Question

In this exercise you are to show that Euclid's fifth postulate implies Playfair's Postulate. Given a line $l$ and a point $A$ not on $l$, we can copy $\angle C B A$ to $A$ to construct a parallel line $n$ to $l$. (Which of Euclid's first 28 Propositions is this based on?) Suppose that there was another line $t$ through A that was not identical to n. Use Euclid's fifth postulate to show that $t$ cannot be parallel to 1 (Figure 2.8).

   In this exercise you are to show that Euclid's fifth postulate implies Playfair's Postulate. Given a line $l$ and a point $A$ not on $l$, we can copy $\angle C B A$ to $A$ to construct a parallel line $n$ to $l$. (Which of Euclid's first 28 Propositions is this based on?) Suppose that there was another line $t$ through A that was not identical to n. Use Euclid's fifth postulate to show that $t$ cannot be parallel to 1 (Figure 2.8).

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Exploring Geometry

Exploring Geometry

Michael Hvidsten 2nd Edition

Chapter 2, Problem 5

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Step 1

We are to use Euclid's fifth postulate (the parallel postulate) to show that Playfair's Postulate holds, which states that through any point not on a given line, there is exactly one line parallel to the given line. Show more…

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In this exercise you are to show that Euclid's fifth postulate implies Playfair's Postulate. Given a line $l$ and a point $A$ not on $l$, we can copy $\angle C B A$ to $A$ to construct a parallel line $n$ to $l$. (Which of Euclid's first 28 Propositions is this based on?) Suppose that there was another line $t$ through A that was not identical to n. Use Euclid's fifth postulate to show that $t$ cannot be parallel to 1 (Figure 2.8).

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