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Find the value of x in the given diagram involving a triangle and an adjacent angle.

Check the final answer first, then review the worked steps.

Problem

Find the value of x in the given diagram involving a triangle and an adjacent angle.

Answer

129

Step-by-step solution

  1. Identify the angles: The diagram shows a triangle with two interior angles given as $49^\circ$ and $80^\circ$. An exterior angle to the triangle is labeled as $x^\circ$.
  2. Understand the relationship between interior and exterior angles: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
  3. Set up the equation: Based on the property of exterior angles, we can write the equation: $x^\circ = 49^\circ + 80^\circ$.
  4. Calculate the value of x: Add the two interior angles: $x = 49 + 80$. $x = 129$.
  5. Alternative method using supplementary angles: The angle adjacent to $x^\circ$ on the straight line is supplementary to $x^\circ$. Let's call this interior angle $y^\circ$. So, $x^\circ + y^\circ = 180^\circ$. The sum of interior angles of a triangle is $180^\circ$. Therefore, $y^\circ + 49^\circ + 80^\circ = 180^\circ$.
  6. Calculate the interior angle y: $y + 49 + 80 = 180$. $y + 129 = 180$. $y = 180 - 129$. $y = 51^\circ$.
  7. Calculate x using the supplementary angle: Now substitute the value of $y$ back into the equation from step 5: $x^\circ + 51^\circ = 180^\circ$. $x = 180 - 51$. $x = 129$.