Write an equation for each of the following and solve it to find the value of the pronumeral.

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Understand the Problem

The question asks for the creation of equations based on the angles in various triangles and to solve for the unknown angle represented as a pronumeral in each case. We need to apply the angle sum property (the sum of the angles in a triangle is 180 degrees) for each triangle to form equations and solve for the pronumeral.

Answer

- Triangle a: $\alpha = 35^\circ$ - Triangle b: $\alpha = 30^\circ$ - Triangle c: $\alpha = 20^\circ$ - Triangle d: $d = 26^\circ$ - Triangle e: $e = 10^\circ$ - Triangle f: $f \approx 16.67^\circ$
Answer for screen readers
  • Triangle a: $\alpha = 35^\circ$
  • Triangle b: $\alpha = 30^\circ$
  • Triangle c: $\alpha = 20^\circ$
  • Triangle d: $d = 26^\circ$
  • Triangle e: $e = 10^\circ$
  • Triangle f: $f \approx 16.67^\circ$

Steps to Solve

  1. Triangle a: Formulate the Equation The angles in the triangle sum to 180 degrees, so we can express this as: $$ 60 + 50 + 2\alpha = 180 $$

  2. Triangle a: Solve for $\alpha$ Rearranging gives: $$ 2\alpha = 180 - 110 $$ Therefore: $$ 2\alpha = 70 $$ Dividing by 2: $$ \alpha = 35 $$

  3. Triangle b: Formulate the Equation The right angle is $90^\circ$, thus: $$ 90 + 2\alpha + \alpha = 180 $$

  4. Triangle b: Solve for $\alpha$ Rearranging gives: $$ 3\alpha + 90 = 180 $$ Thus: $$ 3\alpha = 90 $$ Dividing by 3: $$ \alpha = 30 $$

  5. Triangle c: Formulate the Equation The angles sum to 180 degrees: $$ 3\alpha + 4\alpha + 2\alpha = 180 $$

  6. Triangle c: Solve for $\alpha$ Combining like terms gives: $$ 9\alpha = 180 $$ Dividing by 9: $$ \alpha = 20 $$

  7. Triangle d: Formulate the Equation The sum of the angles is: $$ d + 51 + 103 = 180 $$

  8. Triangle d: Solve for $d$ Rearranging gives: $$ d + 154 = 180 $$ Thus: $$ d = 26 $$

  9. Triangle e: Formulate the Equation The angles sum to: $$ 40 + 120 + 2e = 180 $$

  10. Triangle e: Solve for $e$ Rearranging gives: $$ 2e + 160 = 180 $$ Thus: $$ 2e = 20 $$ Dividing by 2: $$ e = 10 $$

  11. Triangle f: Formulate the Equation The sum of angles in triangle f is: $$ 2f + f + 130 = 180 $$

  12. Triangle f: Solve for $f$ Combining like terms gives: $$ 3f + 130 = 180 $$ Thus: $$ 3f = 50 $$ Dividing by 3: $$ f = \frac{50}{3} \approx 16.67 $$

  • Triangle a: $\alpha = 35^\circ$
  • Triangle b: $\alpha = 30^\circ$
  • Triangle c: $\alpha = 20^\circ$
  • Triangle d: $d = 26^\circ$
  • Triangle e: $e = 10^\circ$
  • Triangle f: $f \approx 16.67^\circ$

More Information

This problem demonstrates the application of the triangle angle sum property, which states that the angles of a triangle always add up to 180 degrees. By carefully setting up an equation for each triangle and solving for the unknown variable, we can easily find the missing angles.

Tips

  • Failing to recognize that a right angle equals $90^\circ$ in triangle b.
  • Miscalculating the sums when combining angles.
  • Forgetting to account for all angles in the triangles, which might lead to incorrect equations.

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