When having two distinct rays on the same plane, there are various ways of positioning them. If it were to be the case that both rays have the same starting point, the geometric figure has its own name, an angle.
An angle is the set of points in the plane formed by two different rays that share the same starting point. This common point is called the vertex of the angle and the rays are the sides of the angle.
There are different ways to denote an angle and all involve the symbol
in front of the name. For the second notation, the vertex always must be in the middle.
|Using the Vertex||Using the Vertex and One Point on Each Ray||Using a Number|
An angle divides the plane into two parts.
interiorof the angle.
exteriorof the angle.
An angle is a figure formed by a ray and its image after being rotated around its endpoint.
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.
The answer is yes. There are infinitely many, and the set of all these points is called a circle.
Since point is not centimeters apart from itself, it does not belong to
Two lines are parallel when one of the lines is the image of the other under a translation.
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 lines are perpendicular when one of the lines is the image of the other under a rotation of around the intersection point.
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
two distinct lines are perpendicular when they form four angles of equal measure.
two distinct lines are perpendicular when they intersect at a 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 it is possible to define perpendicular lines without referencing the slope.
Since a complete turn is each angle must have a measure of 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.
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.
In the final of a math Olympics, two students from different schools had to define what a line segment is.
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.
Consequently, to avoid crashing, the rails must be parallel to each other. Thus, the railroad tracks can be viewed as a pair of parallel lines. Here, the train's axles and the sleepers are perpendicular to each rail.