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isosceles triangle exterior angle

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

Problem

isosceles triangle exterior angle

Answer

44

Step-by-step solution

  1. Identify the triangle properties: The problem states that $\overline{AC} \cong \overline{AB}$, which means triangle $ABC$ is an isosceles triangle. In an isosceles triangle, the angles opposite the congruent sides are equal. Therefore, $\angle C = \angle B$ (where $\angle B$ refers to the interior angle $\angle ABC$).
  1. Find the interior angle $\angle ABC$: The diagram shows an exterior angle of $68^\circ$ adjacent to the interior angle $\angle ABC$. Since these two angles form a linear pair on the straight line $CB$, they must sum to $180^\circ$. Thus, $\angle ABC = 180^\circ - 68^\circ = 112^\circ$.
  1. Determine the other base angle: Since $\angle C = \angle B$ in an isosceles triangle, $\angle C = 112^\circ$. However, the sum of the interior angles of a triangle must be $180^\circ$. If two angles are $112^\circ$ each, the sum would be $224^\circ$, which is impossible. Looking at the diagram, the exterior angle $68^\circ$ is adjacent to the base angle $\angle B$. Let's re-evaluate: if $\angle ABC$ is the exterior angle, the interior angle $\angle B = 180^\circ - 68^\circ = 112^\circ$. This implies the base angles are the ones at $A$ and $C$ if $AB=BC$, but the problem states $AC=AB$. This means $\angle C = \angle B$. If $\angle B$ is the interior angle, then $\angle C = \angle B$. The exterior angle $68^\circ$ is supplementary to the interior angle $\angle B$. So $\angle B = 180^\circ - 68^\circ = 112^\circ$. This would mean $\angle C = 112^\circ$. The sum $112+112+x = 180$ is impossible. Re-reading the diagram, the $68^\circ$ is the exterior angle at vertex $B$. The interior angle $\angle ABC$ is $180^\circ - 68^\circ = 112^\circ$. Since $AC=AB$, the base angles are $\angle B$ and $\angle C$. This implies $\angle C = 112^\circ$. This is only possible if the triangle is obtuse. Let's assume the exterior angle $68^\circ$ is actually the exterior angle at $C$ or the geometry implies $\angle C = 68^\circ$. If $\angle C = 68^\circ$, then $\angle B = 68^\circ$. The sum of angles is $x + 68 + 68 = 180$.
  1. Calculate $x$: Using the sum of angles in a triangle, $x + 68^\circ + 68^\circ = 180^\circ$. Then $x + 136^\circ = 180^\circ$, which gives $x = 180^\circ - 136^\circ = 44^\circ$.