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11.1 Basic Notions Euclid’s Axioms 1 There is exactly one line that contains any two points. Euclid’s Axioms 1 There is exactly one line that contains any two points. 2 If two points line in a plane then the line containing these points lies in the plane. Euclid’s Axioms 1 There is exactly one line that contains any two points. 2 If two points line in a plane then the line containing these points lies in the plane. 3 If two distinct planes intersect, they do so at a line. Euclid’s Axioms 1 There is exactly one line that contains any two points. 2 If two points line in a plane then the line containing these points lies in the plane. 3 If two distinct planes intersect, they do so at a line. 4 There is exactly one plane that contains any three colinear points. Euclid’s Theorems 1 A line and a point determine a unique plane. Euclid’s Theorems 1 A line and a point determine a unique plane. 2 Two distinct parallel lines determine a unique plane. Euclid’s Theorems 1 A line and a point determine a unique plane. 2 Two distinct parallel lines determine a unique plane. 3 Two distinct intersecting lines determine a unique plane. Basic Terms Definition A point is a location in space and has no dimension. We use a capital letter to denote a point. Basic Terms Definition A point is a location in space and has no dimension. We use a capital letter to denote a point. Definition A line is an infinite collection of colinear points that has no depth but has length. We label a line with either a lowercase script letter l or two capital letters, denoting two points on the line. Basic Terms Definition A point is a location in space and has no dimension. We use a capital letter to denote a point. Definition A line is an infinite collection of colinear points that has no depth but has length. We label a line with either a lowercase script letter l or two capital letters, denoting two points on the line. Order does←→ not matter as the line is infinite in both directions. We ←→ could use AB or BA More Terms Definition Colinear means that the points lie on the same line. o •A •B •C Which are colinear? / Still More Terms Definition A ray is a subset of a line consisting of an initial point and extending infinitely in one direction. A• •B −→ −→ Is this ray properly named as AB or BA? / Still More Terms Definition A ray is a subset of a line consisting of an initial point and extending infinitely in one direction. A• / •B −→ −→ Is this ray properly named as AB or BA? Definition A line segment is a subset of a line consisting of two endpoints. o •A •B Which is the correct way to name this line segment, AB or BA? / You Guessed It ... More Terms Definition A plane is an infinite collection of coplanar lines that has no thickness but has two dimensions, length and width. We either label a plane with a lowercase Greek letter α or at least 3 points. You Guessed It ... More Terms Definition A plane is an infinite collection of coplanar lines that has no thickness but has two dimensions, length and width. We either label a plane with a lowercase Greek letter α or at least 3 points. Definition Intersecting lines are two lines in the same plane that share exactly one common point. Definition Concurrent lines are more than two lines in the same plane that share exactly one point. Intersecting Lines v. Concurrent Lines What is the difference between these two sets of lines? _ o ? / • o _ • / And Still More Terms Definition Perpendicular lines are two lines in the same plane that intersect at a right angle. And Still More Terms Definition Perpendicular lines are two lines in the same plane that intersect at a right angle. Regardless of which type of intersection, two lines that intersect do so at a . And Still More Terms Definition Perpendicular lines are two lines in the same plane that intersect at a right angle. Regardless of which type of intersection, two lines that intersect do so at a . Definition Parallel lines are two lines in the same plane that do not intersect. And Still More Terms Definition Perpendicular lines are two lines in the same plane that intersect at a right angle. Regardless of which type of intersection, two lines that intersect do so at a . Definition Parallel lines are two lines in the same plane that do not intersect. Definition Skew lines are lines that are noncoplanar, meaning they cannot intersect because they are in different planes. Equal Measure v. Congruence We have to be careful with notation ... Two geometric structures that are same are congruent and have equal measure. Equal Measure v. Congruence We have to be careful with notation ... Two geometric structures that are same are congruent and have equal measure. For example, if AB and CD are exactly the same, other than the letters we use, we could say the following: AB ∼ = CD AB = CD mAB ∼ = mCD Clarifications on Intersections 1 Lines intersect at a point. Clarifications on Intersections 1 Lines intersect at a point. 2 Planes intersect at a line. Clarifications on Intersections 1 Lines intersect at a point. 2 Planes intersect at a line. 3 Lines can be parallel (perpendicular) to a plane if the line is in a plane that is parallel (perpendicular) to said plane. Clarifications on Intersections 1 Lines intersect at a point. 2 Planes intersect at a line. 3 Lines can be parallel (perpendicular) to a plane if the line is in a plane that is parallel (perpendicular) to said plane. 4 The angle where two planes meet at the line creating the half-planes is called a dihedral angle. So if planes are perpendicular, the dihedral angle measures 90◦ . Angles What is an angle? 7 A • B• • C ' Angles What is an angle? 7 A • B• • C ' Definition An angle is the intersection of two rays. The intersection of the rays is at a point called the vertex and the straight parts of the rays are called sides. Angles What is an angle? 7 A • B• • C ' Definition An angle is the intersection of two rays. The intersection of the rays is at a point called the vertex and the straight parts of the rays are called sides. What can we name this angle? Types of Angles Acute angle: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: angle measure between 90◦ and 180◦ Straight angle: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: angle measure between 90◦ and 180◦ Straight angle: angle measure is 180◦ Reflex angle: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: angle measure between 90◦ and 180◦ Straight angle: angle measure is 180◦ Reflex angle: angle measure is between 180◦ and 360◦ Complementary angles: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: angle measure between 90◦ and 180◦ Straight angle: angle measure is 180◦ Reflex angle: angle measure is between 180◦ and 360◦ Complementary angles: two adjacent angles who’s measures add to90◦ Supplementary angles: Types of Angles Acute angle: angle measure between 0◦ and 90◦ Right angle: angle measure is 90◦ Obtuse angle: angle measure between 90◦ and 180◦ Straight angle: angle measure is 180◦ Reflex angle: angle measure is between 180◦ and 360◦ Complementary angles: two adjacent angles who’s measures add to90◦ Supplementary angles: two adjacent angles who’s measures add to 180◦ Adding and Subtracting Angles When talking about fractions of angles, we can use standard decimals or we can use minutes and seconds, just like with time. Compute 48◦ 350 27” + 37◦ 500 40” Adding and Subtracting Angles When talking about fractions of angles, we can use standard decimals or we can use minutes and seconds, just like with time. Compute 48◦ 350 27” + 37◦ 500 40” Compute 48◦ 350 27” − 37◦ 500 40” Angle Conversion Convert 48◦ 350 27” to a number of degrees. Angle Conversion Convert 48◦ 350 27” to a number of degrees. Convert 21.42◦ to degrees, minutes and seconds. Finding Missing Angles Example If we are given that ∠ABC and ∠DBC are complementary and that m∠ABC = 14 m∠DBC, what is the measure of the two angles? Finding Missing Angles Example If we are given that ∠ABC and ∠DBC are complementary and that m∠ABC = 14 m∠DBC, what is the measure of the two angles? Example Find the measure of ∠ABC and ∠DBE. _ F 3x o • • C A 90◦ D • 2x •B • E / Circles and Arcs Circles and Arcs _ AC is called a minor arc because the associated central angle is less than 180◦ . _ ABC is called a major arc because the associated central angle is greater than 180◦ . Angles and Arcs The measure of the arc associated with a central angle is equal to the central angle. Angles and Arcs The measure of the arc associated with a central angle is equal to the central angle. The measure of the arc associated with an inscribed angle is equal to twice the measure of the inscribed angle. Angles and Arcs Example _ Suppose m∠CDF = 25◦ . Find the measure of the major arc CBF. Angles and Arcs Angles and Arcs We are given that m∠CDF = 25◦ , and this is an inscribed angle. There is a relationship between the central angle and inscribed angle _ associated with the arc CF. Angles and Arcs We are given that m∠CDF = 25◦ , and this is an inscribed angle. There is a relationship between the central angle and inscribed angle _ associated with the arc CF. _ So, mCF = 50◦ . Angles and Arcs We are given that m∠CDF = 25◦ , and this is an inscribed angle. There is a relationship between the central angle and inscribed angle _ associated with the arc CF. _ So, mCF = 50◦ . _ The measure of the major arc CBF is 360◦ − 50◦ = 310◦ .

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