Math Problem Statement

Find the size of angle r in the triangle, given that one angle is 46° and an exterior angle is 112°.

Solution

To find the size of angle rr, we can use the properties of angles in triangles and around straight lines.

Step-by-step solution:

  1. Angles on a straight line: The angle labeled 112112^\circ and the adjacent angle (the exterior angle at the triangle's vertex) are supplementary, meaning their sum is 180180^\circ because they form a straight line. Thus, the exterior angle is: 180112=68180^\circ - 112^\circ = 68^\circ

  2. Sum of angles in a triangle: The interior angles of a triangle must sum to 180180^\circ. The three angles in the triangle are:

    • 4646^\circ (given),
    • rr (unknown), and
    • the exterior angle we just found, which is 6868^\circ.

    Using the sum of interior angles: r+46+68=180r + 46^\circ + 68^\circ = 180^\circ

  3. Solve for rr: Simplifying the equation: r+114=180r + 114^\circ = 180^\circ r=180114=66r = 180^\circ - 114^\circ = 66^\circ

Thus, the size of angle rr is 66\boxed{66^\circ}.


Would you like more details or have any questions? Here are five related questions for further understanding:

  1. How would the solution change if the exterior angle was different?
  2. Can you explain why the sum of angles in a triangle is always 180180^\circ?
  3. How would you find the exterior angle of a polygon with more than three sides?
  4. What are the properties of supplementary angles on a straight line?
  5. How can this method be applied to non-right triangles?

Tip: Always remember the sum of the angles in any triangle is 180180^\circ, which is a key property used to solve for unknown angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles in Triangles
Supplementary Angles

Formulas

Sum of angles in a triangle = 180°
Supplementary angles on a straight line = 180°

Theorems

Sum of angles in a triangle
Supplementary Angles

Suitable Grade Level

Grades 6-8