Geometry Guide. Basic Terms and Definitions

Transcription

1 Geometry Guide Basic Terms and Definitions Term Point Line Plane Space Collinear Coplanar Segment Ray Opposite rays Intersection Congruent Congruent segments Segment midpoint Segment bisector Angle Definition A location or position. A point has no dimension. A point has no length, width, or height. An infinite set of points that extend in two opposite directions. A line has length (infinite), but no width, or thickness. An infinite set of points that form a flat surface extending in all directions. A plane has length and width (infinite), but no thickness. The set of all points. Collinear points lie on the same line. Noncollinear points do not lie on the same line. Coplanar points lie on the same plane. Noncoplanar points do not lie on the same plane. A section of a line designated by two endpoints and the set of all points between them. A section of a line with one endpoint and extending in one direction. Two rays with the same endpoint that form a straight line. The set of points in both objects. Two objects are congruent if they have the same size and shape. Two segments are congruent if they have the same length. A midpoint divides a segment into two segments. A line, ray, or segment that intersects a segment at its midpoint. A figure formed by two rays with the same endpoint. The endpoint is the vertex. The rays form the sides of the angle. Acute angle A is acute if 0 < m A < 90 Obtuse angle A is obtuse if 90 < m A < 180 Right angle A is a right angle if m A = 90 Straight angle A is a straight angle if m A = 180 Congruent angles Angle bisector Adjacent angles Vertical angles Two angles are congruent if they have equal measure. A ray that divides an angle into two congruent angles. Two angles are adjacent if they have: 1. Common vertex 2. Common side 3. No points in common The two angles across from each other at the intersection of lines. 1

2 Basic Postulates and Theorems about Points, Lines, and Planes: A line contains at least 2 points. (P.5) A plane contains at least 3 noncollinear points. (P.5) Space contains at least 4 points not all in one plane. (P.5) Through any 2 points there is exactly one line. (P.6) Through any 3 noncollinear points there is exactly one plane. (P.7) If 2 points are in a plane, then the line formed by the points is in the plane. (P.8) If 2 planes intersect, then their intersection is a line. (P.9) The intersection of two planes is a line. If 2 lines intersect, then they intersect in one point. (1.1) The intersection of two lines is a point. If 2 lines intersect, then one plane contains the lines. (1.3) Through a line and a point not on the line there is exactly one plane. (1.2) 2

3 Segment Concepts: SAP: Segment Addition Postulate: (P.2) AC + CB = AB The sum of the parts equals the whole. Midpoint Theorem: (2.1)/Definition of Midpoint: If M is midpoint of AB, then it divides AB into 2 segments. If M is midpoint of AB, AM MB. If AM MB, then M is midpoint of AB. A C M B If M is midpoint of AB, then, AM = ½ AB and MB = ½ AB. If AM = ½ AB or MB = ½ AB, then M is midpoint of AB. Angle Concepts: A AAP: Angle Addition Postulate: (P.4) m ABX + m XBC = m ABC. The sum of the parts equals the whole. B C Bisector Theorem: (2.2)/Definition of Bisector: If BX bisects ABC, then ABX XBC. If ABX XBC, then BX bisects ABC. If BX bisects ABC, then m ABX = ½ m ABC and m XBC = ½ m ABC. If m ABX = ½ m ABC or m XBC = ½ m ABC, then BX bisects ABC. Definition of Supplementary s: If 1 and 2 supplementary, then m 1 + m 2 = 180. If m 1 + m 2 = 180, then 1 and 2 supplementary. Definition of Complementary s: If 3 and 4 complementary, then m 3 + m 4 = 90. If m 3 + m 4 = 90, then 3 and 4 complementary. Vertical angles are. (2.3) If 2 s are supplements to angles (or to the same angle), then the 2 s are. (2.7) If 2 s are complements to angles (or to the same angle), then the 2 s are. (2.8) X 3

4 Properties from Algebra: Addition/subtraction property of equality. If a = b, then a+ c = b+ c. (add the same thing to both sides). If a = b and c = d, then a+ c = b+ d. (add equals to both sides). Multiplication/division property of equality. If a = b, then ac = bc. (multiply both sides by the same/equal thing.) If a = b, then a = b (divide both sides by the same/equal thing.) c c Distributive property. ab ( + c) = ab+ ac Substitution If a+ b= c and a = d, then d + b= c. (replace d for a) If a+ b= z and x+ y = z, then a+ b= x+ y. (two expressions equal to same thing) Transitive If a = b and b= c, then a = c. (for equality) If a b and b c, then a c. (for congruence) Reflexive a = a Symmetric a = b and b= a. 4

5 Perpendicular Concepts: Definition of lines: If 2 lines are, then they form right s (90 degree s). If 2 lines form right s (90 degree s), then they are. If two lines, then they form adjacent s. (2.4) If 2 lines form adjacent s, then they are. (2.5) If the ext. sides of 2 adjacent acute s are, then the s are complementary. (2.6) Two adjacent angles are complementary if their exterior sides are. 5

6 Parallel Concepts: Definition of parallel lines: 2 coplanar lines that do not intersect (railroad tracks). Properties of parallel lines: If 2 lines, then 1. Corresponding s. (P.10) 2. Alternate interior s. (3.2) 3. Same-side interior s are supplementary. (3.3) 4. If one line is to the transversal, then other line is also to the transversal. (3.4) Proving lines parallel: 1. If corresponding s, then lines. (P.11) 2. If alternate interior s, then lines. (3.5) 3. If same side interior s supplementary, then lines. (3.6) 4. If 2 lines to the same line, then lines. (3.7) 5. If 2 lines to the same line, then lines. (3.10) Through a point outside a line, there is exactly one line to the line. (3.8) Through a point outside a line, there is exactly one line to the line. (3.9) Basic Terms and Definitions Term Parallel lines Transversal Corresponding angles Alternate interior angles Same side interior angles Skew lines Definition 2 coplanar lines that do not intersect (railroad tracks). A line that intersects 2 parallel lines. 2 angles that have the same relative position to lines. 2 angles on alternate sides of the transversal and interior to the lines. 2 angles on the same side of the transversal and interior to the lines. 2 non-coplanar lines that do not intersect. 6

7 Triangle Concepts: Basic Terms and Definitions Term Triangle Vertex Sides Exterior of a Δ Scalene Δ Isosceles Δ Equilateral Δ Definition Figure formed by 3 segments joining 3 noncollinear points. 3 points of a triangle. 3 line segments of a triangle. Angle formed when side of a triangle is extended. Triangle with 3 different length sides. Triangle with at least 2 sides congruent. Triangle with 3 sides congruent. Acute Δ Triangle with 3 acute angles (all < 90). Obtuse Δ Triangle with one obtuse angle (>90 and <180). Equiangular Δ Triangle with 3 congruent angles. Right Δ Triangle with one right angle (=90). Legs of right Δ The sides of a right Δ. The sides that form the right angle. Hypotenuse of right Δ The side opposite the right of a right Δ. Median of Δ Altitude of Δ Perpendicular bisector Segment from vertex to midpoint of opposite side. segment from vertex to opposite side/line. Line, segment, or ray to a segment through its midpoint. Angle Sum Theorem (AST): The sum of the s of a Δ = 180. (3.11) o Corrollaries: 1. If 2 s of one Δ to 2 s of another Δ, then third s are. 2. Each of an equiangular Δ = The acute s of a right Δ are complementary. 4. In a Δ, there can be at most one right or obtuse. Exterior Angle Theorem (EAT): An exterior angle of a triangle equals sum of the 2 remote interior s. (3.12) Definition of congruent triangles: Biconditional definition: Two triangles are congruent if an only if all the corresponding parts (all 3 angles and all 3 sides) of the triangles are congruent. Definition as two statements that are converses: a. If two triangles are congruent, then all the corresponding parts (all 3 angles and all 3 sides) of the triangles are congruent. b. If all the corresponding parts (all 3 angles and all 3 sides) of the triangles are congruent, then the two triangles are congruent. 7

8 Postulates/theorems for proving triangles congruent: 1. SSS (P.12) 2. SAS (P.13) 3. ASA (P.14) 4. AAS (4.3) 5. HL (4.4) (Right Δs only.) CPCTC: Corresponding parts to congruent triangles are congruent. Once 2 triangles are proved congruent, all the corresponding parts (angles and sides) are congruent by CPCTC. Isosceles Triangle Theorem (ITT/BAT) (4.1): Base angles of an isosceles Δ. If 2 sides of a triangle are, then the angles opposite those sides are congruent. o Corollaries: 1. An equilateral triangle is also equiangular. 2. An equilateral triangle has three 60 degree angles. 3. A bisector of the vertex angle of an isosceles triangle is to the base at its midpoint. Isosceles Triangle Theorem Converse (ITTC/BATC) (4.2): If 2 angles of a triangle are congruent, then the opposite sides are congruent. o Corollary: 1. An equiangular triangle is equilateral. Distance from a point to a line: The length of the segment from the point to the line. Segment perpendicular bisector theorems/properties: o If a point is on the bisector of a segment, then it is equidistant from the segment endpoints. (4.5) o If a point is equidistant from the segment endpoints, then it is on the bisector. (4.6) Angle bisector theorems/properties: o If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle. (4.7) o If a point is equidistant from the sides of an angle, then it lies on the bisector of the angle. (4.8) 8