Points, Lines, and Planes Sections 1.1 & 1.2. Definition: Point A point has no dimension. It is represented by a dot. A point is symbolized using an upper-case.

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Presentation transcript:

Points, Lines, and Planes Sections 1.1 & 1.2

Definition: Point A point has no dimension. It is represented by a dot. A point is symbolized using an upper-case letter.

Definition: Line

Definition: Plane A plane has 2 dimensions. It is represented by a shape that looks like a parallelogram. It extends infinitely in length and width. Name a plane using the word plane with 3 non- collinear points in the plane. Plane ABC Also name with an upper-case cursive letter. Plane M

Definition: Collinear Points Points that lie (or could lie) on the same line.

Definition: Coplanar Coplanar points are points that lie (or could lie) in the same plane.

Definition: Line Segment

Definition: Ray

Definition: Opposite Rays If point C lies on line AB between A and B, then ray CA and ray CB are opposite rays. Two opposite rays make a line.

Definition: Intersection The intersection of two or more figures is the set of points the figures have in common. The intersection of 2 different lines is a point. The intersection of 2 different planes is a line.

Definition: Postulate A rule that is accepted without proof.

Definition: Theorem A rule that can be proven.

Definition: Between Between also implies collinear.

Definition: Congruent Segments To show that two segments are congruent in a drawing we use matching tick marks. A B C D

Definition: Distance

Distance Formula  The distance formula is used to compute the distance between two points in a coordinate plane. It is given by:

Finding the Distance  Find the distance between the points (1, 4) and (-2, 8).

Alternative to the Distance Formula  The distance formula comes from the Pythagorean theorem: a 2 + b 2 = c 2  If you are unsure about the distance formula, graph the two points accurately on a graph and use the Pythagorean theorem to find the distance.

Finding distance  Find the distance between (-2, 3) & (10, 8) by graphing and using the Pythagorean theorem.

Compare the two ways  Find the distance between (-7, -3) & (8, 5) using the distance formula.  Graph the same two points and find the distance using the Pythagorean Theorem.

Segment Addition Postulate If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C.

Definition: Midpoint The midpoint of a segment is the point that divides the segment into two congruent pieces.

Midpoint Formula  The coordinates of the midpoint of a segment are the averages of the x- coordinates and of the y-coordinates of the endpoints.

Finding a Midpoint  Find the midpoint between the endpoints (1, 7) & (3, -4).  Find the midpoint between the endpoints (2, 5) & (-3, 9)

Finding an Endpoint  If the midpoint of segment AB is (2, 3) and A is at (-1, 5), where is B located?  If the midpoint of segment CD is (0, - 2) and D is at (3, 4), where is C located?

Definition: Segment Bisector  A segment bisector is a point, ray, line, line segment, or plane, that intersects the segment at its midpoint.

Definition: Angle

Definition: Measure of an angle

Definitions: Angles Classified by Measure An acute angle has a measure between 0 o and 90 o A right angle has a measure of exactly 90 o An obtuse angle has a measure between 90 o and 180 o A straight angle has a measure of 180 o

Angle Addition Postulate The measures of two adjacent angles can be added to represent the large angle they form.

Definition: Angle Bisector An angle bisector is a ray that divides one angle into two congruent angles.

Definition: Congruent Angles Two angle are congruent if they have the same measure. To show that two angles in a diagram are congruent, we put a matching arc inside each angle.

Definition: Complementary Angles Two angles whose measures sum to 90°

Definition: Supplementary Angles  Two angles whose measures sum to180 o.

Definition: Adjacent Angles T wo angles that share a common vertex and side, but have no common interior points.

Definition: Linear Pair Two adjacent angles whose sides form a straight line. The angles in a linear pair are always supplementary.

Definition: Vertical Angle Pairs Formed when two lines intersect. The angle pairs only touch at the vertex. There are two pairs of vertical angles formed whenever two lines intersect.