To determine which groups of interior angle measurements can form a triangle, we need to check if the sum of the three angles in each group equals [tex]\(180^\circ\)[/tex]. A group of angles can form a triangle if and only if their sum is [tex]\(180^\circ\)[/tex].
Let's examine each group step-by-step:
1. Group: [tex]\( 123^\circ, 35^\circ, 22^\circ \)[/tex]
- Sum: [tex]\( 123^\circ + 35^\circ + 22^\circ = 180^\circ \)[/tex]
- Since the sum is [tex]\(180^\circ\)[/tex], this group can form a triangle.
2. Group: [tex]\( 55^\circ, 60^\circ, 60^\circ \)[/tex]
- Sum: [tex]\( 55^\circ + 60^\circ + 60^\circ = 175^\circ \)[/tex]
- Since the sum is not [tex]\(180^\circ\)[/tex], this group cannot form a triangle.
3. Group: [tex]\( 90^\circ, 86^\circ, 40^\circ \)[/tex]
- Sum: [tex]\( 90^\circ + 86^\circ + 40^\circ = 216^\circ \)[/tex]
- Since the sum is not [tex]\(180^\circ\)[/tex], this group cannot form a triangle.
4. Group: [tex]\( 47^\circ, 71^\circ, 62^\circ \)[/tex]
- Sum: [tex]\( 47^\circ + 71^\circ + 62^\circ = 180^\circ \)[/tex]
- Since the sum is [tex]\(180^\circ\)[/tex], this group can form a triangle.
5. Group: [tex]\( 110^\circ, 30^\circ, 40^\circ \)[/tex]
- Sum: [tex]\( 110^\circ + 30^\circ + 40^\circ = 180^\circ \)[/tex]
- Since the sum is [tex]\(180^\circ\)[/tex], this group can form a triangle.
Based on our calculations, the groups of interior angle measurements that can form a triangle are:
- [tex]\( 123^\circ, 35^\circ, 22^\circ \)[/tex]
- [tex]\( 47^\circ, 71^\circ, 62^\circ \)[/tex]
- [tex]\( 110^\circ, 30^\circ, 40^\circ \)[/tex]
These groups are the ones that satisfy the condition for forming a triangle.
