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Here are a few recommended readings before getting started with this lesson.
At a geometry forum, one question made by the audience was to say what an angle is. The only ones who answered this question were Heichi and his friends.
an angle is the set of points formed by two different rays with a common point.
an angle is the set of points formed by two different rays with the same endpoint.
Analyze each definition, indicate whether it is mathematically correct, and point out the flaws, if any.
Among these cases, only the fourth one corresponds to an angle. Consequently, Heichi's definition is not precise since it includes incorrect cases.
Two coplanar lines — lines that are on the same plane — that do not intersect are said to be parallel lines. In a diagram, triangular hatch marks are drawn on lines to denote that they are parallel. The symbol ∥
is used to algebraically denote that two lines are parallel. In the diagram, lines m and ℓ are parallel.
Two lines are parallel when one of the lines is the image of the other under a translation.
Last week, Kevin and his friends learned about parallel lines. Today, the teacher asked them to define, without using their notes, what does it mean for two distinct lines to be parallel.
two distinct lines are parallel when they do not meet.
two distinct lines are parallel when they do not intersect each other and are coplanar.
Analyze each definition and indicate whether it is mathematically correct. Point out the flaws, if any.
Vertical lines have no slope.
Consequently, Kevin's definition fails to cover all possible cases. However, this flaw can be corrected by adding the following premise.
All vertical lines are parallel and no vertical line is parallel to a non-vertical line.
The second flaw in Kevin's definition is that it refers to auxiliary information — the slope of a line. Whereas it is true that two non-vertical lines are parallel when they have the same slope, it is possible to define parallel lines without referencing the slope.
Consequently, Emily's definition is not precise and therefore includes incorrect cases. Her statement can be fixed by adding that the lines are coplanar.
Two coplanar lines — lines that are on the same plane — that intersect at a right angle are said to be perpendicular lines. The symbol ⊥
is used to algebraically denote that two lines are perpendicular. In the diagram, lines m and ℓ are perpendicular.
Three students were asked to write down what it means for two distinct lines to be perpendicular. Analyze each of their answers and indicate whether they are mathematically correct. Point out the flaws, if any.
two distinct lines are perpendicular when the product of their slopes is -1.
two distinct lines are perpendicular when they form four angles of equal measure.
two distinct lines are perpendicular when they intersect at a 90∘ angle.
Vertical lines are perpendicular to horizontal lines.
Secondly, Kevin's answer refers to auxiliary information — the slope of a line. While it is true that two non-vertical lines are perpendicular when the product of their slopes is -1, it is possible to define perpendicular lines without referencing the slope.
Since a complete turn is 360∘, each angle must have a measure of 4360=90∘. That is, the lines form four right angles, and consequently, Heichi's answer is correct. However, his answer has a minor flaw. It requires finding four angle measures when only one is enough.
One last geometric object to be considered is the section or part of a line between any two of its points.
A segment, or line segment, is part of a line bounded by two different points, called endpoints. It is made of all the points on the line between the endpoints. Unlike a line, a segment does not extend infinitely, and then it is drawn without arrowheads.
As with a line, between two different points there is exactly one segment. A segment with endpoints A and B is denoted as AB or BA while its length or measure is written as AB or BA. When two segments have the same length they are said to be congruent.In the final of a math Olympics, two students from different schools had to define what a line segment is.
a line segment is the intersection of two different rays that are on the same line but have opposite directions.
Jefferson High should win the Olympics.
Since the first and second cases do not represent a line segment, Mark's definition is not correct. It seems like Mark did not consider the first two cases. In consequence, Jefferson High should win the Olympics.