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
![### Problem Statement: Geometry - Isosceles Triangle
#### Diagram:
The diagram shows an isosceles triangle \( \triangle HAM \) with the following characteristics:
- Vertex \( A \) is the apex of the isosceles triangle.
- \( \angle A \) measures \( 32^\circ \).
- The sides \( HA \) and \( MA \) are marked as equal (congruent).
The vertices are labeled as follows:
- \( H \) and \( M \) are the base vertices.
- \( A \) is the vertex opposite the base \( HM \).
#### Question:
In isosceles \( \triangle HAM \), \( m \angle A = 32^\circ \). What is \( m \angle H \)?
#### Choices:
- \( 32^\circ \)
- \( 58^\circ \)
- \( 74^\circ \)
- \( 148^\circ \)
#### Solution Explanation:
To solve for \( m \angle H \), we use the properties of isosceles triangles and the fact that the sum of internal angles of a triangle is \( 180^\circ \).
Given that \( \triangle HAM \) is isosceles with \( HA = MA \), the base angles \( \angle H \) and \( \angle M \) are equal. Let \( m \angle H = m \angle M = x \).
Thus, we have:
\[ m \angle A + m \angle H + m \angle M = 180^\circ \]
Substitute the known values:
\[ 32^\circ + x + x = 180^\circ \]
\[ 32^\circ + 2x = 180^\circ \]
\[ 2x = 180^\circ - 32^\circ \]
\[ 2x = 148^\circ \]
\[ x = 74^\circ \]
Therefore, \( m \angle H = 74^\circ \).
#### Correct Answer:
- \( 74^\circ \)
---
This question helps students practice understanding properties of isosceles triangles, specifically the internal angle relationships and sum of angles in a triangle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f766445-6e88-4449-9b12-3a9b363ebf4d%2F0541b1f4-f955-47e1-8a29-fe8ac0b01b28%2Frwlv9nb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Geometry - Isosceles Triangle
#### Diagram:
The diagram shows an isosceles triangle \( \triangle HAM \) with the following characteristics:
- Vertex \( A \) is the apex of the isosceles triangle.
- \( \angle A \) measures \( 32^\circ \).
- The sides \( HA \) and \( MA \) are marked as equal (congruent).
The vertices are labeled as follows:
- \( H \) and \( M \) are the base vertices.
- \( A \) is the vertex opposite the base \( HM \).
#### Question:
In isosceles \( \triangle HAM \), \( m \angle A = 32^\circ \). What is \( m \angle H \)?
#### Choices:
- \( 32^\circ \)
- \( 58^\circ \)
- \( 74^\circ \)
- \( 148^\circ \)
#### Solution Explanation:
To solve for \( m \angle H \), we use the properties of isosceles triangles and the fact that the sum of internal angles of a triangle is \( 180^\circ \).
Given that \( \triangle HAM \) is isosceles with \( HA = MA \), the base angles \( \angle H \) and \( \angle M \) are equal. Let \( m \angle H = m \angle M = x \).
Thus, we have:
\[ m \angle A + m \angle H + m \angle M = 180^\circ \]
Substitute the known values:
\[ 32^\circ + x + x = 180^\circ \]
\[ 32^\circ + 2x = 180^\circ \]
\[ 2x = 180^\circ - 32^\circ \]
\[ 2x = 148^\circ \]
\[ x = 74^\circ \]
Therefore, \( m \angle H = 74^\circ \).
#### Correct Answer:
- \( 74^\circ \)
---
This question helps students practice understanding properties of isosceles triangles, specifically the internal angle relationships and sum of angles in a triangle.
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