Question
The $m\angle B$ is $93^{\circ }$ $m\overparen {BC}$ is $58^{\circ }$ and $m\overparen {CD}$ is $106^{\circ }$ Find the missing angle measures.
(2 points)
$m\angle A=\square ^{\circ },m\angle C=\square ^{\circ }$ and $m\angle D=\square ^{\circ }$
Answers
GPT-4o mini
To find the missing angles in the scenario described, we can use the properties of triangles and the angles in polygons.
1. We are given:
- \(m\angle B = 93^\circ\)
- \(m\overparen{BC} = 58^\circ\)
- \(m\overparen{CD} = 106^\circ\)
We need to find \(m\angle A\), \(m\angle C\), and \(m\angle D\).
2. Assuming points A, B, C, D form a closed figure (e.g., a polygon), we can analyze their relationships:
- The sum of the angles in any quadrilateral is \(360^\circ\).
- If \( \angle A\), \( \angle B\), \( \angle C\), and \( \angle D\) are the angles of a quadrilateral, then:
\[
m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ.
\]
3. We can solve step by step:
- From our problem, we already have \(m\angle B = 93^\circ\).
- We need to clarify what \(m\angle C\) and \(m\angle D\) are in relation to the segments \(BC\) and \(CD\).
4. If \(m\overparen{BC}\) and \(m\overparen{CD}\) indicate the supplementary angles formed at those points (assuming we're forming triangles):
- Let's say \(m\angle C\) is the interior angle facing \(BC\), then it is supplementary to it.
- Therefore, \(m\angle C + m\overparen{BC} = 180^\circ\):
\[
m\angle C + 58^\circ = 180^\circ \implies m\angle C = 180^\circ - 58^\circ = 122^\circ.
\]
- Similarly, for \(m\angle D\), where it is complementary with \(m\overparen{CD}\):
\[
m\angle D + m\overparen{CD} = 180^\circ \implies m\angle D + 106^\circ = 180^\circ \implies m\angle D = 180^\circ - 106^\circ = 74^\circ.
\]
5. Finally, we can find \(m\angle A\) as follows:
\[
m\angle A + 93^\circ + 122^\circ + 74^\circ = 360^\circ.
\]
- Adding the known angles:
\[
93^\circ + 122^\circ + 74^\circ = 289^\circ.
\]
- So, we solve for \(m\angle A\):
\[
m\angle A + 289^\circ = 360^\circ \implies m\angle A = 360^\circ - 289^\circ = 71^\circ.
\]
Thus, summarizing the measures:
- \(m\angle A = 71^\circ\)
- \(m\angle C = 122^\circ\)
- \(m\angle D = 74^\circ\)
Final answers:
\(m\angle A = \boxed{71^\circ}, m\angle C = \boxed{122^\circ}, m\angle D = \boxed{74^\circ}\).
1. We are given:
- \(m\angle B = 93^\circ\)
- \(m\overparen{BC} = 58^\circ\)
- \(m\overparen{CD} = 106^\circ\)
We need to find \(m\angle A\), \(m\angle C\), and \(m\angle D\).
2. Assuming points A, B, C, D form a closed figure (e.g., a polygon), we can analyze their relationships:
- The sum of the angles in any quadrilateral is \(360^\circ\).
- If \( \angle A\), \( \angle B\), \( \angle C\), and \( \angle D\) are the angles of a quadrilateral, then:
\[
m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ.
\]
3. We can solve step by step:
- From our problem, we already have \(m\angle B = 93^\circ\).
- We need to clarify what \(m\angle C\) and \(m\angle D\) are in relation to the segments \(BC\) and \(CD\).
4. If \(m\overparen{BC}\) and \(m\overparen{CD}\) indicate the supplementary angles formed at those points (assuming we're forming triangles):
- Let's say \(m\angle C\) is the interior angle facing \(BC\), then it is supplementary to it.
- Therefore, \(m\angle C + m\overparen{BC} = 180^\circ\):
\[
m\angle C + 58^\circ = 180^\circ \implies m\angle C = 180^\circ - 58^\circ = 122^\circ.
\]
- Similarly, for \(m\angle D\), where it is complementary with \(m\overparen{CD}\):
\[
m\angle D + m\overparen{CD} = 180^\circ \implies m\angle D + 106^\circ = 180^\circ \implies m\angle D = 180^\circ - 106^\circ = 74^\circ.
\]
5. Finally, we can find \(m\angle A\) as follows:
\[
m\angle A + 93^\circ + 122^\circ + 74^\circ = 360^\circ.
\]
- Adding the known angles:
\[
93^\circ + 122^\circ + 74^\circ = 289^\circ.
\]
- So, we solve for \(m\angle A\):
\[
m\angle A + 289^\circ = 360^\circ \implies m\angle A = 360^\circ - 289^\circ = 71^\circ.
\]
Thus, summarizing the measures:
- \(m\angle A = 71^\circ\)
- \(m\angle C = 122^\circ\)
- \(m\angle D = 74^\circ\)
Final answers:
\(m\angle A = \boxed{71^\circ}, m\angle C = \boxed{122^\circ}, m\angle D = \boxed{74^\circ}\).