Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Algebraic geometry wikipedia , lookup
Analytic geometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Multilateration wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Perceived visual angle wikipedia , lookup
Euler angles wikipedia , lookup
History of geometry wikipedia , lookup
Section 2.1 Perpendicularity Geometry Perpendicular () : Lines, rays, or segments that intersect at right angles. b ab a • Oblique : 2 intersecting lines that are not perpendicular. 1 Geometry Section 2.1 Perpendicularity Coordinate Plane: formed by the intersection of the xaxis and the y-axis x-axis - the horizontal number line y-axis - the vertical number line Origin: the point where the number lines y x origin intersect 2 Geometry Section 2.1 Perpendicularity Coordinates : (aka ordered pair) a set of numbers in the form (x,y) that represents a point on the coordinate plane x is the distance from the y-axis (right - left) y (2,7) 7 x 2 y is the distance from the x-axis (up - down) 3 Geometry Given: a b Prove 1 2 Section 2.1 Perpendicularity a 2 1 b Statements Reasons 1. a b 2. 1 is a right angle 1. Given 3. 2 is a right angle 4. 1 2 2. If two lines are perpendicular, they form a right angle. 3. If two lines are perpendicular, they form a right angle. 4. If angles are right angles, they are congruent. 4 Section 2.1 Perpendicularity Geometry Find the area of the rectangle below A(-4,8) D B(10,8) C(10, -2) 5 Answer Geometry Segment AB = 4 + 10 = 14 (x: -4, 10) Segment DC = 4 + 10 = 14 Segment BC = 8 + 2 = 10 (y: 8, -2) Segment AD = 8 + 2 = 10 The coordinate of D is (-4, -2) Area = length x width Area = 14 x 10 = 140 square units 6 Section 2.1 Perpendicularity Geometry Find the perimeter of the rectangle below A(-4,8) D B(10,8) C(10, -2) 7 Geometry Answer Perimeter is the sum of the sides P = 2 (l + w) P = 2 (14 + 10) = 2 (24) = 48 units 8 Discussion Diagram Geometry F A 30 E 45 H G 45 B C D 9 Geometry Section 2.2 Complementary & Supplementary Angles Complementary angles: two angles whose sum is 90 degrees Complement: that which an angle needs to have a measure of 90. (90 - x) Supplementary angles: two angles whose sum is 180 degrees Supplement: that which an angle needs to have a measure of 180. (180 - x) Memory Helper: In alphabetical and numerical order C (90)…… S (180) 10 Geometry Section 2.2 Complementary & Supplementary Angles Sample Problems Find the measure of the complement of an angle whose measure is 30, 79, 19030’ Express the measure of the complement of an angle whose measure is represented by x, (3a), (r - 40), (x+y) 11 Answers Geometry 90-30 = 60 90-79 = 11 900-19030’ = 89060’-19030’ = 70030’ 90-x 90-3a 90-(r-40) 90-(x+y) 12 Geometry Section 2.2 Complementary & Supplementary Angles More Sample Problems Two angles are complementary. The measure of the larger angle is five times the measure of the smaller angle. Find the measure of the larger angle. The supplement of the complement of an acute angle is always (1) an acute angle (2) an obtuse angle (3) a straight angle (4) a right angle. 13 Answers Geometry Let x = measure of smaller angle 5x = measure of larger angle x + 5x = 90 6x = 90 x = 15, so 5x = 75 If two angles are complementary (sum=90), they are each acute, thus the complement of an acute angle must be less than 90. The supplement (sum=180) of an acute angle must be greater than 90. Thus, the supplement of the complement of an acute angle is always obtuse. 14 Geometry Solving Equation Word Problems Steps: 1. Read the problem a few times 2. Line by line, write the definitions of terms in the problem 3. Line by line, write the givens 4. Line by line, write algebraic expressions 5. Set up the equation. Look for key terms, such as exceeds, less than, difference, etc. 6. Solve 7. Check your work 15 Geometry Practice Problem The measure of a supplement of an angle exceeds three times the measure of the complement of the angle by 10. Find the measure of the angle. Steps: 1. Read the problem a few times 2. Line by line, write the definitions of terms in the problem: supplement, complement 16 Geometry 3. Line by line, write the givens: The measure of a supplement of an angle exceeds three times the measure of the complement of the angle by 10. 4. Line by line, write algebraic expressions: Let x = unknown angle Let 180-x = measure of the supplement of an angle Let 90-x = measure of the complement of the angle 17 Geometry 5. Set up the equation Read the given (180-x) + 10 = 3(90-x) 6. Solve (180-x) + 10 = 270-3x 3x - x = 270 – 180 – 10 2x = 80 x = 40 7. Check (180-40) + 10 = 3(90-40) 150 = 150 18 Geometry Section 2.3 Drawing Conclusions Methods, Suggestions Must memorize definitions, theorems, etc. Symbols give away information. Be familiar with them. Draw as much information from each given as possible. Decide what information will make your case. Draw a valid conclusion! 19 Section 2.3 Drawing Conclusions Geometry Sample Proof Given: X XB bis AC XC bis BD Prove: AB CD A Statements XB bis AC AB BC B C Reasons Given Definition of bisector BC CD Given Definition of bisector AB CD Substitution XC bis BD D 20 Geometry Section 2.4 Congruent Supplements & Complements Theorem If angles are supplementary to the same angle, then they are congruent. Assumes only one angle Theorem If angles are supplementary to congruent angles, then they are congruent. Assumes more than one angle 21 Geometry Section 2.4 Congruent Supplements & Complements Theorem If angles are complementary to the same angle, then they are congruent. Assumes only one angle Theorem If angles are complementary to congruent angles, then they are congruent. Assumes more than one angle 22 Geometry Section 2.4 Congruent Supplements & Complements Sample Proof Given: PB AD m1 = m3 QC AD P 2 Prove: m2 = m4 A Statements R B 1 Q 3 4 C D Reasons 23 Sample Answer Geometry R P 2 Statements 1. PB AD 2. PBC is a right angle 3. 2 and 1 are complementary 4. QC AD 5. QCA is a right angle 6. 3 and 4 are complementary 7. m1 = m3 8. 1 and 3 are congruent 9. 2 is congruent to 4 10. m2 = m4 Reasons A 1 B 1. Given 2. Definition of perpendicular 3. If two angles form a right angle, they are complementary. 4. Given 5. Definition of perpendicular 6. Same as 3 7. Given 8. If two angles have the same measure, then they are congruent. 9. If angles are complementary to congruent angles, then they are congruent. 10. Definition of congruent Q 3 4 C D 24 Geometry Section 2.4 Congruent Supplements & Complements Sample Proof K J Given: EJ EK Prove: 1 and 2 are complementary. Statements 1 C 2 E D Reasons 25 Sample Answer Geometry K J 1 C Statements 1. EJ EK 2. JEK is a right angle 3. mJEK = 90º 4. CED is a straight angle 5. mJEK + m1 + m2 = 180 6. 90º + m1 + m2 = 180 7. m1 and m2 = 90 8. 1 and 2 are complementary 2 E D Reasons 1. Given 2. Definition of perpendicular 3. Definition of a right angle 4. Assumption 5. Definition of a straight angle 6. Substitution 7. Subtraction 8. If the sum of two angles is 90º, then they are complementary. 26 Geometry Section 2.5 Addition and Subtraction Properties More Theorems! If a segment is added to congruent segments, the sums are congruent. If congruent segments are added to congruent segments, the sums are congruent. If an angle is added to congruent angles, the sums are congruent. If congruent segments are added to congruent segments, the sums are congruent. 27 Geometry Section 2.5 Addition and Subtraction Properties Subtraction Theorems If a segment (or angle) is subtracted from congruent segments or (angles), the differences are congruent. If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. 28 Geometry Section 2.6 Multiplication and Division Properties Theorems If segments (or angles) are congruent, their like multiples are congruent. If segments (or angles) are congruent, their like divisions are congruent. 29 Geometry Section 2.7 Transitive and Substitution Properties Transitive Theorems If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. Note: The relation “is perpendicular to” is never transitive. Substitution (Same as in algebra) 30 Section 2.8 Vertical Angles Geometry Definitions opposite rays: 2 collinear rays with a common endpoint that extend in opposite directions vertical angles: angles formed when two opposite rays (lines) intersect Theorem Vertical Angles are congruent. 31