Examine the diagram, where quadrilateral HIJK İs inscribed in O C. H K 112° I J © 2016 StrongMind. Created using GeoGebra. If MZIJK = 112°, what is MZIHK? Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 42
Examine the diagram, where quadrilateral HIJK İs inscribed in O C. H K 112° I J © 2016 StrongMind. Created using GeoGebra. If MZIJK = 112°, what is MZIHK? Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 42
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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![### Understanding Inscribed Quadrilaterals
Examine the diagram, where quadrilateral \( H I J K \) is inscribed in circle \( \odot C \).
**Diagram Explanation:**
- The figure shows a circle \( \odot C \) with a quadrilateral \( H I J K \) inscribed in it.
- Points \( H \), \( I \), \( J \), and \( K \) are on the circumference of the circle.
- There is an indicator of an angle \( \angle I J K \) measuring \( 112^\circ \) at vertex \( J \).
- The other given angle is \( \angle H K I \), which is unknown and needs to be calculated.
© 2016 StrongMind. Created using GeoGebra.
#### Problem Statement
**If \( m \angle I J K = 112^\circ \), what is \( m \angle I H K \)?**
Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 42
---
**Solution Approach:**
- Use the property of inscribed angles and the fact that opposite angles of an inscribed quadrilateral sum up to \( 180^\circ \).
- Given \( m \angle I J K = 112^\circ \), the opposite angle \( \angle I H K \) can be found using:
\[
m \angle I H K + m \angle I J K = 180^\circ
\]
\[
m \angle I H K + 112^\circ = 180^\circ
\]
\[
m \angle I H K = 180^\circ - 112^\circ
\]
\[
m \angle I H K = 68^\circ
\]
- Thus, \( m \angle I H K \) is \( 68^\circ \).
### Answer
\( m \angle I H K \) is 68.
Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 68
---
*Note: Understanding these fundamental geometrical principles is crucial for solving problems involving inscribed figures in circles.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38119a94-a193-4725-8b44-bc4499933554%2Fcfffa5f6-b21a-4159-b060-33223918edd7%2F9124isu_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Inscribed Quadrilaterals
Examine the diagram, where quadrilateral \( H I J K \) is inscribed in circle \( \odot C \).
**Diagram Explanation:**
- The figure shows a circle \( \odot C \) with a quadrilateral \( H I J K \) inscribed in it.
- Points \( H \), \( I \), \( J \), and \( K \) are on the circumference of the circle.
- There is an indicator of an angle \( \angle I J K \) measuring \( 112^\circ \) at vertex \( J \).
- The other given angle is \( \angle H K I \), which is unknown and needs to be calculated.
© 2016 StrongMind. Created using GeoGebra.
#### Problem Statement
**If \( m \angle I J K = 112^\circ \), what is \( m \angle I H K \)?**
Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 42
---
**Solution Approach:**
- Use the property of inscribed angles and the fact that opposite angles of an inscribed quadrilateral sum up to \( 180^\circ \).
- Given \( m \angle I J K = 112^\circ \), the opposite angle \( \angle I H K \) can be found using:
\[
m \angle I H K + m \angle I J K = 180^\circ
\]
\[
m \angle I H K + 112^\circ = 180^\circ
\]
\[
m \angle I H K = 180^\circ - 112^\circ
\]
\[
m \angle I H K = 68^\circ
\]
- Thus, \( m \angle I H K \) is \( 68^\circ \).
### Answer
\( m \angle I H K \) is 68.
Enter your answer as the number that correctly fills in the blank in the previous sentence, like this: 68
---
*Note: Understanding these fundamental geometrical principles is crucial for solving problems involving inscribed figures in circles.*
- Point **M** is at the top of the circle.
- Point **N** is to the right.
- Point **O** is at the bottom.
- Point **P** is to the left.
- Circle’s center is marked as **C**.
**Key angles provided:**
- \( \angle M = (8x - 24)^\circ \) near point M.
- \( \angle O = (4x)^\circ \) near point O.
© 2016 StrongMind. Created using GeoGebra.
**Question:**
What is the measure of \( \angle NOP \)?
- [ ] 242°
- [ ] **112°**
- [ ] 68°
- [ ] 136°
Selection: The correct answer is highlighted as 112°.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38119a94-a193-4725-8b44-bc4499933554%2Fcfffa5f6-b21a-4159-b060-33223918edd7%2F47cwgbv_processed.png&w=3840&q=75)
Transcribed Image Text:**Study the diagram provided, where quadrilateral \( MNOP \) is inscribed in circle \( C \) such that \( m \angle M = (8x - 24)^\circ \) and \( m \angle O = (4x)^\circ \).**
- Point **M** is at the top of the circle.
- Point **N** is to the right.
- Point **O** is at the bottom.
- Point **P** is to the left.
- Circle’s center is marked as **C**.
**Key angles provided:**
- \( \angle M = (8x - 24)^\circ \) near point M.
- \( \angle O = (4x)^\circ \) near point O.
© 2016 StrongMind. Created using GeoGebra.
**Question:**
What is the measure of \( \angle NOP \)?
- [ ] 242°
- [ ] **112°**
- [ ] 68°
- [ ] 136°
Selection: The correct answer is highlighted as 112°.
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