Question
Explain how the exterior angle relates to the interior angles.
A triangle has angles A, B, C. The exterior angle to angle C is angle D.
A triangle has angles A, B, C. The exterior angle to angle C is angle D.
Answers
Answered by
GPT-4o mini
In a triangle, the relationship between an exterior angle and the interior angles is based on the property that the exterior angle is equal to the sum of the two remote interior angles.
For a triangle with angles \( A \), \( B \), and \( C \), the exterior angle \( D \) corresponding to angle \( C \) is defined as the angle formed by extending one side of the triangle beyond the vertex \( C \).
The key relationship is:
\[
D = A + B
\]
This means that the measure of the exterior angle \( D \) is equal to the sum of the measures of the two interior angles \( A \) and \( B \) that are not adjacent to it (the remote angles).
### Explanation:
1. **Triangle Sum Theorem**: In any triangle, the sum of all interior angles is always \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
2. **Exterior Angle Definition**: Angle \( D \), being the exterior angle at vertex \( C \), can be represented as:
\[
D = 180^\circ - C
\]
because it is supplementary to the interior angle \( C \).
3. **Relating Interior and Exterior Angles**:
Now, if we combine the equations, we can express angle \( D \):
\[
D = 180^\circ - C = A + B
\]
Thus, the exterior angle \( D \) is directly related to the interior angles \( A \) and \( B \) by the relationship that it is equal to their sum. This property is true for all triangles and is a fundamental aspect of triangle geometry.
For a triangle with angles \( A \), \( B \), and \( C \), the exterior angle \( D \) corresponding to angle \( C \) is defined as the angle formed by extending one side of the triangle beyond the vertex \( C \).
The key relationship is:
\[
D = A + B
\]
This means that the measure of the exterior angle \( D \) is equal to the sum of the measures of the two interior angles \( A \) and \( B \) that are not adjacent to it (the remote angles).
### Explanation:
1. **Triangle Sum Theorem**: In any triangle, the sum of all interior angles is always \( 180^\circ \):
\[
A + B + C = 180^\circ
\]
2. **Exterior Angle Definition**: Angle \( D \), being the exterior angle at vertex \( C \), can be represented as:
\[
D = 180^\circ - C
\]
because it is supplementary to the interior angle \( C \).
3. **Relating Interior and Exterior Angles**:
Now, if we combine the equations, we can express angle \( D \):
\[
D = 180^\circ - C = A + B
\]
Thus, the exterior angle \( D \) is directly related to the interior angles \( A \) and \( B \) by the relationship that it is equal to their sum. This property is true for all triangles and is a fundamental aspect of triangle geometry.