Construction works and also re-planning of the apartment and preparation for its repair demand not only construction skills, but also knowledge of mathematics, geometry and so forth. So, often happens it is necessary to find an internal corner of a triangle.
Instruction
1. For finding of an internal corner of a triangle remember the theorem of the sum of corners of a triangle. Theorem: the sum of corners of a triangle is equal 180 °. From this theorem mark out five consequences which can help with calculation of an internal corner. 1. The sum of acute angles of a rectangular triangle is equal 90 °.2. Each acute angle is equal in an isosceles rectangular triangle 45 °.3. Each corner is equal in an equilateral triangle 60 °.4. In any triangle either all acute angles, or two corners sharp, and the third stupid or direct.5. The external corner of a triangle is equal to the sum of two internal corners. Example 1: To find AVS triangle corners, knowing that the corner With on 15 ° is more, and the corner And on 30 ° is less than corner And. Decision: Designate a-degree measure of a corner And through X, then the-degree measure of a corner With is equal to X +15 °, and the corner In is equal to H-30 °. As the sum of internal corners of a triangle is equal 180 °, you receive the equation: X + (X +15)+ (H-30) =180reshaya it, you will find X =65 °. Thus the corner And is equal 65 °, the corner In is equal 35 °, the corner With is equal 80 °.
2. Work with a bisector. The corner And is equal in AVS triangle 60 °, the corner In is equal 80 °. The bisector of AD of this triangle cuts from it ACD triangle. Try to find corners of this triangle. Construct the schedule for descriptive reasons. The corner of DAB is equal 30 ° as AD is a bisector And, the corner of ADC is equal 30 °+80 °=110 ° as the external corner of a triangle of ABD (the investigation 5), a corner With is equal 180 ° - (110 °+30 °) =40 ° according to the theorem of the sum of corners of a triangle of ACD.
3. Even for finding of an internal corner you can use equality of triangles: Theorem 1: If two parties and a corner between them one triangle are respectively equal to two parties and a corner between them other triangle, then such triangles are equal. On the basis of the Theorem 1 the Theorem 2 is established. Theorem 2: Sum of any two internal corners of a triangle less than 180 °. From the previous theorem the Theorem 3 follows. Theorem 3: The external corner of a triangle is more than any internal corner not adjacent to it. Also for calculation of an internal corner of a triangle it is possible to use the theorem of cosines, but only in case all three parties are known.
4. Remember the theorem of cosines: The square of the party of a triangle is equal to the sum of squares of two other parties minus the doubled work of these parties on a cosine of the angle between them: a2=b2+c2-2bc cos Aili b2=a2+c2 - 2ac cos Bilis2=a2+b2-2ab cos C