Chapter 1. Foundations of Geometry: Points, Lines, and Planes


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1 Chapter 1 Foundations of Geometry: Points, Lines, and Planes
2 Objectives(Goals) Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space. Use algebra to compute segment measures Find the distance between two points Find the midpoint of a segment. Measure and classify angles. Identify and use congruent angles and the bisector of an angle. Identify and use special pairs of angles. Find the measures of angles.
3 Undefined Terms: most basic figures in geometry that can t be defined using other figures Term Facts Diagram Name Point Line Plane Names a location and has no shape or size. A straight path that has no thickness and extends forever. A flat surface that has no thickness and extends forever.
4 Collinear: points that lie on the same line Collinear: Noncollinear:
5 Coplanar: points that lie in the same plane Coplanar: Noncoplanar:
6 Postulate (axiom): a statement that is accepted as true without proof. P11: Through any two points there is exactly one line. P12: Through any three noncollinear points there is exactly one plane containing them. P13: If two points lie in a plane, then the line containing those points lie in the plane. P14: If two lines intersect, then they intersect in exactly one point. P15: If two planes intersect, then they intersect in exactly one line.
7 Name a line that contains point T. Name three collinear points. Name three noncollinear points. Name the intersection of line n and line m. What is another name for line n?
8 Name the intersection of plane R and plane P. Name three points that are nonplanar. What is another name for plane P?
9 Line Segment: part of a line consisting of two endpoints and all the points between them. Diagram: Name: Fact: line segments can be measured. Ruler Postulate: The points on a line can be put into a oneone to correspondence with the real numbers.
10 Betweenness of Points Point D is between points C and E if and only if C, D, and E are collinear and CD + DE = CE. Segment Addition Postulate: If B is between A and C, then AB + BC = AC Example 1 of Using Segment Addition Postulate B is between A and C, AC =14 and BC =11.4. Find AB.
11 Example 2 of Using the Segment Addition Postulate S is between R and T. Find the value of x and RT if RS = 2x+7, ST = 28, and RT = 4x Congruent Segments: segments that have the same length(measure). Symbol: is congruent to Tick marks (slashes) on figures also indicate congruence.
12 Distance Between Two Points Number Line The distance between any two points is the absolute value of the distance of the coordinates. Coordinate Plane The distance d between two points with coordinates (x 1, y 1 ) and (x 2, y 2 ) is d ( x y x1 ) ( y2 1) Example: CD = Example: (5, 1) and (3, 3) AC = CF =
13 Midpoint of Segment The point halfway between two endpoints. A midpoint is the point that bisects (divides) into two congruent segments. If K is the midpoint of AB, then AK = KB. Number Line. The coordinate of the midpoint of a segment is the mean of the endpoints. Coordinate Plane The coordinates of the midpoint of a segment whose endpoints (x 1, y 1 ) and (x 2, y 2 ) are x 1 2 y1 y2 x 2, 2 Find the midpoint of (12, 9) and (7, 4).
14 Finding the coordinates of a missing endpoint given the coordinates of one endpoint and the midpoint. Given that S is the midpoint of S(1, 5). RT. T(4, 3) and
15 Using Algebra to Find Measures Example 1: E is the midpoint of DF, DE = 2x + 4, and EF = 3x 1. Find DE, EF, and DF. PR Example 2: Q bisects, PQ = 3y, and PR = 42. Find y and QR.
16 Rays and Angles Ray: part of a line that starts at an endpoint and extends forever in one direction. Diagram: Name: Opposite Rays: two rays that have a common endpoint and form a line.
17 Angle: figure formed by two rays that have a common endpoint called the vertex. Diagram: Name: 4 Types of Angles 1. Acute 2. Right 3. Obtuse 4. Straight
18 Congruent Angles: angles that have the same measure. Diagram: Name: Arc Marks are used to indicate that two angles are congruent.
19 Angle Addition Postulate If S is in the interior of PQR, m PQS m SQR m PQR. then the Example: Find the m CAB
20 Angle Bisector: a ray that divides an angle into two congruent angles. Diagram: Finding the measure of an Angle BD bisects ABC, m ABD 6x 3, and m DBC 8x 7. Find m ABD.
21 Angle Relationships 5 types of special angle pairs 1. Adjacent Angles: two angles that have a common vertex, and a common side, but no common interior points. 2. Vertical Angles: two nonadjacent angles formed by two intersecting lines. Vertical Angles are congruent.
22 3. Linear Pair: a pair of adjacent angles whose noncommon sides are opposite angles. In others, adjacent angles that form a line. 4. Complementary Angles: two angles whose measures have a sum of 90 degrees. 5. Supplementary Angles: two angles who measures have a sum of 180 degrees.
23 Angle Problems 1. ABD and BDE are supplementary. m ABD 3x 12 and m BDE 7x 32. Find the measure of each angle. 2. and BDC are complementary. and ABD m BDC m ABD 5y 1 3y 7. Find the measures of both angles.
24 3. Solve for x. Then find the measure of each angle. 4. Find the measure of each angle.
25 5. Solve for x and y. Find the measure of each angle. 6. m BGC 35. Find the measure of each remaining angle.
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