Question
Examine the diagram. It is not drawn to scale.
A triangle has angles 28 degrees, A, C. The exterior angle to the angle with a measure of 28 degrees is D, the exterior angle to angle A is B, the exterior angle to angle C is 125 degrees.
Choose the equation that cannot be used to find the measure of angle B.
m∠B = m∠C + 28°.
m∠B = 360° – (m∠D + 125°).
m∠B = 180° – m∠A.
m∠B = m∠D + 125°.
A triangle has angles 28 degrees, A, C. The exterior angle to the angle with a measure of 28 degrees is D, the exterior angle to angle A is B, the exterior angle to angle C is 125 degrees.
Choose the equation that cannot be used to find the measure of angle B.
m∠B = m∠C + 28°.
m∠B = 360° – (m∠D + 125°).
m∠B = 180° – m∠A.
m∠B = m∠D + 125°.
Answers
GPT-4o mini
To determine which equation cannot be used to find the measure of angle \( B \), let’s examine the relationships between the angles based on the properties of exterior angles and triangle angles.
1. **Exterior Angle Theorem**: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Therefore, for angle \( D \), we have:
\[
m\angle D = m\angle A + m\angle C
\]
2. For the triangle with angles \( 28^\circ \), \( A \), and \( C \):
- The sum of interior angles in a triangle is \( 180^\circ \):
\[
m\angle A + m\angle C + 28^\circ = 180^\circ
\]
- We can rearrange this to express \( m\angle A + m\angle C \):
\[
m\angle A + m\angle C = 180^\circ - 28^\circ = 152^\circ
\]
3. Since \( m\angle D \) is an exterior angle, we know:
\[
m\angle D = m\angle A + m\angle C
\]
Substituting the known sum of angles, we can find \( m\angle D \):
\[
m\angle D = 152^\circ
\]
Now, let's analyze the equations:
- **Equation 1: \( m\angle B = m\angle C + 28^\circ \)**:
This equation is valid if we use the exterior angle theorem because \( m\angle B \) (the exterior at angle A) should equal \( m\angle C + 28^\circ \).
- **Equation 2: \( m\angle B = 360^\circ - (m\angle D + 125^\circ) \)**:
This equation doesn’t automatically derive from any specific triangle relationships. It mainly depends on using a 360-degree framework, and while it could technically yield a result, it's an unorthodox approach.
- **Equation 3: \( m\angle B = 180° - m\angle A \)**:
This makes sense as \( B \) (the exterior opposite angle) can indeed relate to \( A \) within certain constructs.
- **Equation 4: \( m\angle B = m\angle D + 125^\circ \)**:
If we consider exterior relationships and absolute angle definitions, this may yield a logical conclusion based on angle sum relationships, specifically since \( m\angle D = 152^\circ \).
With all the analysis put forth, the **equation that cannot be used** in a straightforward, geometrically valid manner to find \( m\angle B \) would be:
\[
\textbf{m∠B = 360° – (m∠D + 125°).}
\]
1. **Exterior Angle Theorem**: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Therefore, for angle \( D \), we have:
\[
m\angle D = m\angle A + m\angle C
\]
2. For the triangle with angles \( 28^\circ \), \( A \), and \( C \):
- The sum of interior angles in a triangle is \( 180^\circ \):
\[
m\angle A + m\angle C + 28^\circ = 180^\circ
\]
- We can rearrange this to express \( m\angle A + m\angle C \):
\[
m\angle A + m\angle C = 180^\circ - 28^\circ = 152^\circ
\]
3. Since \( m\angle D \) is an exterior angle, we know:
\[
m\angle D = m\angle A + m\angle C
\]
Substituting the known sum of angles, we can find \( m\angle D \):
\[
m\angle D = 152^\circ
\]
Now, let's analyze the equations:
- **Equation 1: \( m\angle B = m\angle C + 28^\circ \)**:
This equation is valid if we use the exterior angle theorem because \( m\angle B \) (the exterior at angle A) should equal \( m\angle C + 28^\circ \).
- **Equation 2: \( m\angle B = 360^\circ - (m\angle D + 125^\circ) \)**:
This equation doesn’t automatically derive from any specific triangle relationships. It mainly depends on using a 360-degree framework, and while it could technically yield a result, it's an unorthodox approach.
- **Equation 3: \( m\angle B = 180° - m\angle A \)**:
This makes sense as \( B \) (the exterior opposite angle) can indeed relate to \( A \) within certain constructs.
- **Equation 4: \( m\angle B = m\angle D + 125^\circ \)**:
If we consider exterior relationships and absolute angle definitions, this may yield a logical conclusion based on angle sum relationships, specifically since \( m\angle D = 152^\circ \).
With all the analysis put forth, the **equation that cannot be used** in a straightforward, geometrically valid manner to find \( m\angle B \) would be:
\[
\textbf{m∠B = 360° – (m∠D + 125°).}
\]