An isosceles triangle has an angle that measures 106.8°. What measures are possible for the other two angles? Choose all that apply.

To find the measures of the other two angles in an isosceles triangle with one angle measuring $106.8^\circ$, we'll follow these steps:

1. Break Down the Problem

• In an isosceles triangle, two angles are equal.
• The sum of the angles in any triangle is $180^\circ$.

2. Relevant Concepts

• Let's denote the angles of the triangle as $A$, $B$, and $C$.
• Given that one angle is $A = 106.8^\circ$, we need to find the measures for the other two angles $B$ and $C$.

3. Analysis and Detail

• If $A$ is the unique angle, then the two equal angles $B$ and $C$ would be: $B = C$ Using the sum of angles in triangle: $A + B + C = 180^\circ$ Substituting $A = 106.8^\circ$: $106.8^\circ + B + B = 180^\circ$ Simplifying gives: $106.8^\circ + 2B = 180^\circ$ $2B = 180^\circ - 106.8^\circ$ $2B = 73.2^\circ$ $B = \frac{73.2^\circ}{2} = 36.6^\circ$

• If $A$ is one of the equal angles, then: $A = B = 106.8^\circ$ and using the sum of angles: $106.8^\circ + 106.8^\circ + C = 180^\circ$ $2 \times 106.8^\circ + C = 180^\circ$ $C = 180^\circ - 213.6^\circ$ This results in $C$ being negative, which is not possible.

4. Verify and Summarize

The only valid scenario occurs when $A = 106.8^\circ$ is the unique angle. The measures of the other two angles are both $36.6^\circ$.

Final Answer

The possible measures for the other two angles are:

• $36.6^\circ$ and $36.6^\circ$ (both angles).