Question
Examine the diagram. It is not drawn to scale.
A triangle has angles 1, 2, 3. The exterior angle to angle 1 is 4, to angle 2 is 5, to angle 3 is 6.
In the diagram shown, m∠1 = 30° and m∠5 = 120°.
What is the sum of the measures of ∠4, ∠5, and ∠6?
degrees
Identify the following angle measures.
m∠2 =
degrees
m∠4 =
degrees
m∠6 =
degrees
A triangle has angles 1, 2, 3. The exterior angle to angle 1 is 4, to angle 2 is 5, to angle 3 is 6.
In the diagram shown, m∠1 = 30° and m∠5 = 120°.
What is the sum of the measures of ∠4, ∠5, and ∠6?
degrees
Identify the following angle measures.
m∠2 =
degrees
m∠4 =
degrees
m∠6 =
degrees
Answers
GPT-4o mini
To solve this problem, we need to use the relationships between the angles of a triangle and their corresponding exterior angles.
1. **Given Information**:
- \( m\angle 1 = 30^\circ \)
- \( m\angle 5 = 120^\circ \)
2. **Using the Exterior Angle Theorem**:
The theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles. We can express this as follows:
- \( m\angle 4 = m\angle 2 + m\angle 3 \)
- \( m\angle 5 = m\angle 1 + m\angle 3 \)
- \( m\angle 6 = m\angle 1 + m\angle 2 \)
3. **Finding \( m\angle 3 \)**:
From the exterior angle \( m\angle 5 \):
\[
m\angle 5 = m\angle 1 + m\angle 3
\]
Plugging in the given values:
\[
120^\circ = 30^\circ + m\angle 3
\]
Solving for \( m\angle 3 \):
\[
m\angle 3 = 120^\circ - 30^\circ = 90^\circ
\]
4. **Finding \( m\angle 2 \)**:
Since the sum of the angles in a triangle is \( 180^\circ \):
\[
m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ
\]
Plugging in \( m\angle 1 \) and \( m\angle 3 \):
\[
30^\circ + m\angle 2 + 90^\circ = 180^\circ
\]
Therefore:
\[
m\angle 2 = 180^\circ - 30^\circ - 90^\circ = 60^\circ
\]
5. **Finding \( m\angle 4 \)**:
We know \( m\angle 4 \) is the exterior angle to \( m\angle 1 \):
\[
m\angle 4 = m\angle 2 + m\angle 3
\]
Substituting in \( m\angle 2 \) and \( m\angle 3 \):
\[
m\angle 4 = 60^\circ + 90^\circ = 150^\circ
\]
6. **Finding \( m\angle 6 \)**:
Using the formula for \( m\angle 6 \):
\[
m\angle 6 = m\angle 1 + m\angle 2
\]
Thus:
\[
m\angle 6 = 30^\circ + 60^\circ = 90^\circ
\]
7. **Finding the Sum of \( m\angle 4, m\angle 5, m\angle 6 \)**:
Now we calculate the total:
\[
m\angle 4 + m\angle 5 + m\angle 6 = 150^\circ + 120^\circ + 90^\circ = 360^\circ
\]
**Final Answers**:
- The sum of the measures of angles \( \angle 4\), \( \angle 5\), and \( \angle 6\) is \( 360^\circ \).
- \( m\angle 2 = 60^\circ \)
- \( m\angle 4 = 150^\circ \)
- \( m\angle 6 = 90^\circ \)
1. **Given Information**:
- \( m\angle 1 = 30^\circ \)
- \( m\angle 5 = 120^\circ \)
2. **Using the Exterior Angle Theorem**:
The theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles. We can express this as follows:
- \( m\angle 4 = m\angle 2 + m\angle 3 \)
- \( m\angle 5 = m\angle 1 + m\angle 3 \)
- \( m\angle 6 = m\angle 1 + m\angle 2 \)
3. **Finding \( m\angle 3 \)**:
From the exterior angle \( m\angle 5 \):
\[
m\angle 5 = m\angle 1 + m\angle 3
\]
Plugging in the given values:
\[
120^\circ = 30^\circ + m\angle 3
\]
Solving for \( m\angle 3 \):
\[
m\angle 3 = 120^\circ - 30^\circ = 90^\circ
\]
4. **Finding \( m\angle 2 \)**:
Since the sum of the angles in a triangle is \( 180^\circ \):
\[
m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ
\]
Plugging in \( m\angle 1 \) and \( m\angle 3 \):
\[
30^\circ + m\angle 2 + 90^\circ = 180^\circ
\]
Therefore:
\[
m\angle 2 = 180^\circ - 30^\circ - 90^\circ = 60^\circ
\]
5. **Finding \( m\angle 4 \)**:
We know \( m\angle 4 \) is the exterior angle to \( m\angle 1 \):
\[
m\angle 4 = m\angle 2 + m\angle 3
\]
Substituting in \( m\angle 2 \) and \( m\angle 3 \):
\[
m\angle 4 = 60^\circ + 90^\circ = 150^\circ
\]
6. **Finding \( m\angle 6 \)**:
Using the formula for \( m\angle 6 \):
\[
m\angle 6 = m\angle 1 + m\angle 2
\]
Thus:
\[
m\angle 6 = 30^\circ + 60^\circ = 90^\circ
\]
7. **Finding the Sum of \( m\angle 4, m\angle 5, m\angle 6 \)**:
Now we calculate the total:
\[
m\angle 4 + m\angle 5 + m\angle 6 = 150^\circ + 120^\circ + 90^\circ = 360^\circ
\]
**Final Answers**:
- The sum of the measures of angles \( \angle 4\), \( \angle 5\), and \( \angle 6\) is \( 360^\circ \).
- \( m\angle 2 = 60^\circ \)
- \( m\angle 4 = 150^\circ \)
- \( m\angle 6 = 90^\circ \)