1.2 Informal Geometry


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1 1.2 Informal Geometry Mathematical System: (xiomatic System) Undefined terms, concepts: Point, line, plane, space Straightness of a line, flatness of a plane point lies in the interior or the exterior of an angle Definitions: give meaning to new terms xioms or Postulates: statements accepted as true Theorems: If P, then Q need to be proved. Undefined terms: Point Line; collinear points Plane Space Terms that will be formally defined latter ngle Triangle Rectangle Line Segment Definition: line segment is the part of a line that consists of two end points, and all the points between them. Notation Measuring The Midpoint ongruent line segments Measuring angles ongruent angles isect an angle (trisect ) Straight angle Right angle ngles formed by intersection of two lines Perpendicular lines Parallel lines (informally) Empty set onstructions: onstruct a segment congruent to a given segment onstruct the midpoint of a segment
2 1.3 Early Definitions and Postulates Definition: gives meaning to new terms; described precisely a term. Postulate: a statement assumed to be true Postulate 1. Two points determine a line Through two distinct points, there is exactly one line. Postulate 2. Ruler Postulate The measure of any line segment is a unique positive number. Postulate 3 Segment ddition Postulate If X is a point on and X, then X+X=. Postulate 4 If two lines intersect, they intersect at a point. Postulate 5 If two lines intersect, they intersect at a point. Postulate 6 If two distinct planes intersect, then their intersection is a line. Postulate 7 Given two distinct points in a plane, the line containing these points also lies in the plane. Definition: The distance between two points and is the line segment that joins the two points. Definition: ongruent line segments are two segments that have the same length. Definition: The midpoint of a line segment is the point that separates the line segment into two congruent parts. Definition: Ray denoted is the union of and all points X on such that is between and X. Definition: Parallel lines are lines that lie in the same plane but do not intersect. Definition: oplanar points are points that lie in the same plane. Theorem The midpoint of a line segment is unique
3 1.4 ngles and their relationships Definition: n angle is the union of two rays that share a common endpoint. Postulate 8 The measure of an angle is a unique positive number. Types of angles: cute angles Right angles Obtuse angles Straight angles Reflex angles Interior, Exterior, and the sides of an angle. Postulate 9 ngleddition Postulate If a point D lies in the interior of an angle, then m D+m D=m. Definition: Two angles are adjacent (adj. s) if they have a common vertex and a common side between them. Definition: Two angles are congruent ( s) if they have the same measure. Definition: The bisector of an angle is the ray that separate the given angle into two congruent angles. Definition: Two angles are complementary if the sum of their measures is 90 Definition: Two angles are supplementary if the sum of their measures is 180. Vertical ngles onstruct an angle congruent to a given angle. onstruct the bisector for a given angle.
4 1.5 Introduction to geometric proof To believe certain geometric principles, it is necessary to have proof Two column proof: 1. Statement: State the theorem to be proved 2. Drawing: Represents the hypothesis of the theorem 3. Given: Describes the drowning according to the information found in the hypothesis of the theorem 4. Prove: Describes the drowning according to the claim found in the conclusion of the theorem 5. Proof: Orders in a logical flow, a list of claims (Statements) and justifications (Reasons), beginning with the Given and ending with the Prove Selected properties from lgebra are used as reasons to justify statements: Properties of equality (a, b, and c are real numbers) ddition Property of Equality If =, then + = + Subtraction Property of Equality If =, then = Multiplication Property of Equality If =, then = Division Property of Equality If =, then = Reflexive Property = Symmetric Property If =, then = Distributive Property + = + Substitution Property If =, then replaces b in any equation Transitive Property If =, and = then = Properties of inequality (a, b, and c are real numbers) ddition Property of Inequality If >, then + > + (<) Subtraction Property of Inequality If >, then > (<)
5 lgebraic Proof Ex: Given: 2 1 = 3 Prove: = 2 Proof Statements Reasons = 3 1. Given = ddition Property of Equality 3. 2 = 4 3. Substitution 4. = 4. Division Property of Equality 5. = 2 5. Substitution Geometric Proof Given P on the segment, prove that = Drawing Given: Prove: = P Proof Statements Reasons Given 2. + = 2. Segment ddition Postulate 3 3. = 3. Subtraction property of addition
6 1.6 Perpendicular Lines Definition: Perpendicular lines are two lines that meet to form congruent adjacent angles. Theorem If two lines are perpendicular, then they meet to form right angles. Drawing Given: intersecting at E Prove: is a right angle E D Proof Statements Reasons Given Definition: 3. = 3.Two congruent angles have equal measurements 4. = Measure of a straight angle 5. + = 5. ngle addition postulate 6. + = Substitution 7. + = Substitution 8. 2 = Substitution 9. = Division Property 10. is a right angle 10. Definition of right angle
7 Theorem If two lines intersect, then the vertical angles formed are congruent. Given: intersecting at O Prove: 2 4 Drawing 2 3 O 1 4 D Proof Statements Reasons 1. intersects at O 1. Given 2. = Definition: measure of straight angles 3. = 3. Substitution = and = 4. ngle addition Postulate = Substitution 6. 4 = 2 7. Subtraction Property If two angles are equal in measure the angles are congruent Symmetric Property of ongruence of angles
8 onstruction 1. onstruct a perpendicular to a given line at a specified point. Theorem (NO Proof) In a plane, there is exactly one line perpendicular to a given line at any point on the line. onstruction 2. onstruct a perpendicular bisector to a line segment.
9 1.7 Geometric Proof of a Theorem Theorems: Statements that can be proved Hypothesis: given onclusion: what we need to establish onverse of a Theorem The converse of the statement If P, then Q is If Q, then P. Interchange the hypothesis with the conclusion. Exemple: Theorem If two lines are perpendicular, then they meet to form right angles. The converse: Theorem If two lines meet to form right angles, then they are perpendicular lines.
10 Proof of Theorem Theorem If two lines meet to form right angles, then they are perpendicular lines. Drawing Given: and intersecting at E so that is a right angle Prove: E D Proof Statements Reasons 1. and intersecting at E so that is a right angle 1. Given 2. = If an is a right angle, its measure is is a straight, so = If an is a straight angle, its measure is = 4. ngle addition postulate = Substitution (2), (3), (4) 6. = Subtraction Property of = (5) 7. = 7. Substitution (2), (^) If two have = measures, the s are If two lines form adjacent s the lines are
11 Proof of Theorem Theorem If two angles are complementary to the same angle ( or to congruent angles), then these angles are congruent. Given: 1 and 3are complementary 2 and 4 are complementary 1 = 2 Prove: 3 4 Plan: 1 and 3=90 2 and 4= = = 4 Drawing Proof Statements Reasons 1. 1 and 3 are complementary, 2 and 4 are complem. 1. Given 2. m 1 + 3=90 and 2 + 4= Defn of complementary angles 3. m 1 + 3= Substitution ( 2) 4. 1 = 2 4. Given 5. m 1 + 3= Substitution, (3), (4) 6. 3 = 4 6. Subtraction Property of = (5) Defn
12 Theorem If two angles are supplementary to the same angle ( or to congruent angles), then these angles are congruent. Theorem ny two right angles are congruent. Theorem If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. Picture Proof of Theorem Given: Prove: 1 and 2 are complementary Proof: With, we see that 1 and 2 are parts of a right angle. Then m 1 + m 2 = 90, so 1 and 2 are complementary. Drawing 1 2 D
13 Proof of Theorem Theorem If the exterior sides of two adjacent acute angles form a straight line, then these angles are supplementary. Given: 3 and 4 and Prove: 3 and 4 are supplementary Plan: Drawing H E 3 4 F G Proof Statements Reasons 1. 3 and 4 and 1. Given 2. m = 2. ngle  ddition Postulate 3. is a straight angle 3. Defn of a straight angle 4. = The measure of a straight angle 5. m 3 + 4= Substitution, (2), (4) 6. 3 and 4 are supplementary 6. Defn of supplementary angles Theorem If two line segments are congruent, then their midpoint separate these segments into four congruent segments. Theorem If two angles are congruent, then their bisector separate these angles into four congruent angles.
14 2.1: The Parallel Postulate 2.2: Indirect Proof 2.3: Proving Lines Parallel 2.4: The ngles of a Triangle hapter 2 2.1: The Parallel Postulate 1. Perpendicular Lines 2. Parallel Lines 3. Euclidian Geometry 4. Parallel Lines and ongruent ngles 5. Parallel Lines and ongruent ngles 1. Perpendicular Lines Recall:. Definition: Two lines are perpendicular if they meet to form congruent adjacent angles. Theorem 1.6.1: Perpendicular lines meet to form right angles. onstruction 5: onstruct a perpendicular line to a given line at a given point D. onstruct a bisector to a given line onstruction 6: onstruct a perpendicular line to a given line from a point not on the given line Theorem From a point not on a given line, there is exactly one line perpendicular to the given line. 2. Parallel Lines Parallel Lines and planes Definition: Parallel lines are lines in the same plane that do not intersect 3. Euclidian Geometry the plane is flat, two dimensional surface in which the line segment joining any two points of the plane lies entirely within the plane Postulate 10 Through a point not on a line, exactly one line is parallel to the given line.
15 Special ngles Transversal: a line that intersect two (or more) other lines at distinct points Interior angles: ngles formed between line m and n: 3, 4, 5, 6 Exterior angles: ngles formed outside line m and n: 1, 2, 7, l m orresponding angles: ngles that lie in the same relative position: left, right, above, below 1 and 5 above, left 3 and 7 below, left 2 and 6 above, right 4 and 8 below, right n lternate Interior ngles: interior angles that have different vertices and lie on opposite sides of the transversal: lternate Exterior ngles: exterior angles that have different vertices and lie on opposite sides of the transversal: 3 and 6 3 and 7 1 and 8 2 and 7
16 Postulate 11 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. orresponding angles: l 1 and 5, 3 and 7, 2 and 6, 4 and m Ex: If 1 = 110 and, find 2, 4, 5, and n Theorem If two parallel lines are cut by a transversal line, then the pairs of alternate interior angles are congruent. Given:, transversal l Prove: l m n Proof Statements Reasons 1., transversal l 1. Given Postulate Vertical ngles, Theorem Substitution
17 Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. Given:, transversal T W Prove: 1 and 3 are supplementary R U X S V Y Proof Statements Reasons 1., transversal 1. Given lternate interior angles 3. m 1 = 2 3. Defn of congruent angles 4. m = Straight angle 5. m = 5. ngle ddition Postulate 6. m = Substitution, (4) and (5) 7. m = Substitution, (3) and (6) 8. 1 and 3 are supplementary 8. Definition Suppl angles
18 Theorem If two parallel line are cut by a transversal, then the pairs of exterior angles on the same side of the transversal are supplementary.
19 2.2 Indirect Proof Let represents the conditional statement If P then Q and ~ =not P. The following statements are related to the given statement onditional If P then Q onverse If Q then P Inverse ~ ~ If not P then not Q ontrapositive ~ ~ If not Q then not P Example: If Juan lives in L, then he lives in onditional: : If Juan lives in L, then he lives in onverse: If Juan lives in, then he lives in L Inverse : If Juan doesn t live in L, then he doesn t lives in ontrapositive : If Juan doesn t lives in, then he doesn t live in L Given : P Prove : Q Indirect Proof : 1. Suppose that ~Q is true. 2. Reason from the supposition until you reach a contradiction 3. Note that the supposition claiming that ~ is true must be false and that Q must therefore be true
20 Ex1: Given is not perpendicular to prove that 1 and 2 are not complementary angles Paragraph Proof: Writing like an essay; still each statement has to be justified. Suppose that 1 and 2 are complementary angles. Then m = 90 because the sum of the measures of two compl. angles is 90. We also know that m = by the ngle ddition Postulate. In turn, =90 by substitution. Then is a right angle. Thus,. ut this contradicts the given hypothesis; therefore, the supposition must be false, nd it follows that 1 and 2 are not complementary angles. 1 2 D Statements Reasons 1. Suppose that 1 and 2 are complementary angles 1. ontradict the hypothesis 2. m = The sum of the measures of two compl. angles is m = 3. ngle ddition Postulate 4. =90 4. substitution 5. is a right angle 5. Definition of right angle Theorem of perpendicular lines 7. ontradiction 7. Given is not perpendicular to 8. 1 and 2 are not complementary angles 8. ecause of contradiction
21 Ex2: omplete a formal proof of the following theorem: If two lines are cut by a transversal so that the corresponding angles are not congruent, then the two lines are not parallel lines Given: m and l are cut by the transversal t and 1 5 Prove: m ssume that. When these lines are cut by transversal t, any two corresponding angles (including 1 and 5 are congruent. ut 1 5 by hypothesis. Thus, the assumed statement, which claims that must be false. It follows that m t m l Ex3: omplete a formal proof of the following theorem: The bisector of an angle is unique Given: bisect Prove: is the only bisector for D ssume that : is also a bisector of and that m =. Given that bisect, it follows that m =. y the ngle ddition Postulate, m = + m. y substitution, = + ; but then = 0, by subtraction. n angle with measure 0 contradicts the Protractor Postulate, therefore, the assumed statement must be false, and it follows that the bisector of an angle is unique. E D
22 Recall from 2.1 the relevant Postulate and Theorems: 2.3 Proving line parallel Postulate 11 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Theorem If two parallel lines are cut by a transversal line, then the pairs of alternate interior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. Theorem If two parallel line are cut by a transversal, then the pairs of exterior angles on the same side of the transversal are supplementary. Note: The Postulate and each Theorem has 1. The same hypothesis: If two parallel line are cut by a transversal, and 2. conclusion involving angle relationship. If we wish to prove that two lines are parallel instead of a relation between angles we can use onverse. 1. hypothesis: an angle relationship, and 2. conclusion:, then the two lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of corresponding angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of the alternate interior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of the alternate exterior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the alternate interior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the alternate exterior angles on the same side of the transversal are supplementary, then these lines are parallel lines.
23 Theorem If two lines are cut by a transversal such that the pairs of corresponding angles are congruent, then these lines are parallel lines. Given: and Prove: cut by transversal ; 1 2 r t l Proof: 2 m Suppose that m. Then a line r can be drown through point P that is parallel to m; this follows from the Parallel Postulate. If, then 3 2 because these angles correspond. ut 1 2 by hypothesis. Now 3 1 by the Transitive Property of ongruence; therefore, m 3 = 1. ut m = 1 (see fig.). Substituting 1 for 3 leads to m = 1, and by subtraction, 4 = 0. This contradicts the Protractor Postulate, therefore r and l must coincide, and it follows that.
24 Theorem If two lines are cut by a transversal so that the pairs of the alternate interior angles are congruent, then these lines are parallel. Given: and cut by transversal ; 2 3 Prove: Paragraph Proof: t 1 3 l 2 m Proof Statements Reasons 1. and and transversal Given If two lines intersect, vertical angles are congruent Transitive property Theorem 2.3.1
25 Theorem If two lines are cut by a transversal such that the pairs of the alternate exterior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the interior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Given: and Prove: cut by transversal ; 1 is supplementary to 2 t 3 l 1 2 m Proof Statements Reasons 1. and and transversal ; 1 is supplementary to 2 1. Given 2. 1 is supplementary to 3 2. Straight line If two angles are supplementary to the same angle they are Theorem 2.3.1
26 Theorem If two lines are cut by a transversal such that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Theorem If two lines are each parallel to a third line, then these lines are parallel to each other. Theorem If two coplanar lines are each perpendicular to a third line, then these lines are parallel to each other. onstruction 7 onstruct the line parallel to a given line from a point not on that line.
27 2.4 The ngles of a Triangle Definition: triangle (symbol ) is the union of three line segments that are determined by three noncollinear points. or or or or,, are vertices are sides of the triangle Points: D a point inside the triangle E point on the triangle F a point outside the triangle D E... F Triangles classified by ongruent sides Scalene Isosceles Equilateral Triangles classified by ongruent angles cute Obtuse Equilateral Right
28 Theorem In a triangle the sum of the measures of the interior angles is 180 Given: fig (a) Prove: + + = 180 (a) Picture Proof Through in fig (a), draw. We see that = 180. ut 1 = and 3 = (alternate interior angles) (b) D E Then + + = 180 in fig (a) orollary Each angle of an equiangular triangle measures 60 Proof: y Theorem m = 180 ; Since = =m by hypothesis, it follows that 3m = 180. Therefore m = 60, and = =m = 60 ; orollary The acute angles of a right triangle are complementary. orollary If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent orollary The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles
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