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
![**Title: Finding Interior Angles of a Trapezoid**
**Objective:**
Learn how to find the interior angles of a trapezoid.
**Problem:**
Given a trapezoid \(EFGH\), find each interior angle based on the provided diagram.
**Diagram Description:**
In the provided trapezoid diagram, the vertices are labeled as \(E\), \(F\), \(G\), and \(H\). There is an angle \(H\) at the bottom left of the trapezoid which is marked as \(117^\circ\). The sides \(EH\) and \(FG\) are parallel to each other.
**Task:**
Find the angles at vertices \(E\), \(F\), and \(G\).
**Solution Steps:**
1. **Understanding Trapezoid Angle Properties:**
- The sum of the internal angles in any quadrilateral is \(360^\circ\).
- In a trapezoid with one pair of parallel sides, consecutive angles between the parallel sides are supplementary (they add up to \(180^\circ\)).
2. **Given Information:**
- \( \angle H = 117^\circ \)
3. **Find \( \angle G \):**
- Since \(H\) and \(G\) are consecutive angles between the parallel sides, they are supplementary.
\[
\angle H + \angle G = 180^\circ
\]
\[
117^\circ + \angle G = 180^\circ
\]
\[
\angle G = 180^\circ - 117^\circ = 63^\circ
\]
4. **Find \( \angle E \) and \( \angle F \):**
- Angles \(E\) and \(F\) are also consecutive angles between the parallel sides, hence:
\[
\angle E + \angle F = 180^\circ
\]
- Since \(EH\) and \(FG\) are parallel, \( \angle E \) and \( \angle G \) are equal because \(EF \parallel HG\) (alternate interior angles). Thus,
\[
\angle E = \angle G = 63^\circ
\]
- Use the quadrilateral angle sum property to find \( \angle F \):
\[
\angle E +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c42cb7f-ddef-4115-a77a-7e64e5b29e59%2Ff7095eb5-1a73-4465-a055-af5d9eac190e%2F8jazrca_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding Interior Angles of a Trapezoid**
**Objective:**
Learn how to find the interior angles of a trapezoid.
**Problem:**
Given a trapezoid \(EFGH\), find each interior angle based on the provided diagram.
**Diagram Description:**
In the provided trapezoid diagram, the vertices are labeled as \(E\), \(F\), \(G\), and \(H\). There is an angle \(H\) at the bottom left of the trapezoid which is marked as \(117^\circ\). The sides \(EH\) and \(FG\) are parallel to each other.
**Task:**
Find the angles at vertices \(E\), \(F\), and \(G\).
**Solution Steps:**
1. **Understanding Trapezoid Angle Properties:**
- The sum of the internal angles in any quadrilateral is \(360^\circ\).
- In a trapezoid with one pair of parallel sides, consecutive angles between the parallel sides are supplementary (they add up to \(180^\circ\)).
2. **Given Information:**
- \( \angle H = 117^\circ \)
3. **Find \( \angle G \):**
- Since \(H\) and \(G\) are consecutive angles between the parallel sides, they are supplementary.
\[
\angle H + \angle G = 180^\circ
\]
\[
117^\circ + \angle G = 180^\circ
\]
\[
\angle G = 180^\circ - 117^\circ = 63^\circ
\]
4. **Find \( \angle E \) and \( \angle F \):**
- Angles \(E\) and \(F\) are also consecutive angles between the parallel sides, hence:
\[
\angle E + \angle F = 180^\circ
\]
- Since \(EH\) and \(FG\) are parallel, \( \angle E \) and \( \angle G \) are equal because \(EF \parallel HG\) (alternate interior angles). Thus,
\[
\angle E = \angle G = 63^\circ
\]
- Use the quadrilateral angle sum property to find \( \angle F \):
\[
\angle E +
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