4) An isosceles triangle has an angle that measures $$ 28^{\circ} $$. What measures are possible for the other two angles? Choose all that apply.

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UpStudy AI · Accepted Answer

In an isosceles triangle, two angles are equal, and the sum of all angles in a triangle is always $$ 180^{\circ} $$.

Given that one angle measures $$ 28^{\circ} $$, we can consider two cases:

1. **Case 1:** The $$ 28^{\circ} $$ angle is one of the equal angles.
2. **Case 2:** The $$ 28^{\circ} $$ angle is the unique angle.

### Case 1: $$ 28^{\circ} $$ is one of the equal angles

Let the equal angles be $$ x $$. Then we have:
$$
x + x + 28^{\circ} = 180^{\circ}
$$
This simplifies to:
$$
2x + 28^{\circ} = 180^{\circ}
$$
Subtracting $$ 28^{\circ} $$ from both sides gives:
$$
2x = 180^{\circ} - 28^{\circ}
$$
Calculating the right side:
$$
2x = 152^{\circ}
$$
Dividing by 2:
$$
x = 76^{\circ}
$$

So, in this case, the angles are $$ 76^{\circ}, 76^{\circ}, 28^{\circ} $$.

### Case 2: $$ 28^{\circ} $$ is the unique angle

Let the equal angles be $$ y $$. Then we have:
$$
y + y + 28^{\circ} = 180^{\circ}
$$
This simplifies to:
$$
2y + 28^{\circ} = 180^{\circ}
$$
Subtracting $$ 28^{\circ} $$ from both sides gives:
$$
2y = 180^{\circ} - 28^{\circ}
$$
Calculating the right side:
$$
2y = 152^{\circ}
$$
Dividing by 2:
$$
y = 76^{\circ}
$$

So, in this case, the angles are $$ 76^{\circ}, 76^{\circ}, 28^{\circ} $$.

### Conclusion

The possible measures for the other two angles in the isosceles triangle are $$ 76^{\circ} $$ and $$ 76^{\circ} $$. Thus, the angles of the triangle can be $$ 76^{\circ}, 76^{\circ}, 28^{\circ} $$.
The other two angles are both $$ 76^{\circ} $$.

4) An isosceles triangle has an angle that measures \( 28^{\circ} \). What measures are possible for the other two angles? Choose all that apply.

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