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
![### Proving Triangle Congruence: Example Problem
**Objective:**
Prove that triangle \( \triangle ABC \cong \triangle DEF \).
**Description of Graphs:**
The image consists of two coordinate planes with triangle \( \triangle ABC \) on the left and triangle \( \triangle DEF \) on the right.
**Triangles and Coordinates:**
- **Triangle \( \triangle ABC \):**
- Point \( A \) is at coordinates \((-6, 4)\).
- Point \( B \) is at coordinates \((-4, 2)\).
- Point \( C \) is at coordinates \((-2, 3)\).
- **Triangle \( \triangle DEF \):**
- Point \( D \) is at coordinates \((0, 2)\).
- Point \( E \) is at coordinates \((4, 2)\).
- Point \( F \) is at coordinates \((2, 5)\).
**Graph Observations:**
- Both triangles are positioned on a grid with the x- and y-axis labeled and marked with equal intervals.
- Triangle \( \triangle ABC \) appears vertically flipped and shifted left compared to \( \triangle DEF \).
**Steps for Proving Congruence:**
1. **Verify Corresponding Sides:**
- Calculate the lengths of sides \( AB \), \( BC \), and \( AC \) for triangle \( \triangle ABC \).
- Calculate the lengths of sides \( DE \), \( EF \), and \( DF \) for triangle \( \triangle DEF \).
2. **Use Distance Formula:**
- For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance is given by:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
3. **Check for Congruent Sides:**
- If the lengths of corresponding sides are equal, the triangles are congruent by the SSS (Side-Side-Side) criterion.
This graphical representation can be used to practice calculating distances in the coordinate plane and understanding triangle congruence properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1730815b-ba67-4b2f-bdff-29eae8ea9422%2F24595cdb-747c-4abe-8e58-4342ae00bde1%2Fvv122t9_processed.png&w=3840&q=75)
Transcribed Image Text:### Proving Triangle Congruence: Example Problem
**Objective:**
Prove that triangle \( \triangle ABC \cong \triangle DEF \).
**Description of Graphs:**
The image consists of two coordinate planes with triangle \( \triangle ABC \) on the left and triangle \( \triangle DEF \) on the right.
**Triangles and Coordinates:**
- **Triangle \( \triangle ABC \):**
- Point \( A \) is at coordinates \((-6, 4)\).
- Point \( B \) is at coordinates \((-4, 2)\).
- Point \( C \) is at coordinates \((-2, 3)\).
- **Triangle \( \triangle DEF \):**
- Point \( D \) is at coordinates \((0, 2)\).
- Point \( E \) is at coordinates \((4, 2)\).
- Point \( F \) is at coordinates \((2, 5)\).
**Graph Observations:**
- Both triangles are positioned on a grid with the x- and y-axis labeled and marked with equal intervals.
- Triangle \( \triangle ABC \) appears vertically flipped and shifted left compared to \( \triangle DEF \).
**Steps for Proving Congruence:**
1. **Verify Corresponding Sides:**
- Calculate the lengths of sides \( AB \), \( BC \), and \( AC \) for triangle \( \triangle ABC \).
- Calculate the lengths of sides \( DE \), \( EF \), and \( DF \) for triangle \( \triangle DEF \).
2. **Use Distance Formula:**
- For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance is given by:
\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
3. **Check for Congruent Sides:**
- If the lengths of corresponding sides are equal, the triangles are congruent by the SSS (Side-Side-Side) criterion.
This graphical representation can be used to practice calculating distances in the coordinate plane and understanding triangle congruence properties.
Expert Solution
Step 1
Compare the lengths of both the triangles.
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