Answer :
To determine if the given angle measures form valid triangles, we first focus on each pair of angles provided. Recall that the sum of the internal angles in any triangle should be exactly [tex]\(180^{\circ}\)[/tex].
### 1. Case 1:
- Given: [tex]\( m \angle PQR = 54^{\circ} \)[/tex] and [tex]\( m \angle PR = 54^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 54^{\circ} + 54^{\circ} = 108^{\circ} \][/tex]
- To form a valid triangle, the total sum should be [tex]\(180^{\circ}\)[/tex]:
[tex]\[ 108^{\circ} \text{ (This does not form a valid triangle as it does not sum to 180 degrees.)} \][/tex]
### 2. Case 2:
- Given: [tex]\( m \angle PQR = 84^{\circ} \)[/tex] and [tex]\( m \angle PS = 96^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 84^{\circ} + 96^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### 3. Case 3:
- Given: [tex]\( m \angle PQR = 90^{\circ} \)[/tex] and [tex]\( m \angle PR = 90^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 90^{\circ} + 90^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### 4. Case 4:
- Given: [tex]\( m \angle PQR = 96^{\circ} \)[/tex] and [tex]\( m \angle PR = 84^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 96^{\circ} + 84^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### Final Validation
- Case 1: Sum = 108 degrees -> Not a valid triangle
- Case 2: Sum = 180 degrees -> Valid triangle
- Case 3: Sum = 180 degrees -> Valid triangle
- Case 4: Sum = 180 degrees -> Valid triangle
Thus, among the given cases:
- Case 2, Case 3, and Case 4 form valid triangles as their internal angles add up to [tex]\(180^{\circ}\)[/tex].
- Case 1 does not form a valid triangle since the sum is [tex]\(\mathbf{108^{\circ}}\)[/tex], which is less than [tex]\(180^{\circ}\)[/tex].
So the answer breakdown is:
- Sum of Angles: (108, 180, 180, 180)
- Valid Triangles: (False, True, True, True)
Therefore, thus we infer that specific configurations of angles satisfy the triangle condition and form valid triangles while others do not.
### 1. Case 1:
- Given: [tex]\( m \angle PQR = 54^{\circ} \)[/tex] and [tex]\( m \angle PR = 54^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 54^{\circ} + 54^{\circ} = 108^{\circ} \][/tex]
- To form a valid triangle, the total sum should be [tex]\(180^{\circ}\)[/tex]:
[tex]\[ 108^{\circ} \text{ (This does not form a valid triangle as it does not sum to 180 degrees.)} \][/tex]
### 2. Case 2:
- Given: [tex]\( m \angle PQR = 84^{\circ} \)[/tex] and [tex]\( m \angle PS = 96^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 84^{\circ} + 96^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### 3. Case 3:
- Given: [tex]\( m \angle PQR = 90^{\circ} \)[/tex] and [tex]\( m \angle PR = 90^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 90^{\circ} + 90^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### 4. Case 4:
- Given: [tex]\( m \angle PQR = 96^{\circ} \)[/tex] and [tex]\( m \angle PR = 84^{\circ} \)[/tex]
- Calculate the sum of these angles:
[tex]\[ 96^{\circ} + 84^{\circ} = 180^{\circ} \][/tex]
- Since the sum is [tex]\(180^{\circ}\)[/tex], this forms a valid triangle.
### Final Validation
- Case 1: Sum = 108 degrees -> Not a valid triangle
- Case 2: Sum = 180 degrees -> Valid triangle
- Case 3: Sum = 180 degrees -> Valid triangle
- Case 4: Sum = 180 degrees -> Valid triangle
Thus, among the given cases:
- Case 2, Case 3, and Case 4 form valid triangles as their internal angles add up to [tex]\(180^{\circ}\)[/tex].
- Case 1 does not form a valid triangle since the sum is [tex]\(\mathbf{108^{\circ}}\)[/tex], which is less than [tex]\(180^{\circ}\)[/tex].
So the answer breakdown is:
- Sum of Angles: (108, 180, 180, 180)
- Valid Triangles: (False, True, True, True)
Therefore, thus we infer that specific configurations of angles satisfy the triangle condition and form valid triangles while others do not.
