Which of the following is true in neutral geometry (axioms are the  three incidence axioms, the four betweenness axiom, and the six congruence axioms)  - Choose all that apply:

 

(a) There exist a triangle with all angles of 60 degrees

(b) In a right triangle, the hypotenuse is larger than any other side of the triangle

(c) In any triangle, the sum of two sides is greater than the third side'

(d) if two different lines m and m are both perpendicular to line g, then m and n are parallel

Let's analyze each statement based on the principles of neutral geometry, which includes the three incidence axioms, the four betweenness axioms, and the six congruence axioms.

### (a) There exists a triangle with all angles of 60 degrees.
- **False**: In neutral geometry (which is common to both Euclidean and hyperbolic geometries), there is no guarantee that a triangle with all angles of 60 degrees exists. This is true in Euclidean geometry (an equilateral triangle), but in hyperbolic geometry, the sum of the angles in a triangle is always less than 180 degrees, making it impossible for all angles to be 60 degrees.

### (b) In a right triangle, the hypotenuse is larger than any other side of the triangle.
- **True**: This is true in both Euclidean and hyperbolic geometries. The hypotenuse is the side opposite the right angle and is always longer than either of the legs in a right triangle, as guaranteed by the triangle inequality.

### (c) In any triangle, the sum of two sides is greater than the third side.
- **True**: This is the **triangle inequality theorem**, which holds in both Euclidean and hyperbolic geometries. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### (d) If two different lines \( m \) and \( n \) are both perpendicular to line \( g \), then \( m \) and \( n \) are parallel.
- **True**: This is true in both Euclidean and hyperbolic geometries and follows from the definition of parallel lines in neutral geometry. Two lines that are both perpendicular to the same line do not intersect and are thus parallel.

### Conclusion:
The true statements in neutral geometry are:
- (b) In a right triangle, the hypotenuse is larger than any other side of the triangle.
- (c) In any triangle, the sum of two sides is greater than the third side.
- (d) If two different lines \( m \) and \( n \) are both perpendicular to line \( g \), then \( m \) and \( n \) are parallel.

Would you like further explanation on any of these points?

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Here are five questions to expand on this:

1. What is the difference between neutral geometry and Euclidean geometry?
2. How does the triangle inequality theorem apply to other geometries?
3. Why can we not assume that all angles of a triangle sum to 180 degrees in neutral geometry?
4. What implications do the congruence axioms have on the properties of triangles?
5. How do the incidence axioms define the relationship between lines and points?

**Tip:** When studying different types of geometries, pay attention to the underlying assumptions, especially regarding parallel lines and the sum of angles in triangles. These vary between Euclidean and non-Euclidean geometries.

Which of the following is true in neutral geometry (axioms are the three incidence axioms, the four betweenness axiom, and the six congruence axioms) - Choose all that apply:

(a) There exists a triangle with all angles of 60 degrees
(b) In a right triangle, the hypotenuse is larger than any other side of the triangle
(c) In any triangle, the sum of two sides is greater than the third side
(d) If two different lines m and n are both perpendicular to line g, then m and n are parallel

Explore which statements hold true in neutral geometry, involving the three incidence axioms, the four betweenness axioms, and the six congruence axioms. This covers triangle properties like the triangle inequality theorem and the concept of parallel lines. Suitable for high school students in Grades 10-12.

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