Question
Use the diagram to find the angle measures that satisfy each case.
E
H
1
2
4
3
G
F
Find the measures of all four angles if m<2-m<1=30°.
Answer:
m<1 =
, m<2 =
,m<3=
,m<4=
E
H
1
2
4
3
G
F
Find the measures of all four angles if m<2-m<1=30°.
Answer:
m<1 =
, m<2 =
,m<3=
,m<4=
Answers
Answered by
GPT-4o mini
To solve the problem, we first note the relationship given: \( m \angle 2 - m \angle 1 = 30^\circ \).
Let's define:
- \( m \angle 1 = x \)
- \( m \angle 2 = y \)
According to the information given, we can express the first relationship as:
\[
y - x = 30^\circ
\]
or rearranged:
\[
y = x + 30^\circ
\]
Next, we typically have relationships between the angles that might depend on their setup in the diagram, which we don't have direct access to, but we can consider common geometric relationships. Assuming that angles 1, 2, 3, and 4 are related in a way typically found in geometric configurations (for instance, vertical angles or angles around a point), we can get some additional equations.
If we also assume that angles 1 and 2 are supplementary to angles 3 and 4, let's say:
\[
m \angle 1 + m \angle 3 = 180^\circ \quad (1)
\]
\[
m \angle 2 + m \angle 4 = 180^\circ \quad (2)
\]
Since \( m \angle 2 = y \) and \( m \angle 1 = x \), we can substitute into the equations (1) and (2):
1. From equation \( (1) \):
\[
x + m \angle 3 = 180^\circ \implies m \angle 3 = 180^\circ - x
\]
2. From equation \( (2) \):
\[
(x + 30^\circ) + m \angle 4 = 180^\circ \implies m \angle 4 = 180^\circ - (x + 30^\circ) = 150^\circ - x
\]
Now we have \( m \angle 3 \) and \( m \angle 4 \) in terms of \( x \):
- \( m \angle 3 = 180^\circ - x \)
- \( m \angle 4 = 150^\circ - x \)
Now we need to find a consistent value for \( x \) ensuring all angle measures stay positive.
If we sum all angles (assuming they all form a full circle, which is typical in planar geometry) it equals 360 degrees:
\[
x + (x + 30^\circ) + (180^\circ - x) + (150^\circ - x) = 360^\circ
\]
Simplifying the left side:
\[
x + x + 30^\circ + 180^\circ - x + 150^\circ - x = 360^\circ
\]
\[
30^\circ + 180^\circ + 150^\circ = 360^\circ
\]
\[
360^\circ = 360^\circ
\]
This equation holds true for any \( x \) under the constraints we set.
Now let's check for specific possible values for \( x \) given that it must stay positive.
**Example Calculation:**
Let's assume \( x = 60^\circ \) (just an example — you may adjust based on actual angle configuration).
- \( m \angle 1 = x = 60^\circ \)
- \( m \angle 2 = x + 30^\circ = 90^\circ \)
- \( m \angle 3 = 180^\circ - x = 120^\circ \)
- \( m \angle 4 = 150^\circ - x = 90^\circ \)
So a possible solution would be:
- \( m \angle 1 = 60^\circ \)
- \( m \angle 2 = 90^\circ \)
- \( m \angle 3 = 120^\circ \)
- \( m \angle 4 = 90^\circ \)
You can adjust according to specific conditions from the diagram mentioned if that differs.
But as per the simplest checks based on provided relationships, ensure all angle values are correct within defined positive bounds, and check the original angle conditions, including the supplementary nature mentioned.
If you have additional specifics regarding the angle relationship depicted in the diagram, feel free to share!
Let's define:
- \( m \angle 1 = x \)
- \( m \angle 2 = y \)
According to the information given, we can express the first relationship as:
\[
y - x = 30^\circ
\]
or rearranged:
\[
y = x + 30^\circ
\]
Next, we typically have relationships between the angles that might depend on their setup in the diagram, which we don't have direct access to, but we can consider common geometric relationships. Assuming that angles 1, 2, 3, and 4 are related in a way typically found in geometric configurations (for instance, vertical angles or angles around a point), we can get some additional equations.
If we also assume that angles 1 and 2 are supplementary to angles 3 and 4, let's say:
\[
m \angle 1 + m \angle 3 = 180^\circ \quad (1)
\]
\[
m \angle 2 + m \angle 4 = 180^\circ \quad (2)
\]
Since \( m \angle 2 = y \) and \( m \angle 1 = x \), we can substitute into the equations (1) and (2):
1. From equation \( (1) \):
\[
x + m \angle 3 = 180^\circ \implies m \angle 3 = 180^\circ - x
\]
2. From equation \( (2) \):
\[
(x + 30^\circ) + m \angle 4 = 180^\circ \implies m \angle 4 = 180^\circ - (x + 30^\circ) = 150^\circ - x
\]
Now we have \( m \angle 3 \) and \( m \angle 4 \) in terms of \( x \):
- \( m \angle 3 = 180^\circ - x \)
- \( m \angle 4 = 150^\circ - x \)
Now we need to find a consistent value for \( x \) ensuring all angle measures stay positive.
If we sum all angles (assuming they all form a full circle, which is typical in planar geometry) it equals 360 degrees:
\[
x + (x + 30^\circ) + (180^\circ - x) + (150^\circ - x) = 360^\circ
\]
Simplifying the left side:
\[
x + x + 30^\circ + 180^\circ - x + 150^\circ - x = 360^\circ
\]
\[
30^\circ + 180^\circ + 150^\circ = 360^\circ
\]
\[
360^\circ = 360^\circ
\]
This equation holds true for any \( x \) under the constraints we set.
Now let's check for specific possible values for \( x \) given that it must stay positive.
**Example Calculation:**
Let's assume \( x = 60^\circ \) (just an example — you may adjust based on actual angle configuration).
- \( m \angle 1 = x = 60^\circ \)
- \( m \angle 2 = x + 30^\circ = 90^\circ \)
- \( m \angle 3 = 180^\circ - x = 120^\circ \)
- \( m \angle 4 = 150^\circ - x = 90^\circ \)
So a possible solution would be:
- \( m \angle 1 = 60^\circ \)
- \( m \angle 2 = 90^\circ \)
- \( m \angle 3 = 120^\circ \)
- \( m \angle 4 = 90^\circ \)
You can adjust according to specific conditions from the diagram mentioned if that differs.
But as per the simplest checks based on provided relationships, ensure all angle values are correct within defined positive bounds, and check the original angle conditions, including the supplementary nature mentioned.
If you have additional specifics regarding the angle relationship depicted in the diagram, feel free to share!