(
a
)
 ﻿Show that in neutral geometry, if two lines cut by a transversal line have a pair of congruent alternate interior angles, then they are parallel.
(
b
)
 ﻿Explain why spherical geometry is not a neutral geometry.
Hint: For the first part, suppose the assertion is false for some pair of lines. Find a triangle that violates the conclusion of the previous problem.
For the second, draw a picture.

### (a) Proof in Neutral Geometry: Alternate Interior Angles and Parallel Lines

#### Problem Statement:
In neutral geometry (which includes both Euclidean and hyperbolic geometries), we need to prove that if two lines \( l \) and \( m \) are cut by a transversal \( t \) and have a pair of congruent alternate interior angles, then the lines \( l \) and \( m \) are parallel.

#### Proof:

1. **Assume the Hypothesis:**
   - Let \( l \) and \( m \) be two lines.
   - Let \( t \) be a transversal that cuts \( l \) and \( m \).
   - Assume the alternate interior angles \( \angle 1 \) and \( \angle 2 \) formed by \( t \) are congruent: \( \angle 1 \cong \angle 2 \).

2. **Suppose the Assertion is False:**
   - Assume for contradiction that \( l \) and \( m \) are not parallel. Then, by definition, \( l \) and \( m \) must intersect at some point \( P \).

3. **Construct a Triangle:**
   - Consider the points of intersection of \( t \) with \( l \) and \( m \), say \( A \) and \( B \), respectively.
   - Let \( P \) be the point where \( l \) and \( m \) intersect (assuming they are not parallel).
   - Now, consider the triangle \( \triangle ABP \).

4. **Apply the Angle Sum Property:**
   - In neutral geometry, the sum of the interior angles of a triangle \( \triangle ABP \) must be less than or equal to \( 180^\circ \) (with equality in Euclidean geometry).
   - Let's denote the angles \( \angle 1 \) and \( \angle 2 \) as mentioned, and the angle at point \( P \) by \( \angle 3 \).

5. **Contradiction with the Angle Sum Property:**
   - Since \( \angle 1 \cong \angle 2 \) by hypothesis and if \( \angle 1 \) and \( \angle 2 \) are angles on the same side of the transversal \( t \), then \( \angle 1 + \angle 2 \) should be equal to \( 180^\circ \) (because \( t \) would be parallel to \( m \) in Euclidean geometry).
   - But, if \( l \) and \( m \) are not parallel, \( \angle 3 \) would exist and should contribute to the angle sum of \( \triangle ABP \). Hence \( \angle 1 + \angle 2 + \angle 3 > 180^\circ \), which contradicts the angle sum property.

6. **Conclusion:**
   - Therefore, the assumption that \( l \) and \( m \) are not parallel leads to a contradiction.
   - Hence, \( l \) and \( m \) must be parallel if they have congruent alternate interior angles when cut by a transversal.

### (b) Why Spherical Geometry is Not a Neutral Geometry

#### Explanation:

Neutral geometry refers to geometrical systems that adhere to the first four postulates of Euclidean geometry but do not necessarily include the parallel postulate. These include Euclidean geometry (where the parallel postulate holds) and hyperbolic geometry (where it does not).

**Spherical Geometry** is different:

1. **Lines in Spherical Geometry:**
   - In spherical geometry, "lines" are great circles on a sphere.
   - Any two distinct great circles always intersect at two points (antipodal points).

2. **Lack of Parallel Lines:**
   - Since any two great circles intersect, there are no parallel lines in spherical geometry.
   - This violates the concept of parallelism crucial to neutral geometry, where at least one parallel line must exist through a given point not on a given line.

3. **Triangle Angle Sum:**
   - In spherical geometry, the sum of the angles of a triangle is always greater than \( 180^\circ \), whereas in neutral geometry, the sum of the angles is always less than or equal to \( 180^\circ \).

4. **Illustration:**
   - Consider the Earth as a sphere. The equator and a meridian are great circles. They intersect at the poles, showing that no parallel lines exist in this geometry.

Therefore, spherical geometry cannot be considered neutral geometry because it fails to satisfy key aspects, particularly concerning parallel lines and the sum of the angles in a triangle.

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Here are 5 related questions to explore further:

1. How does the concept of parallel lines differ in hyperbolic geometry compared to Euclidean geometry?
2. Can we have a consistent geometry system if the angle sum of a triangle is less than \( 180^\circ \)?
3. How does the fifth postulate influence the development of non-Euclidean geometries?
4. What are the implications of the absence of parallel lines in spherical geometry?
5. How do the properties of triangles differ between Euclidean, hyperbolic, and spherical geometries?

**Tip:** Understanding the differences between various geometries helps in grasping the limitations and extensions of Euclidean principles and exploring alternative models of space.

Explore a proof in neutral geometry that demonstrates if two lines cut by a transversal have congruent alternate interior angles, then they are parallel. This problem delves into key concepts of neutral geometry and triangle properties.

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