INSIDE the circle. The angle is MADE BY. The angle EQUALS
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1 ANGLES IN A CIRCLE The VERTEX is located At the CENTER of the circle. ON the circle. INSIDE the circle. OUTSIDE the circle. The angle is MADE BY Two Radii Two Chords, or A Chord and a Tangent, or A Chord and a Secant. Two chords Two Secants, or A Secant and a Tangent, or Two Tangents. The angle EQUALS The measure of the Intercepted Arc Half the measure of the Intercepted Arc Half the SUM of the measures of the Intercepted Arcs. Half the DIFFERENCE of the measures of the intercepted arcs.
2 SEGMENTS IN A CIRCLE The VERTEX is located INSIDE the circle OUTSIDE the circle Two chords Two Secant Segments A Tangent and a Secant Segment
3 CLASSIFY QUADRILATERALS NO Pairs of Opposite Sides Are Parallel Exactly ONE Pair of Opposite Sides Are Parallel BOTH Pair of Opposite Sides Are Parallel
4 POLYGONS DEFINITION
5 SOME PROPERTIES PROPERTY ADDITION MULTIPLICATION COMMUTATIVE A + B = B + A A! B = B! A ASSOCIATIVE A + ( B + C ) = (A + B ) + C A! ( B! C ) = ( A! B )! C IDENTITY A + 0 = A A! 1 = A INVERSE A + B B = A A! B! (1/B) = A PROPERTIES OF EQUALITY REFLEXIVE Addition Property of Equality: If A = B, Then A + C = B + C Subtraction Property of Equality If A = B, Then A C = B C A = A Multiplication Property of Equality: If A = B, Then A! C = B! C Division Property of Equality: If A = B and C " 0, then (A / C) = (B / C) DISTRIBUTIVE A! ( B + C ) = A! B + A! C TRANSITIVE If A = B And If B = C, Then A = C
6 Triangle Classifications We Classify Triangles By Angles Obtuse one angle is obtuse Classifying Triangles Right one angle is right Acute all 3 angles are acute Equilangular All 3 angles are 60 AND Find point C that makes!abc : a) equilateral b) isosceles c) right d) acute e) obtuse f) scalene By Sides Scalene No sides are congruent Isosceles At leat two sides are congruent Equilateral All 3 sides are congruent Problem: Given: AB is a side of!abc Draw the loci of all points for C so that!abc is: a) equilateral b) isosceles c) right d) acute e) obtuse f) scalene
7 CLASSIFYING TRIANGLES BY ANGLES AND SIDES ISOSCELES (at least two congruent sides) SCALENE (no congruent sides) RIGHT (one right angle) OBTUSE (one obtuse angle) >90 >90 >90 <90 ACUTE (all acute angles) <90 EQUILATERAL (all congruent sides) Three angle-side combinations make no sense: EQUIANGULAR (all 60 angles) Equiangular Scalene Obtuse Equilateral Right -- Equilateral
8 CLASSIFY NUMBERS REAL NUMBERS RATIONAL NUMBERS: -1, -!, 0,!, 1, (ratio of integers, no zero in the denominator) INTEGERS: -3, -2, -1, 0, 1, 2, 3, WHOLE NUMBERS: 0, 1, 2, 3, NATURAL (COUNTING) NUMBERS: 1, 2, 3, IRRATIONAL NUMBERS: (non-repeating decimals cannot be written as a ratio of two integers) Transcendental (can be multiplied by another irrational number to make it rational): example 2, 3 etc Non-Transcendental Example:!, e IMAGINARY NUMBERS Example:!1
9 ANGLE RELATIONSHIPS CONGRUENT:!1 "!2 CORRESPONDING ANGLES SUPPLEMENTARY:!3+!4 = 180 SAME-SIDE INTERIOR ALTERNATE INTERIOR ANGLES COMPLEMENTARY:!5 +!6 = 90 SAME SIDE EXTERIOR ALTERNATE EXTERIOR ANGLES VERTICAL PAIR Adjacent Adjacent (LINEAR PAIR) Non-Adjacent Non-Adjacent
10 Angle-Angle Similarity Postulate (AA Sim) If two angles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar. C F A PROVING TRIANGLES CONGRUENT C B F Side-Side-Side Triangle Congruence Postulate (SSS) If the sides of one triangle are congruent to the sides of second triangle, then the two triangles are congruent. C Side-Angle-Side Triangle Congruence Postulate (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. C A B D!ABC "!DEF Extension problem: How would you draw two NONcongruent triangles that have 5 congruent parts? 8 E D E AB! DE!A "!D BC! EF!B "!E AC! DF!C "!F!ABC "!DEF CPCTC: Corresponding Parts of Congruent Triangles are Congruent. 1 A D B F This is one of 4 shortcuts to prove triangles congruent that are shown in this brochure. 2 E A D B F Once you prove two triangles are congruent, you can justify that all the corresponding parts are congruent. How? CPCTC. 3 E
11 Angle-Side-Angle Triangle Congruence Postulate (ASA) Angle-Angle-Side or Side-Angle-Angle Triangle Congruence Theorem (AAS or SAA) Hypotenuse Leg Triangle Congruence Theorem (HL) Side-Side-Angle or Angle-Side-Side (SSA or ASS) also known as the Donkey Theorem If two angles and the included side of one triangle are congruent to two angles and the included side of second triangle, then the two triangles are congruent. If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of a second right triangle, then the two triangles are congruent. It is NOT a valid method to prove triangles are congruent to say that since two sides and a non-included angle of each are congruent that the triangles are congruent. C C C It ain t necessarily so! Check it out: C 4 A D B F E A D B F Do you see the difference between this and the sketch on page 4? 5 E A Notice how this is the exception or special case where SSA actually works. If the triangle is NOT a right triangle, you get the Donkey Theorem. (See page 7) 6 B A A B C B 7
12 REMEMBER YOUR VOWELS A + E + I + O = U AND Y When submitting math work, always ask yourself: A. Did I clearly identify and label or circle an ANSWER that makes sense for what I did? E. Is my reasoning clearly and efficiently EXPLAINED? I. Does my work reflect all the necessary INFORMATION I used to solve the problem? Can others tell what letter, number and symbol represents? O. Is my work well ORGANIZED with steps that follow a logical sequence? U. If so, my work demonstrates that I UNDERSTAND the concepts involved. and always Y. Ask WHY. Why did I select the strategies I did? Why might others argue the benefits of a different strategy? Why is the concept significant? GRAPHIC ORGANIZER: RULES FOR PRESENTING 1. COME PREPARED. 2. VOLUNTEER TO PRESENT. 3. ADDRESS YOUR CLASSMATES. 4. SHOW RESPECT. 5. HELP FIND ERRORS. 6. CONTRIBUTE YOUR IDEAS. 7. POINT OUT CONNECTIONS. 8. CITE YOUR SOURCES. GRAPHIC ORGANIZER: PROBLEM SOLVING STRATEGY: Identify the relationship. In math, how is what is given and what you are asked to solve related? Devise a plan. In math, write an equation. Execute the plan. In math, solve the equation. Answer the question that is asked. In math, this may or may not be the solution to your equation. Look at the question again.
13 STEPS FOR PROOF (shorthand used to explain your reasoning) 1. Write the GIVEN and PROVE statements. (Identify and state the given information and what it is you are trying to prove). 2. Draw a SKETCH. 3. MARK the drawing. (the most important step) A. First, mark only what is explicitly GIVEN. B. Secondly, mark what is implied or implicitly true. 4. Set up a STATEMENT and REASON table. 5. Fill in the table. Justify each statement with a reason or rule. SIX RULE CATEGORIES (used in this course) 1. Given Information 2. Definitions 3. Properties 4. Postulates 5. Theorems 6. Corollaries IMPLICIT INFORMATION to watch for Vertical Pairs Common Side or Common Angle (Reflexive Property) Parallel Lines (and angle relationships formed by a transversal crossing parallel lines). Straight Angle Triangle Angle Sum
14 ENGLISH VOLUME CONVERSION
15 WRITE SYMBOLS SAY WORDS DRAW FIGURES MODEL REAL WORLD OBJECTS A Point A Tip of a pen l Line L Pointer!## " AB Line AB Intersection of ceiling and wall CD Segment CD Pencil AB The measure of segment AB The length of the pencil!abc Angle ABC Bridge supports m!abc The measure of angle ABC The degrees of an angle!!! " AB Ray AB Pointer!EFG Triangle EFG Hill AB! CD AB CD!##"!##" EF! GH Segment AB is congruent to segment CD Segment AB is parallel to segment CD Segment EF is perpendicular to segment GH Parallelogram ABCD Two people s height Rows of corn Stairs and the railing Street layout Circle M Gear ABC Plane ABC White Board
16 TRANSFORMATIONS
17 BASIC TRIGONOMETRIC FUNCTIONS (SO-CA-TOA) T o a
18 EXPONENTS RULE EXAMPLE
19 CIRCLES
20 THE CARTESIAN COORDINATE PLANE
21 PROOF
22 QUADRATIC EQUATIONS
23 MULTIPLE REPRESENTATIONS NUMERICAL, ALGEBRAIC, GRAPHICAL (NAG)
24 PROPORTION
25 SIMPLIFYING RADICALS
26 SIMILAR TRIANGLESGRAPHIC ORGANIZER: SOLVING LINEAR SYSTEMS Methods to Solve Systems of LINEAR SYSTEMS 2x + 3y = 5 4x y = 17 GRAPHING SUBSTITUTION COMBINATIONS Intersecting lines have exactly one solution which is the point of intersection (x, y) 1. Graph Line Graph Line Visually identify the point of intersection. Parallel lines have no solution because there is no point of intersection Coinciding lines have an infinite number of solutions (all the points on the line are solutions of the system) 1. Isolate one variable in one of the equations. 2. Substitute that expression into the second equation and solve for the variable. 3. Substitute the value found in step 2 into either of the original equations and solve for the remaining variable. 4. Write the ordered pair. 1. Rewrite each equation in standard form. 2. Choose a variable to eliminate and multiply by appropriate number to eliminate it. 3. Solve for remaining variable either by substituting into one of the original equations or by repeating step 2 for the other variable. 4. Write the ordered pair. What if both of the variables cancel out? Look at the resulting arithmetic equation. *False statement indicates the lines are parallel so there is no solution. *True statement indicates the lines coincide so there are infinite solutions.
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