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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Question
![**Finding the Measure of Each Angle in a Trapezoid**
In the given trapezoid \(PQRS\), you are tasked with finding the measures of the angles \(\angle LQ\), \(\angle LR\), and \(\angle LS\). Below is the breakdown of this problem.
**Diagram Explanation:**
The diagram shows a trapezoid \(PQRS\) with the following vertices:
- \(P\) (top-left)
- \(Q\) (top-right)
- \(R\) (bottom-right)
- \(S\) (bottom-left)
Additionally, the exterior angle at vertex \(Q\) is given as \(112^\circ\).
**Mathematical Interpretation:**
To find the interior angles at each vertex, we can use the properties of angles in trapezoids and the fact that the interior angle sum of any four-sided polygon (quadrilateral) is \(360^\circ\).
Given:
\[
\angle PQS = 112^\circ
\]
Since \(\angle PQS\) is an exterior angle, the adjacent interior angle \(\angle PQR\) can be found using:
\[
\angle PQR + \angle PQS = 180^\circ \implies \angle PQR = 180^\circ - 112^\circ = 68^\circ
\]
In this specific trapezoid, if \(PQ\) and \(RS\) are parallel, each pair of corresponding angles \(\angle SRQ\) and \(\angle RPS\) will sum up to \(180^\circ \).
However, based on the question, further details or additional information about specific angles are needed to come up with the following formulas.
Use the supplementary rule to find the complementary angle:
\(
\angle LQ + \angle LR + \angle LS = 360^\circ \text.
)
With this information, we shall calculate the angles and fill in the values as required by the learning activity:
\(\angle LQ =\)\_\_\_
\(\angle LR =\)\_\_\_
\(\angle LS =\)\_\_\_
Use these details and the relationships between angles to solve for the missing measures and ensure all conditions (like parallel lines and sum of angles in a quadrilateral) are used.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3285f48-0c32-44fd-8ae1-1a69587ed78f%2F824343bb-7254-49ab-aad3-b8c5ac9a81d7%2Fkd6826p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Measure of Each Angle in a Trapezoid**
In the given trapezoid \(PQRS\), you are tasked with finding the measures of the angles \(\angle LQ\), \(\angle LR\), and \(\angle LS\). Below is the breakdown of this problem.
**Diagram Explanation:**
The diagram shows a trapezoid \(PQRS\) with the following vertices:
- \(P\) (top-left)
- \(Q\) (top-right)
- \(R\) (bottom-right)
- \(S\) (bottom-left)
Additionally, the exterior angle at vertex \(Q\) is given as \(112^\circ\).
**Mathematical Interpretation:**
To find the interior angles at each vertex, we can use the properties of angles in trapezoids and the fact that the interior angle sum of any four-sided polygon (quadrilateral) is \(360^\circ\).
Given:
\[
\angle PQS = 112^\circ
\]
Since \(\angle PQS\) is an exterior angle, the adjacent interior angle \(\angle PQR\) can be found using:
\[
\angle PQR + \angle PQS = 180^\circ \implies \angle PQR = 180^\circ - 112^\circ = 68^\circ
\]
In this specific trapezoid, if \(PQ\) and \(RS\) are parallel, each pair of corresponding angles \(\angle SRQ\) and \(\angle RPS\) will sum up to \(180^\circ \).
However, based on the question, further details or additional information about specific angles are needed to come up with the following formulas.
Use the supplementary rule to find the complementary angle:
\(
\angle LQ + \angle LR + \angle LS = 360^\circ \text.
)
With this information, we shall calculate the angles and fill in the values as required by the learning activity:
\(\angle LQ =\)\_\_\_
\(\angle LR =\)\_\_\_
\(\angle LS =\)\_\_\_
Use these details and the relationships between angles to solve for the missing measures and ensure all conditions (like parallel lines and sum of angles in a quadrilateral) are used.
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