Find the value of each variable. If your answer is not an integer, round to the nearest tenth. 11. 140 60 of
Find the value of each variable. If your answer is not an integer, round to the nearest tenth. 11. 140 60 of
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Title: Solving for Unknown Variables in Geometric Diagrams**
**Objective:**
Find the value of each variable. If your answer is not an integer, round to the nearest tenth.
**Problem Statement:**
11.
**Diagram Explanation:**
The diagram shows a circle and a triangle with angles at two intersecting lines.
- There is a circle with a central angle of 60 degrees.
- A straight line intersecting the circle, creating an external angle of 140 degrees.
- A triangle is formed outside the circle, with one of its angles labeled as \( x^\circ \).
- The external angle (140 degrees) and part of the triangle share a straight line segment.
**Solution Explanation:**
1. **Understanding the External Angle:**
- The external angle given is 140 degrees.
- The sum of the internal angles on a straight line is 180 degrees.
- Therefore, the internal adjacent angle to the 140-degree angle is:
\[
180^\circ - 140^\circ = 40^\circ
\]
2. **Using the Central Angle:**
- The central angle inside the circle (opposite to the segment extending outward) is 60 degrees.
3. **Finding the Value of \( x \):**
- In the triangle formed, the internal angle adjacent to the 140-degree angle is found to be 40 degrees (as calculated).
- Let’s use the triangle sum property, which states that the sum of angles in a triangle is always 180 degrees.
- Since one angle is part of a right-angled section within the circle (60 degrees):
\[
x + 60^\circ + 40^\circ = 180^\circ
\]
- Simplifying,
\[
x + 100^\circ = 180^\circ
\]
- Solving for \( x \):
\[
x = 180^\circ - 100^\circ = 80^\circ
\]
Therefore, the value of the variable \( x \) is \( 80^\circ \).
**Note:** This problem requires knowledge of the properties of angles in a triangle and the angle sum property. The key concept used is that the sum of the angles in any triangle is always 180 degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d85d9da-75db-48bf-a3cb-c4a1ef66ebdb%2Ffd388bdd-675a-4781-965b-ffd81e88c24c%2Fvbtlue_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving for Unknown Variables in Geometric Diagrams**
**Objective:**
Find the value of each variable. If your answer is not an integer, round to the nearest tenth.
**Problem Statement:**
11.
**Diagram Explanation:**
The diagram shows a circle and a triangle with angles at two intersecting lines.
- There is a circle with a central angle of 60 degrees.
- A straight line intersecting the circle, creating an external angle of 140 degrees.
- A triangle is formed outside the circle, with one of its angles labeled as \( x^\circ \).
- The external angle (140 degrees) and part of the triangle share a straight line segment.
**Solution Explanation:**
1. **Understanding the External Angle:**
- The external angle given is 140 degrees.
- The sum of the internal angles on a straight line is 180 degrees.
- Therefore, the internal adjacent angle to the 140-degree angle is:
\[
180^\circ - 140^\circ = 40^\circ
\]
2. **Using the Central Angle:**
- The central angle inside the circle (opposite to the segment extending outward) is 60 degrees.
3. **Finding the Value of \( x \):**
- In the triangle formed, the internal angle adjacent to the 140-degree angle is found to be 40 degrees (as calculated).
- Let’s use the triangle sum property, which states that the sum of angles in a triangle is always 180 degrees.
- Since one angle is part of a right-angled section within the circle (60 degrees):
\[
x + 60^\circ + 40^\circ = 180^\circ
\]
- Simplifying,
\[
x + 100^\circ = 180^\circ
\]
- Solving for \( x \):
\[
x = 180^\circ - 100^\circ = 80^\circ
\]
Therefore, the value of the variable \( x \) is \( 80^\circ \).
**Note:** This problem requires knowledge of the properties of angles in a triangle and the angle sum property. The key concept used is that the sum of the angles in any triangle is always 180 degrees.
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