To identify the possible measures for the third angle in an isosceles triangle, we need to use the sum of the angles in a triangle, which is always 180º. Additionally, an isosceles triangle has at least two angles that are equal.
We are given:
- Angle 1 = 34°
- Angle 2 = 112°
Let's analyze the scenarios where the triangle can be isosceles:
1. Case where the third angle equals the first angle (34°):
- If the third angle (Angle 3) is 34°, the sum of these angles would be:
[tex]\[
34° + 34° + Angle 3 = 180°
\][/tex]
Simplifying, we get:
[tex]\[
68° + Angle 3 = 180°
\][/tex]
Thus,
[tex]\[
Angle 3 = 180° - 68° = 112°
\][/tex]
So, one possible measure for the third angle is 112°.
2. Case where the third angle equals the second angle (112°):
- If the third angle (Angle 3) is 112°, the sum of these angles would be:
[tex]\[
112° + 112° + Angle 1 = 180°
\][/tex]
Simplifying, we get:
[tex]\[
224° + Angle 1 = 180°
\][/tex]
This scenario is not possible as the sum exceeds 180°.
3. Case where the third angle forms an isosceles with the given angles 34° and 112°:
- If the triangle is isosceles and the third angle forms part of the two equal angles with either of the given angles, we need to verify if it could be one of these angles that differ from 34° and 112°.
First, consider that the third angle might create an isosceles triangle with the first angle:
- If so, then [tex]\( Angle 1 = 34° \)[/tex]:
[tex]\[
34° + 34° + Angle 3 = 180°
\][/tex]
Simplifying, we get:
[tex]\[
68° + Angle 3 = 180°
\][/tex]
Thus,
[tex]\[
Angle 3 = 180° - 68° = 112°
\][/tex]
- Now let's consider [tex]\( Angle 2 = 112° \)[/tex]:
- This implies two angles being both [tex]\( 112° \)[/tex]:
[tex]\[
112° + 112° + Angle 3 = 180°
\][/tex]
Simplifying:
[tex]\[
224° + Angle 3 = 180°
\][/tex]
This results similarly not possible.
Finally, verify if the third angle forms an isosceles with both given angles being part of differing sectors:
Assumption:
[tex]\[34° + 112° + Angle 3 = 180° \][/tex]
Simplifying, we get:
[tex]\[
146° + Angle 3 = 180°
\][/tex]
Thus,
[tex]\[
Angle 3 = 180° - 146° = 34°
\][/tex]
In conclusion, the measures for the third angle could be [tex]\(34°\)[/tex] or [tex]\(112°\)[/tex] thus leading to:
1. (A). 112°
2. (B). 34°
Hence, the answers that satisfy the given conditions for an isosceles triangle are:
- A. 112°
- B. 34°
