Parallel Lines

page revision: 0, last edited: 15 Sep 2008 08:57

Parallel Lines

page revision: 0, last edited: 15 Sep 2008 08:57

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'angles a and b" should be "angles 1 and 2"

ReplyOptions"angle 2" should be "angle 4"

ReplyOptions$\S 70$ - misspelling, "In writing, parallel lineD"

ReplyOptions$\S 78$ - From the two theorems: direct ($\S 73$) and its converse ($\S 75$)… the converse section is $\S 77$ .

ReplyOptionsExercise 140:

Divide the plane by infinite straight lines into 5 parts, using as few lines as possible.Zero lines renders just the plane; 1 part.

One line divides the plane into 2 parts; one part on either side of this line.

Two lines in the plane are either parallel or intersect. If parallel the plane is divided into 3 parts. If intersecting, the plane is divided into the 4 angles formed at the point of intersection. $\S 4$) Straight line; any pair of lines intersects at most at one point.

Three lines are parallel, or a combination of parallel and/or intersecting. If parallel, the plane is divided into 4 parts. In the other cases the lines must intersect at 1, 2, or 3 points. $\S 4$ Straight line - any pair of lines intersects at most at one point. These figures divide the plane into a minimum of 6 parts.

With only parallel lines in the plane, every additional parallel line added to the plane divides the plane adding an additional part. In the case of 3 parallels there are 4 parts, so adding a fourth parallel results in 5 parts in the plane. Therefore the fewest number of lines if 4.

ReplyOptionsExercise 141:

In the interior of a given angle, construct an angle congruent to it.Given angle assumed less than 2d.

Let the given angle be $\angle ABC$. $\S 79$) Angles with respectivel parallel sides in the same direction are congruent. Construct a new angle with one side parallel to $AB$ and on the same side of $BC$. The other side is along and in the same direction as $BC$.

Select a point $D$ along side $BC$. Use the method in $\S 74$ to construct a side parallel to $AB$ through point $D$, label a point $E$ along the new side. $\angle EDC$ is the angle to be constructed.

ReplyOptionsExercise 142:

Using a protractor, straight edge, and drafting triangle, measure an angle whose vertex does not fit the page of the diagram.Label arbitrary points $A$ and $B$ on one of the angle's sides. On the other side label points $M$ and $N$. Use the straight edge and drafting triangle as depicted in figure 76 to construct a line parallel to $AB$ through point $M$. This constructed line is $MB'$. The angle with sides $AB$ and $MN$ is congruent to $\angle NMB'$ by $\S 79$) angles with parallel sides in the same direction. Use the protractor to measure $\angle NMB'$, which has the same measure as the given angle.

ReplyOptionsExercise 143:

How many axes of symmetry does a pair of parallel lines have? How about three parallel lines?TWO PARALLEL LINES

Two parallel lines have two axes of symmetry. The first axis is the line parallel to the two given lines and halfway between them. This line divides the plane into 2 parts, each with one of the given parallel lines the same distance from the axis of symmetry. The constant distance proved by $\S 77$ corollary) perpendicular to one of two parallels is parallel to the other.

The second axis of symmetry is a perpendicular to the parallel lines. By $\S 23$ and $\S 22$) all angles formed about the intersection of this perpendicular with the parallel lines are right and congruent. Further the plane is divided in half about this perpendicular with half of each parallel on either side of it.

THREE PARALLEL LINES

Three parallel lines have one axis of symmetry. It is possible that there is a second axis of symmetry if one of the parallel lines is equidistant between the other two and that line itself can be considered the axis of symmetry. In this case the explanation is the first case of my "two parallel lines" answer above.

The other possible axis is a perpendicular to the parallel lines for the same reason as the "two parallel lines" case. Any other line would not form right angles at the intersections with the parallel lines. Thus the parts of each parallel line on either side of the transverse would not be equidistant from it. $\S 51$) distance from point to line.

ReplyOptionsExercise 144:

Two parallel lines are intersected by a transversal, and one of the eight angles thus formed is 72 degrees. Find the measures of the remaining seven angles.$\S 77$) angles about parallel lines and transverse intersection. Using figure 74 angle labels.

$\angle 1 = \angle 5 = \angle 3 = \angle 7 = 72^\circ$

$\angle 2 = \angle 4 = \angle 6 = \angle 8 = 2d - 72^\circ = 108^\circ$

ReplyOptionsExercise 145:

One of the interior angles formed by a transversal with one of two given parallel lines is (4d)/5 . What angle does its bisector make with the other of the two parallel lines?The sub-angles of the bisected interior angle are $\frac{2d}{5}$. The bisector itself is a transversal of the two given parallel lines. Thus $\frac{2d}{5}$ is the measure of an interior angle it forms with one of the parallel lines. $\S 77$) Sum of same-side interior angles formed by transversal of parallel lines is $2d$. The interior same-side angle formed with the other parallel line is $2d - \frac{2d}{5} = \frac{8d}{5}$.

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