Trigonometry and Triangles Study Guide (page 2)
Trigonometry and Triangles
This lesson examines triangles. We see why the angles of a triangle add up to 180° (or p radians). We also see how two triangles with the same three angles (similar triangles) will always be scaled or enlarged versions of one another.
A triangle consists of three sides and three angles.
In Figure 2.1, has three sides: , , and . The three angles could be written as (or ), (or ), and (or ). Sometimes we just write to represent with vertex A. Thus, and .
In trigonometry, we often put a variable in an angle to represent its measure. For some reason, the Greek letter θ, pronounced theta, is often used to represent the measure of an angle. In Figure 2.1, the measure of is θ.
The Sum of the Angles of a Triangle
Imagine that we draw a line through point B that is parallel to . This means that these lines would never intersect, no matter how far they were drawn in either direction (Figure 2.2).
It is a result of basic geometry that the measure of will be the same as and the measure of equals the measure of . This means that the three angles of the triangle have the same measures as three angles that make up a straight angle. Thus, the three angles of any triangle add up to the measure of a straight angle, either 180° or π radians.
+ + = 180° = π radians
If two angles of a triangle measure 40° and 105°, then what is the third angle of the triangle? If the third angle is θ, then because the angles of a triangle add up to 180°:
- θ + 40° + 105° = 180°
- θ = 180° – 145° = 35°
Thus, the third angle measures 35°.
A right triangle is a triangle with a right angle (90°, or ).
If one angle of a right triangle has a radian measure of , what is the measure f the third angle? We know that the triangle has angles of and (the right angle). If the third angle is θ, then the sum of the three must be π radians.
- + + θ = π
Thus, the third angle measures .
Angles that add up to 90° (or ) like this are called complements of one another.
If a triangle has one right angle, the other two angles must be less than 90°, or .
Isosceles and Equilateral Triangles
An equilateral triangle has all three sides of the same length. Equilateral triangles also have all three angles of the same measure. Because the angles of a triangle must add up to 180°, or π, radians, each angle of an equilateral triangle measures 60°, or , radians. Equilateral triangles with sides 5 and 8 are shown in Figure 2.4.
An isosceles triangle has two sides of the same length. The two angles that are not between these two sides must also have the same measure.
In the triangle shown in Figure 2.5, what is the measure of angle θ?
Because this triangle is isosceles, with two sides of length 7, the third angle must also have measure θ. Because the three angles sum to 180°, we know that
- θ + θ + 30° = 180°
- θ = 75°
A triangle is scaled, magnified, or enlarged if the length of each side is multiplied by the same number. For example, in Figure 2.6, ΔDEF is a magnified version of ΔABC because the three sides of ΔDEF are twice as big as the three sides of ΔABC. The scale factor in this example is thus 2.
We can construct ΔDEF with four copies of ΔABC as illustrated in Figure 2.7.
This means that the area of ΔDEF is four times the area of ΔABC . It also means that the angles of ΔDEF are the same as the angles of ΔABC.
If a triangle is enlarged by a scale factor of 3, then the new triangle will have sides three times longer, will have nine times the area, and will have all the same angles. The most important detail for us is that the angles remain the same. Two triangles are similar if they have the same three angles.
A scaled version of a triangle is always similar to the original, no matter what the scale factor may be. If each side of the second triangle is k times longer than each side in the first, then the angles are all the same. This is depicted in Figure 2.8.
It can also be proven that if two triangles are similar, then one of the triangles is a scaled version of the other. This is illustrated in Figure 2.9. Because the triangles are similar, the sides of the second triangle must be some multiple k of the sides from the first triangle.
Suppose there is a triangle with sides 8, 9, and 11 inches in length. Another triangle has the same angles θ, α, and β, but the side between angles α and θ has a length of 6 inches. What are the lengths x and y of the other two sides of the triangle, as shown in Figure 2.10?
Because the two triangles have all the same angles, they are similar. Thus, the sides of the second triangle are some multiple k of the sides of the first triangle.
- 6 = k · 9
- x = k · 8
- y = k · 11
The first equation can be solved k = . Thus,
Notice that if two triangles have two angles in common, then they must have all three angles in common, because the third angle must make up the difference to 180°. This is shown in Figure 2.11.
Practice problems for this study guide can be found at:
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