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
Related questions
Question
![Use the image, given information, and prove statement to construct a two-column proof. You must include a two-column proof with a column for statements and a column for reasons that justify your statements.
[Image: Diagram showing four angles formed by two intersecting lines. The angles are labeled 1, 2, 3, and 4 in a clockwise manner starting from the top left.]
Given: m∠1 = (2x - 3)°
---
### Explanation of the Diagram:
The diagram depicts two straight lines intersecting at a point, which forms four angles: ∠1, ∠2, ∠3, and ∠4. The given measure of angle 1 is expressed as (2x - 3)°. This information can be used to set up and solve a two-column proof, a common structure in geometry used to verify that a particular mathematical statement is true.
### Two-Column Proof Structure:
A two-column proof consists of two main parts: the Statements and the Reasons. Each step in the proof is written in the Statements column, and the justification or logical reasoning for each step is articulated in the Reasons column. This method helps in systematically proving mathematical theorems.
### Example:
Below is a generic example of how you might start constructing a two-column proof for given information.
| Statements | Reasons |
|----------------------------------------------|----------------------------------------------|
| 1. m∠1 = (2x - 3)° | Given |
| 2. m∠1 + m∠2 = 180° | Linear Pair Theorem |
| 3. m∠2 = 180° - m∠1 | Subtraction Property of Equality |
| 4. Substitute (2x - 3)° for m∠1 in Step 3 | Substitution |
| 5. m∠2 = 180° - (2x - 3)° | Simplification |
This table provides the framework, and additional steps will be required depending on what is to be proven.
---
To complete the proof, more specific steps tailored to the unique problem will need to be devised, ensuring logical consistency and adherence to geometric principles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe53cea77-c966-4d47-ae93-d2c22a57e8a1%2F5fb3e438-ae9f-419d-8148-486d6dbaa1d7%2Forhem5g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use the image, given information, and prove statement to construct a two-column proof. You must include a two-column proof with a column for statements and a column for reasons that justify your statements.
[Image: Diagram showing four angles formed by two intersecting lines. The angles are labeled 1, 2, 3, and 4 in a clockwise manner starting from the top left.]
Given: m∠1 = (2x - 3)°
---
### Explanation of the Diagram:
The diagram depicts two straight lines intersecting at a point, which forms four angles: ∠1, ∠2, ∠3, and ∠4. The given measure of angle 1 is expressed as (2x - 3)°. This information can be used to set up and solve a two-column proof, a common structure in geometry used to verify that a particular mathematical statement is true.
### Two-Column Proof Structure:
A two-column proof consists of two main parts: the Statements and the Reasons. Each step in the proof is written in the Statements column, and the justification or logical reasoning for each step is articulated in the Reasons column. This method helps in systematically proving mathematical theorems.
### Example:
Below is a generic example of how you might start constructing a two-column proof for given information.
| Statements | Reasons |
|----------------------------------------------|----------------------------------------------|
| 1. m∠1 = (2x - 3)° | Given |
| 2. m∠1 + m∠2 = 180° | Linear Pair Theorem |
| 3. m∠2 = 180° - m∠1 | Subtraction Property of Equality |
| 4. Substitute (2x - 3)° for m∠1 in Step 3 | Substitution |
| 5. m∠2 = 180° - (2x - 3)° | Simplification |
This table provides the framework, and additional steps will be required depending on what is to be proven.
---
To complete the proof, more specific steps tailored to the unique problem will need to be devised, ensuring logical consistency and adherence to geometric principles.
In the provided diagram, there are four angles formed at the intersection of two lines. These angles are labeled as ∠1, ∠2, ∠3, and ∠4.
**Given:**
m ∠1 = (2x - 3)°
m ∠3 = 25°
**Objective:**
Prove that x = 14
---
**Solution:**
1. **Understand the relationships between the angles:**
- Angle ∠1 and angle ∠3 are vertical angles.
- The measure of vertical angles is equal.
2. **Set up the equation based on the given information:**
Since vertical angles are equal:
\[ m ∠1 = m ∠3 \]
Given:
\[ m ∠1 = (2x - 3)° \]
\[ m ∠3 = 25° \]
Therefore:
\[ (2x - 3)° = 25° \]
3. **Solve for x:**
\[ 2x - 3 = 25 \]
\[ 2x = 25 + 3 \]
\[ 2x = 28 \]
\[ x = 14 \]
4. **Verification:**
To verify, substitute x = 14 back into the expression for m ∠1:
\[ m ∠1 = (2(14) - 3)° \]
\[ m ∠1 = (28 - 3)° \]
\[ m ∠1 = 25° \]
Since m ∠1 = 25°, which is equal to m ∠3, our solution is correct.
**Conclusion:**
We have proven that x = 14.
---
This explanation should clarify the problem for students and guide them through the logical steps required to solve for x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe53cea77-c966-4d47-ae93-d2c22a57e8a1%2F5fb3e438-ae9f-419d-8148-486d6dbaa1d7%2F32493ea_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
In the provided diagram, there are four angles formed at the intersection of two lines. These angles are labeled as ∠1, ∠2, ∠3, and ∠4.
**Given:**
m ∠1 = (2x - 3)°
m ∠3 = 25°
**Objective:**
Prove that x = 14
---
**Solution:**
1. **Understand the relationships between the angles:**
- Angle ∠1 and angle ∠3 are vertical angles.
- The measure of vertical angles is equal.
2. **Set up the equation based on the given information:**
Since vertical angles are equal:
\[ m ∠1 = m ∠3 \]
Given:
\[ m ∠1 = (2x - 3)° \]
\[ m ∠3 = 25° \]
Therefore:
\[ (2x - 3)° = 25° \]
3. **Solve for x:**
\[ 2x - 3 = 25 \]
\[ 2x = 25 + 3 \]
\[ 2x = 28 \]
\[ x = 14 \]
4. **Verification:**
To verify, substitute x = 14 back into the expression for m ∠1:
\[ m ∠1 = (2(14) - 3)° \]
\[ m ∠1 = (28 - 3)° \]
\[ m ∠1 = 25° \]
Since m ∠1 = 25°, which is equal to m ∠3, our solution is correct.
**Conclusion:**
We have proven that x = 14.
---
This explanation should clarify the problem for students and guide them through the logical steps required to solve for x.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning