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
![### Geometry Proof Example
#### Problem 14
**Given:**
A geometric figure includes two intersecting triangles, △ABC and △ABD, intersecting at point X.
**Prove:**
\[ BX \cdot CD = XC \cdot AB \]
**Diagram:**
A diagram depicts two triangles, △ABC and △ABD, intersecting each other at point X. The vertices are labeled as follows:
- Triangle △ABC: Vertices A, B, C
- Triangle △ABD: Vertices A, B, D
- Intersection point: X
Here is the structure of the proof:
| Statement | Reason |
|----------------------|-------------------|
| 1. ∠A = ∠D | 1. Given |
**Explanation:**
In the problem, it is given that angle ∠A is equal to angle ∠D.
To start the proof, this information will be used to show that the following relationship holds true: \( BX \cdot CD = XC \cdot AB \).
This is a standard geometry proof problem involving intersecting triangles and the use of similar triangles or other geometric properties to establish the equality.
**Steps to Prove:**
1. Highlight the congruent angles and corresponding sides.
2. Utilize properties of similar triangles, ratios, or other geometric theorems to prove the given relationship.
This is a classical exercise in geometry that demonstrates the utilization of congruency and proportionality in problem-solving.
Feel free to continue the proof by examining the properties of the intersecting triangles and applying relevant geometric theorems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ec0482a-ffdc-4040-bf9f-ac7e87cdc73c%2F51c7c76f-e69a-4641-a989-55361a047c30%2Fbdyxy1ha_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Proof Example
#### Problem 14
**Given:**
A geometric figure includes two intersecting triangles, △ABC and △ABD, intersecting at point X.
**Prove:**
\[ BX \cdot CD = XC \cdot AB \]
**Diagram:**
A diagram depicts two triangles, △ABC and △ABD, intersecting each other at point X. The vertices are labeled as follows:
- Triangle △ABC: Vertices A, B, C
- Triangle △ABD: Vertices A, B, D
- Intersection point: X
Here is the structure of the proof:
| Statement | Reason |
|----------------------|-------------------|
| 1. ∠A = ∠D | 1. Given |
**Explanation:**
In the problem, it is given that angle ∠A is equal to angle ∠D.
To start the proof, this information will be used to show that the following relationship holds true: \( BX \cdot CD = XC \cdot AB \).
This is a standard geometry proof problem involving intersecting triangles and the use of similar triangles or other geometric properties to establish the equality.
**Steps to Prove:**
1. Highlight the congruent angles and corresponding sides.
2. Utilize properties of similar triangles, ratios, or other geometric theorems to prove the given relationship.
This is a classical exercise in geometry that demonstrates the utilization of congruency and proportionality in problem-solving.
Feel free to continue the proof by examining the properties of the intersecting triangles and applying relevant geometric theorems.
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