180º, right? Correct. That means you can also figure out the value of the third angle: 180º – 35º – 35º = 110º.
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