Please help and explain everything you do. This is from the textbook Euclidean a
ID: 2900038 • Letter: P
Question
Please help and explain everything you do. This is from the textbook Euclidean and Non-Euclidean Geometries 4th Edition. Chapter 3 Congruence exercise number 35.
In the real Euclidean plane of Example 3 in this chapter, we have defined the length of any segment by the Pythagorean formula. We will now distort that interpretation as follows: For segments on the x-axis only, redefine their length as twice what t was previously (e.g., the length of the segment from (1, 0) to (4, 0) is now 6 in- stead of 3). Reinterpret congruence of segments to mean that two segments in the plane have the same "length" in this perverse way of measuring (e.g., the segment from (0, 0) to (0, 6) on the y-axis is now congruent to the segment from, 0) to (4, 0) on thex-axis) Points, lines, incidence, and betweenness will have the same mean ing as before and satisfy the same axioms as before. Congruence of angles will mean that the angles have the same number of degrees, i.e., the same meaning as in high school geometry (something we have not defined, but treat this example informally). Show infor mally that the first five congruence axioms and angle addition (Proposition 3.19) still hold in this interpretation but that SAS fails for certain pairs of triangles (see Figure 3.41). This shows that Ax- iom C-6 (SAS) is independent of the other 12 axioms for a Hilbert x axis Figure 3.41 plane can neither be proved nor disproved from them). Draw diagrams to show that SSS and ASA also fail for certain pairs of tri- angles. Draw a diagram of a circle with center on the x-axis in this interpretation and use that diagram to show that the circle-circle continuity principle and the segment-circle continuity principle fail in this interpretation.Explanation / Answer
I do'nt kno exactly what is in your example 3. The Hilbert's sisth axiom for congruence is
" If, in the two triangles ABC and ABC the congruences AB AB, AC AC, BAC BAC hold, then the congruence ABC ABC holds (and, it follows that ACB ACB also holds)."
As per above discussions that if the measure parallel to x-axis, if doubled then whether the test for the congruency remains invariant.
Note that in above case if the two sides equal to corresponding two sides in other triangle, and one corresponding angle is also equal. Now if, angle is 900 which is formed by the line parallel to x- and y-axes and the angles are deformed due to measurements madon abve discussion of varying scales for the axis of x. Now if we compute the third angle measured by subtraction 900 the invariance will not be maintained, if we have the sides on x axis. Since above example one of the triangle has one of its side on the x-axis and in other triangle it is not. .