Math Problem Statement

Bestimme die Dreiecksart und die Größe der Winkel α und β?

Solution

The problem asks to determine the type of triangle and the sizes of the angles α\alpha and β\beta, based on the provided diagram.

Step-by-Step Solution:

  1. Identify Known Angles:

    • From the diagram, we know that the exterior angle at BB is 142142^\circ.
  2. Use of Exterior Angle Theorem: The exterior angle theorem states that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. In this case, the exterior angle 142142^\circ is equal to the sum of α\alpha and β\beta: α+β=142\alpha + \beta = 142^\circ

  3. Determine Third Angle: Since the sum of all angles in any triangle is 180180^\circ, the angle at CC is given by: C=180142=38\angle C = 180^\circ - 142^\circ = 38^\circ

  4. Solve for α\alpha and β\beta: Now that we know the angle at CC, we can use it in the triangle ABCABC. The sum of the interior angles of a triangle is also 180180^\circ, so we have: α+β+38=180\alpha + \beta + 38^\circ = 180^\circ Substituting the value α+β=142\alpha + \beta = 142^\circ into this equation: 142+38=180142^\circ + 38^\circ = 180^\circ This equation confirms that our values are consistent.

Thus, the triangle ABCABC is scalene (since all angles are different), and the angles are as follows:

  • α+β=142\alpha + \beta = 142^\circ
  • C=38\angle C = 38^\circ

Would you like further details on how to handle these types of geometry problems?

Related Questions:

  1. How can you calculate the length of the sides of this triangle if given?
  2. What is the exterior angle theorem in more detail?
  3. Can the type of triangle change if one of the angles is altered?
  4. How can you apply trigonometry to find the side lengths given angles and one side?
  5. How do we classify triangles based on their angles?

Tip: Always remember that the sum of the angles in any triangle must be 180180^\circ, which is helpful for finding unknown angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Angles

Formulas

Sum of interior angles of a triangle: α + β + γ = 180°
Exterior angle theorem: Exterior angle = sum of non-adjacent interior angles

Theorems

Exterior Angle Theorem

Suitable Grade Level

Grades 7-9