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Exploring Geometry

Michael Hvidsten

Chapter 2

Euclidean Geometry - all with Video Answers

Educators

Section 1

ANGLES, LINES, AND PARALLELS

01:17

Problem 1

Finish the proof of Theorem 2.9. That is, show that $\angle D A B$ is congruent to $\angle E B A$ and that $m \angle D A B+m \angle A B F=180$.

Jay Patel
Jay Patel
Numerade Educator
02:16

Problem 2

Use the Exterior Angle Theorem to show that the sum of the measures of two interior angles of a triangle is always less than 180 degrees.
It is critical to have a clear understanding of the terms of an axiomatic system when working on a mathematical proof. One must use only the facts given in the definitions and not impose preconceived notions on the terms.

Jay Patel
Jay Patel
Numerade Educator
02:17

Problem 3

Each of the following statements are about a specific term of this section. Determine, solely on the basis of the definition of that term, if the statement is true or false.
(a) A right angle is an angle whose measure is 90 degrees.
(b) An angle is the set of points lying between two rays that have a common vertex.
(c) An exterior angle results from a line crossing two other lines.
(d) A line is the union of two opposite rays.

Allison Knapp
Allison Knapp
Numerade Educator
01:28

Problem 4

In this exercise we will practice defining terms. Be careful to use only previously defined terms, and take care not to use imprecise and colloquial language in your definitions.
(a) Define the term midpoint of a segment.
(b) Define the term perpendicular bisector of a segment.
(c) Define the tem triangle defined by three non-collinear points $A, B, C$.
(d) Define the term equilateral triangle.
In the next set of exercises, we consider the logical equivalence of Euclid's fifth postulate with Playfair's Postulate. You may use any of the first 28 Propositions of Euclid (found in Appendix B) and/or any of the results from this section for these exercises.

Luca Alexander
Luca Alexander
Numerade Educator

Problem 5

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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Problem 6

Now, we will prove the converse of the preceding exercise, that Playfair's Postulate implies Euclid's fifth postulate. Consider Figure 2.8. Suppose line $m$ intersects lines $t$ and $l$ such that the measures of angles $\angle C B A$ and $\angle B A D$ add up to less than 180 degrees. Copy $\angle C B A$ to $A$, creating line $n$, and then use Playfair's Postulate to angue that lines $t$ and l must intersect. We now need to show that the lines intersect on the same side of $m$ as $D$ and C. We will prove this by contradiction. Assume that $t$ and $l$ intersect on the other side of $m$ from point $C$, say at some point E. Use the exterior angle $\angle C B A$ to triangle $\triangle A B E$ to produce a contradiction.

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Problem 7

Show that Playfair's Postulate is equivalent to the statement, Whenever a line is perpendicular to one of two parallel lines, it must be perpendicular to the other.

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01:38

Problem 8

Given triangle $\triangle A B C$, construct a parallel to $\overleftrightarrow{B C}$ at $A$. (How would we do this?) Use this construction to show that Playfair's Postulate (or Euclid's fifth) implies that the angle sum in a triangle is 180 degrees, namely, equal to two right angles. (The converse to this statement is also true: If the angle sum of a triangle is always 180 degrees, then Playfair's Postulate is true. For a proof see [43], (pages 21-29].)

Jay Patel
Jay Patel
Numerade Educator

Problem 9

Show that Playfair's Postulate is equivalent to the statement, Two lines that are parallel to the same line are coincident (the same) or themselves parallel.

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Problem 10

Show that Playfair's Postulate is equivalent to the statement, If a line intersects but is not coincident with one of two parallel lines, it must intersect the other.

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