Chapter 1-2 Points, Lines, and Planes

Transcription

1 Chapter 1-2 Points, Lines, and Planes Undefined Terms: A point has no size but is often represented by a dot and usually named by a capital letter.. A A line extends in two directions without ending. Lines can be named by using two points on the line or a single lowercase letter. A B m A plane is a flat surface with no edges but is often represented by a four sided figure. A plane is named by a capital letter. R

2 Definitions: Collinear points are points all in one line. Coplanar points are points all in one plane.... The intersection of two figures is the set of points that are in both figures.

3 Chapter 1 3 Segments Definitions: Segment AC consists of points A and C and all the points between A and C. A C The length of AC (written AC) is the distance between A and C. Ray ED consists of segment ED and all other points P such that D is between E and P. E D Segments that have the same length are called congruent segments, written as AB CD. A B C D

4 The midpoint of a segment is the point that divides the segment into 2 congruent segments. A M B A bisector of a segment is a line, ray, or plane that intersects the segment at its midpoint. A B A statement that is accepted without proof is called a Postulate. Segment Addition Postulate: If B is between A and C, then AB + BC = AC A B C Example: If B is between A and C, with AB = 2x, BC = x + 10, and AC = 37, write and solve an equation for x. Is B the midpoint of AC?

5 Chapter 1 4 Angles An angle is a figure formed by two rays that have the same endpoint. A B C The two rays are called sides and the common endpoint is called the vertex. Angles can be named, ABC, CBA, or B. Angles will be measured in degrees, written as m ABC = 40 Angle Names: Acute angle has measure between 0 & 90 Right angle has measure 90 Obtuse angle has measure between 90 & 180 Straight angle has measure 180 Congruent angles have equal measure.

6 Adjacent angles have a common endpoint and a common side but no common interior points. The bisector of an angle is a ray that divides the angle into two congruent adjacent angles. Angle Addition Postulate: If point B is interior of AOC, then m AOB + m BOC = m AOC A O B Example: Find x, if m AOB = 2x + 16, m BOC = 3x 1, and m AOC = 60. Does OB bisect AOC? C

7 Chapter 2 4 Special Angle Pairs Complementary angles are two angles whose measures sum to 90. Supplementary angles are two angles whose measures sum to 180. Vertical angles are two angles such that the sides of one angle are opposites rays to the sides of the other angle. 1 2 Vertical Angle Theorem: Vertical angles are congruent. 1 2

8 Example: Write and solve an equation for x. 2x 3x + 5 Example: Write and solve an equation for x. 2x 15 3x 5 Example: Write and solve an equation for x. 2(x + 4) 4(x 5)

9 Chapter 3 1 Parallel Lines Parallel lines are coplanar lines that do not intersect, m // n. m n Perpendicular lines are coplanar lines that intersect to form four right angles, m n. m n Skew lines are non-coplanar lines. n m A transversal, t, is a line that intersects two or more coplanar lines in different points. m t n m n t

10 Alternate interior angles are nonadjacent interior angles on opposite sides of the transversal. t n m n t m Same-side interior angles are two interior angles on the same side of the transversal. t n m n t Corresponding angles are two angles in corresponding positions relative to the two lines. t n m n t m m

11 Chapter 3 2 Properties of Parallel Lines Corresponding Angle Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent

12 Same Side Interior Angle Theorem If two parallel lines are cut by a transversal, then same side interior angles are supplementary. 2 1 m 1 + m 2 = 180 Example: Solve for x and y. Explain your reasoning x 3y + 6

13 Chapter 3 4 Angles of a Triangle A triangle is the figure formed by three segments joining three noncollinear points. Each point is called a vertex. The segments are the sides of the triangle, ABC. A B C Names of Triangles: Scalene triangle has no congruent sides. Isosceles triangle has two congruent sides. Equilateral triangle has three congruent sides. Acute triangle has three acute angles. Obtuse triangle has one obtuse angle. Right triangle has one right angle. Equiangular triangle has all angles congruent.

14 Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180º. x y z x + y + z = 180 Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of its remote interior angles. x y z z = x + y

15 Chapter 3 5 Angles of a Polygon A polygon is a figure formed by n coplanar segments such that each segment intersects exactly two other segments at the endpoints. quadrilateral pentagon hexagon octagon n = 4 n = 5 n = 6 n = 8 Find the sum of the interior angles of each polygon. number of sides interior angle sum Polygon Interior Sum Theorem: The sum of the measures of the interior angles of a polygon with n sides is (n 2)180. Interior Sum = (n 2)180

16 Polygon Exterior Sum Theorem: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. Exterior Sum = 360 A regular polygon has all interior angles congruent and all sides congruent. Interior Angle Measure: An interior angle of a regular polygon with n sides is (n 2)180 divided by n. Interior Angle = Exterior Angle Measure: An exterior angle of a regular polygon with n sides is 360 divided by n Exterior Angle =

17 Example: A polygon has 10 sides, find its interior angle sum and its exterior angle sum. Example: Find the measure of each interior and each exterior angle of a regular polygon with 9 sides.

18 Chapter 10-1 Constructions A Straightedge is used to construct lines or parts of a line. A Compass is used to construct circles or parts of a circle. Construction 1: Congruent Segments Given segment AB construct congruent segment CD: Construction Steps A B Construction 2: Congruent Angles Given angle ABC construct congruent angle XYZ: Construction Steps A B C

19 Construction 3: Angle Bisector Given angle ABC construct angle bisector BZ: Construction Steps A B C Construction 4: Segment Bisector Given segment AB construct segment bisector CD: Construction Steps A B