During the course of geometry the concept of "angle", "vertical angles", "adjacent angles" are fairly common. Understanding each of these terms will help you understand the task at hand and properly solve it. What are adjacent angles and how to define them?
Adjacent angles - definition
The term "adjacent angles" characterizes the two angles formed by a common beam and two additional half-lines which lie on a straight line. All three beams out from a single point. Total is also the half-side of both one and the second corner.
Adjacent angles - basic properties
1. Based on the formulation of adjacent angles is easy to see that the sum of these angles always forms a straight angle, degree measure is equal to 180 °:
- If μ and η are adjacent corners, then η \u003d μ + 180 °.
- Knowing the value of one of the adjacent angles (e.g., μ), you can easily calculate the measure of the second-degree angle (η), using the expression η \u003d 180 ° - μ.
2. This property enables the corners to the following conclusion: the angle being adjacent the right angle, is also straightforward.
3. Considering the trigonometric functions (sin, cos, tg, ctg), based on the above formulas for adjacent angles μ and η following is true:
- sinη \u003d sin (180 ° - μ) \u003d sinμ,
- cosη \u003d cos (180 ° - μ) \u003d -cosμ,
- tgη \u003d tg (180 ° - μ) \u003d -tgμ,
- ctgη \u200b\u200b\u003d ctg (180 ° - μ) \u003d -ctgμ.
Adjacent angles - examples
EXAMPLE 1
Triangle with vertices set M, P, Q - ΔMPQ. Find the angles of adjacent corners ∠QMP, ∠MPQ, ∠PQM.
- We extend each of the straight sides of the triangle.
- Knowing that the adjacent angles are complementary to a straight angle, we find out that:
adjacent to the angle ∠QMP will ∠LMP,
adjacent to ∠MPQ angle will ∠SPQ,
adjacent to the angle ∠PQM will ∠HQP.
EXAMPLE 2
The value of one of the adjacent angle is 35 °. What is the degree measure of an adjacent second corner?
- Two adjacent corner to form the sum of 180 °.
- If ∠μ \u003d 35 °, then it adjacent ∠η \u003d 180 ° - 35 ° \u003d 145 °.
EXAMPLE 3
Determine the values \u200b\u200bof adjacent angles, if it is known that the degree of one of the bottom three times more degree of the other angle.
- Denote the value of one (smaller) angle through - ∠μ \u003d λ.
- Then, according to the condition of the problem, the value of the second angle will be equal to ∠η \u003d 3λ.
- Based on the basic properties of adjacent angles, μ + η \u003d 180 ° follows
λ + 3λ \u003d μ + η \u003d 180 °,
4λ \u003d 180 °,
λ \u003d 180 ° / 4 \u003d 45 °.
Hence a first angle ∠μ \u003d λ \u003d 45 °, and the second angle ∠η \u003d 3λ \u003d 135 °.
The ability to appeal the terminology, as well as the knowledge of the basic properties of adjacent angles will help to cope with the solution of many geometric tasks.
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