DOCSLIB.ORG

LESSON 18.1: Geometry Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
LESSON 18.1: Geometry Review LESSON 18.1: Geometry Review 1.1: ANGLES Angle Measure Angles are measured using two different systems: degree or radian. We will be using degrees, minutes, and seconds to measure angles. 1 minute = 1 degree 1 degree = 60 minutes 60 1 minute = 60 seconds 1 second = 1 minute 60 1° = 60' Symbolically: 1” = 60” 1 degree = 60 × 60 seconds 1 second = 1 degrees 3600 = 3600 seconds ° 1" = 1 1° = 3600" 3600 Example 1: ° ° a) 36.5°= 36° + 0.5 x 60’ b) 68°45’18” = 68° + 45 + 18 60 3600 = 36°30’ = 68° + 0.75° + 0.005° = 68.755° Special Angles Acute Angle: An angle is acute Obtuse Angle: An angle is Right Angle: A right angle if it measures between 0° and obtuse if it measures between measures exactly 90°. 90° 90° and 180°. In a diagram, the square marking at an angle indicates a right angle. Straight Angle: an angle which Complementary Angles: Two Supplementary Angles: Two measures exactly 180°. angles are complementary if angles are supplementary if they they sum to 90°. sum to 180°. ∠ABC = 180° ∠ x + ∠ y = 90° ∠ α + ∠ β = 180° C B y A x α β Chapter 1: Geometry Review Page 1 Example 2: a) If ∠A = 23° and ∠B = 67° then ∠A and ∠B are complementary since ∠A + ∠B = 90° b) If ∠X = 15°, ∠Y = 65° and ∠Z = 10°, even though ∠X + ∠Y + ∠Z = 90° they are NOT complementary angles. A complementary angle refers only to pairs of angles. c) What is the complement of the supplement of the angle 145°? Solution: Supplement of 145° = (180 – 145)° = 35° Complement of 35° = (90 – 35)° = 55° The complement of the supplement of the angle 145° is 55° d) Twice the supplement of a certain angle is 5 times its complement. Find the angle. Solution: Let the angle be x Then: supplement of x = (180 – x) complement of x = (90 – x) and: 2 (supplement of x) = 5 (complement of x) Supplement of x Hence: (180 – x) x 2()180 − x = 5(90 − x) (90 – x) 360 − 2x = 450 − 5x Complement of x − 2x + 5x = 450 − 360 3x = 90 x = 30 Hence the angle is 30°. Chapter 1: Geometry Review Page 2 Exercises 18.1.1: 1. Convert to degrees and minutes: (a) 57.5° (b) 114.2° (c) 34.75° 2. Convert to decimal degrees: (a) 74°15’ (b) 17°48’ (c) 56°60’ 3. State whether each angle given is acute, obtuse, right, or straight. (a) 97°55’ (b) 48.6° (c) 180° (d) 88.6° (e) 90° (f) the supplement of 45° (g) the complement of 45° (h) the complement of any acute angle (i) the supplement of any acute angle (j) the supplement of a right angle 4. What angle has the same measure as its supplement? 5. What angle has the same measure as its complement? 6. Find the complement of each angle: (a)74° (b)56° (c)34.5° (d)88.6° (e)17°25’ (f)6°34’ 7. Find the supplement of each angle: (a)104° (b)8° (c)132.2° (d)88.6° (e)97°55’ (f)16°14’ 8. What angle has a supplement that measures three times its complement? 9. What angle has a supplement that measures ten times its complement? 10. Find the supplement of the complement of the angle measuring 40°. 11. Find the complement of the supplement of the angle measuring 120°. 12. Find the complement of a right angle. Chapter 1: Geometry Review Page 3 18.1.2: TRIANGLES General A triangle has three angles, or vertices, and three sides. Each vertex of the triangle is named using capital letters: A, B, C, etc. The angle-sum of a triangle is 180°. That is, the sum of the 3 angles of the triangle is 180°. The longest side of the triangle is always opposite the largest angle of the triangle; the shortest side is opposite the smallest angle. Example 1: a) b) 65° 67° x 90° 35° 78° Since x° + 65° + 90° = 180° 67° + 35° + 78° = 180° x° + 155° = 180° x° + 155° = 180° x° = 25° The third angle of the triangle is 25° c) d) β 63°43’ 44.8° 46°39’ β x Since 44.8° + β + β = 180° 46°39’ + 63°43’ + x = 180° 44.8° + 2β = 180° 110°22’ + x = 180° 2β = 180° − 44.8° 2β = 135.2° ⎛Since 46°39' + 63°43' = 109°82' ⎞ ⎜ ⎟ 135.2° ⎝ = 110°22' ⎠ β = 2 x = 180° - 110°22’ β = 67.6° The third angle of the triangle is 67.6° ⎛Since 180° −110°22' = 179°60'−110°22' ⎞ ⎜ ⎟ ⎝ = 69°38' ⎠ x = 69°38' Chapter 1: Geometry Review Page 4 The Components of a Triangle Median: A line from the vertex Altitude: A line from the vertex Angle Bisector: A line from the of an angle to the midpoint of of an angle perpendicular to the vertex of an angle to the the opposite side. opposite side. opposite side which bisects that angle. AD is a median of ∆ABC. XW is an altitude of ∆XYZ. PT is an angle bisector of ∆PQR. A Y P W B Q Z D T X C R Naming the Sides and Angles of a Triangle Angles: There are three common systems of naming the angles of a triangle: A 1. Using the vertices, with the vertex of the angle in the center. α 2. Using a number or letter (often a Greek letter) inside the angle. 3. According to the vertex of the triangle at that angle. C In the diagram the angle at the vertex A can be called ∠BAC, ∠ α, or ∠ A. B Sides: There are two common ways to name the sides of a triangle: 1. According to two vertices it connects. The sides of ∆ABC (diagram above) are AB, BC and AC. 2. According to the vertex of the angle opposite the side. Each side of the triangle is named using the lower case letter corresponding to the letter of the vertex of the triangle opposite that side. B In the triangle ∆ABC the vertices of the triangle are A, B, and C and the sides of this triangle must be a, b, and c, such that: c The side opposite the vertex A is called a. a The side opposite the vertex B is called b. The side opposite the vertex C is called c. A b C We often use this second method in trigonometry because the name of each side of a triangle indicates its specific relationship to the angles of the triangle. Chapter 1: Geometry Review Page 5 Classifying Triangles 1. Types of Triangles Defined by Angle Size Acute Angled Triangle: All angles of the triangle are acute. Obtuse Angled Triangle: One angle of the triangle is obtuse. Acute Triangle Obtuse Triangle Right Triangle: One angle of the triangle a right angle. A The other two angles of the triangle are acute and complementary. hypotenuse The side of the triangle opposite the right angle is called the c hypotenuse. The other sides are called the legs of the b triangle. The hypotenuse is always the longest side of the triangle. Pythagoras’ Theorem states the important relationship C a B between the lengths of the sides of a right triangle. 2. Types of Triangles Defined by Side Length Scalene Triangle: A triangle with three sides of different lengths. apex Isosceles Triangle: A triangle with two sides of equal length. Properties: ; The angles opposite to the equal sides are congruent. ; The third (unequal) angle of the triangle is called the apex. A line from the apex to the midpoint of the opposite side of the triangle bisects the apex angle from which it passes and is perpendicular to the opposite side. Any altitude of the triangle from the apex to the base is also a base median and an angle bisector. Equilateral Triangle: A triangle with three sides of equal length. Properties: All properties of the isosceles triangle, and: 60° ; All angles measure 60°. ; Any altitude of the triangle is also a median and an angle bisector. 60° ; Any median or altitude of the triangle divides the triangle into 2 congruent right triangles, with angles of 30°, 60°, and.90°. We will use this special property in the study of 60° Trigonometry. 30° 30° 60° 60° Chapter 1: Geometry Review Page 6 Relationships between Triangles Congruent Triangles: Two triangles are congruent if their three sides and the three corresponding angles are congruent. Example 2: The two triangles ∆ABC and ∆RPQ are congruent, C written ∆ABC ≅ ∆RPQ, since x three corresponding angles are equal: b ∠ A = ∠ R, ∠ B = ∠ P, ∠ C = ∠ Q, a R and the corresponding sides are equal: p A q a = r, b = p, and c = q. c x Q B (Note: the order in which the vertices are written when we name the triangles reflect the corresponding angles.) r P Similar Triangles Two triangles are similar if their angles are equal. The three corresponding pairs of sides are then in fixed proportion. Example 3: The triangles ∆DEF and ∆ZXY are similar, written ∆ABC ≈ E ∆ZXY, since ∠ D = ∠ Z, ∠ F = ∠ Y, then (using the angle sum of the triangle) we must have ∠ E = 50° = ∠ X. 1.7 cm 2.0 cm So three corresponding angles are equal. and: 75° ED 1.7 1 55° F = = D 1.8 cm XZ 5.1 3 DF 1.8 1 = = YZ 5.4 3 Y EF 2.0 1 = = XY 6.0 3 55° so the sides are in fixed proportion of 1:3. 5.4 cm 6.0 cm 75° X Z 5.1 cm Chapter 1: Geometry Review Page 7 Exercise 18.1.2: 1.
Load more

Recommended publications
  • Circle Theorems
    Circle theorems A LEVEL LINKS Scheme of work: 2b. Circles – equation of a circle, geometric problems on a grid Key points • A chord is a straight line joining two points on the circumference of a circle. So AB is a chord. • A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90°. • Two tangents on a circle that meet at a point outside the circle are equal in length. So AC = BC. • The angle in a semicircle is a right angle. So angle ABC = 90°. • When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference. So angle AOB = 2 × angle ACB. • Angles subtended by the same arc at the circumference are equal. This means that angles in the same segment are equal. So angle ACB = angle ADB and angle CAD = angle CBD. • A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. Opposite angles in a cyclic quadrilateral total 180°. So x + y = 180° and p + q = 180°. • The angle between a tangent and chord is equal to the angle in the alternate segment, this is known as the alternate segment theorem. So angle BAT = angle ACB. Examples Example 1 Work out the size of each angle marked with a letter. Give reasons for your answers. Angle a = 360° − 92° 1 The angles in a full turn total 360°. = 268° as the angles in a full turn total 360°.
    [Show full text]
  • Angles ANGLE Topics • Coterminal Angles • Defintion of an Angle
    Angles ANGLE Topics • Coterminal Angles • Defintion of an angle • Decimal degrees to degrees, minutes, seconds by hand using the TI-82 or TI-83 Plus • Degrees, seconds, minutes changed to decimal degree by hand using the TI-82 or TI-83 Plus • Standard position of an angle • Positive and Negative angles ___________________________________________________________________________ Definition: Angle An angle is created when a half-ray (the initial side of the angle) is drawn out of a single point (the vertex of the angle) and the ray is rotated around the point to another location (becoming the terminal side of the angle). An angle is created when a half-ray (initial side of angle) A: vertex point of angle is drawn out of a single point (vertex) AB: Initial side of angle. and the ray is rotated around the point to AC: Terminal side of angle another location (becoming the terminal side of the angle). Hence angle A is created (also called angle BAC) STANDARD POSITION An angle is in "standard position" when the vertex is at the origin and the initial side of the angle is along the positive x-axis. Recall: polynomials in algebra have a standard form (all the terms have to be listed with the term having the highest exponent first). In trigonometry, there is a standard position for angles. In this way, we are all talking about the same thing and are not trying to guess if your math solution and my math solution are the same. Not standard position. Not standard position. This IS standard position. Initial side not along Initial side along negative Initial side IS along the positive x-axis.
    [Show full text]
  • 20. Geometry of the Circle (SC)
    20. GEOMETRY OF THE CIRCLE PARTS OF THE CIRCLE Segments When we speak of a circle we may be referring to the plane figure itself or the boundary of the shape, called the circumference. In solving problems involving the circle, we must be familiar with several theorems. In order to understand these theorems, we review the names given to parts of a circle. Diameter and chord The region that is encompassed between an arc and a chord is called a segment. The region between the chord and the minor arc is called the minor segment. The region between the chord and the major arc is called the major segment. If the chord is a diameter, then both segments are equal and are called semi-circles. The straight line joining any two points on the circle is called a chord. Sectors A diameter is a chord that passes through the center of the circle. It is, therefore, the longest possible chord of a circle. In the diagram, O is the center of the circle, AB is a diameter and PQ is also a chord. Arcs The region that is enclosed by any two radii and an arc is called a sector. If the region is bounded by the two radii and a minor arc, then it is called the minor sector. www.faspassmaths.comIf the region is bounded by two radii and the major arc, it is called the major sector. An arc of a circle is the part of the circumference of the circle that is cut off by a chord.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Foundations of Geometry
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.
    [Show full text]
  • "Perfect Squares" on the Grid Below, You Can See That Each Square Has Sides with Integer Length
    April 09, 2012 "Perfect Squares" On the grid below, you can see that each square has sides with integer length. The area of a square is the square of the length of a side. A = s2 The square of an integer is a perfect square! April 09, 2012 Square Roots and Irrational Numbers The inverse of squaring a number is finding a square root. The square-root radical, , indicates the nonnegative square root of a number. The number underneath the square root (radical) sign is called the radicand. Ex: 16 = 121 = 49 = 144 = The square of an integer results in a perfect square. Since squaring a number is multiplying it by itself, there are two integer values that will result in the same perfect square: the positive integer and its opposite. 2 Ex: If x = (perfect square), solutions would be x and (-x)!! Ex: a2 = 25 5 and -5 would make this equation true! April 09, 2012 If an integer is NOT a perfect square, its square root is irrational!!! Remember that an irrational number has a decimal form that is non- terminating, non-repeating, and thus cannot be written as a fraction. For an integer that is not a perfect square, you can estimate its square root on a number line. Ex: 8 Since 22 = 4, and 32 = 9, 8 would fall between integers 2 and 3 on a number line. -2 -1 0 1 2 3 4 5 April 09, 2012 Approximate where √40 falls on the number line: Approximate where √192 falls on the number line: April 09, 2012 Topic Extension: Simplest Radical Form Simplify the radicand so that it has no more perfect-square factors.
    [Show full text]
  • Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
    Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers.
    [Show full text]
  • Foundations for Geometry
    Foundations for Geometry 1A Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes Lab Explore Properties Associated with Points 1-2 Measuring and Constructing Segments 1-3 Measuring and Constructing Angles 1-4 Pairs of Angles 1B Coordinate and Transformation Tools 1-5 Using Formulas in Geometry 1-6 Midpoint and Distance in the Coordinate Plane 1-7 Transformations in the Coordinate Plane Lab Explore Transformations KEYWORD: MG7 ChProj Representations of points, lines, and planes can be seen in the Los Angeles skyline. Skyline Los Angeles, CA 2 Chapter 1 Vocabulary Match each term on the left with a definition on the right. 1. coordinate A. a mathematical phrase that contains operations, numbers, and/or variables 2. metric system of measurement B. the measurement system often used in the United States 3. expression C. one of the numbers of an ordered pair that locates a point on a coordinate graph 4. order of operations D. a list of rules for evaluating expressions E. a decimal system of weights and measures that is used universally in science and commonly throughout the world Measure with Customary and Metric Units For each object tell which is the better measurement. 5. length of an unsharpened pencil 6. the diameter of a quarter __1 __3 __1 7 2 in. or 9 4 in. 1 m or 2 2 cm 7. length of a soccer field 8. height of a classroom 100 yd or 40 yd 5 ft or 10 ft 9. height of a student’s desk 10. length of a dollar bill 30 in.
    [Show full text]
  • 08. Non-Euclidean Geometry 1
    Topics: 08. Non-Euclidean Geometry 1. Euclidean Geometry 2. Non-Euclidean Geometry 1. Euclidean Geometry • The Elements. ~300 B.C. ~100 A.D. Earliest existing copy 1570 A.D. 1956 First English translation Dover Edition • 13 books of propositions, based on 5 postulates. 1 Euclid's 5 Postulates 1. A straight line can be drawn from any point to any point. • • A B 2. A finite straight line can be produced continuously in a straight line. • • A B 3. A circle may be described with any center and distance. • 4. All right angles are equal to one another. 2 5. If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles. � � • � + � < 180∘ • Euclid's Accomplishment: showed that all geometric claims then known follow from these 5 postulates. • Is 5th Postulate necessary? (1st cent.-19th cent.) • Basic strategy: Attempt to show that replacing 5th Postulate with alternative leads to contradiction. 3 • Equivalent to 5th Postulate (Playfair 1795): 5'. Through a given point, exactly one line can be drawn parallel to a given John Playfair (1748-1819) line (that does not contain the point). • Parallel straight lines are straight lines which, being in the same plane and being • Only two logically possible alternatives: produced indefinitely in either direction, do not meet one 5none. Through a given point, no lines can another in either direction. (The Elements: Book I, Def.
    [Show full text]
  • Classify Each Triangle by Its Side Lengths and Angle Measurements
    GRADE 4 | MODULE 4 | TOPIC D | LESSONS 12–16 KEY CONCEPT OVERVIEW In Lessons 12 through 16, students explore lines of symmetry and characteristics of triangles and quadrilaterals. You can expect to see homework that asks your child to do the following: ▪ Find and draw lines of symmetry. ▪ Given half of a figure and the line of symmetry, draw the other half of the figure. ▪ Classify triangles by side lengths (e.g., equilateral, isosceles, scalene) and by angle measurements (e.g., acute, right, obtuse). ▪ Draw triangles that fit different classifications (e.g., acute and scalene). ▪ Name quadrilaterals, identify attributes (i.e., characteristics) that define them, and construct them based on given attributes. SAMPLE PROBLEM (From Lesson 13) Classify each triangle by its side lengths and angle measurements. Circle the correct names. Additional sample problems with detailed answer steps are found in the Eureka Math Homework Helpers books. Learn more at GreatMinds.org. For more resources, visit » Eureka.support GRADE 4 | MODULE 4 | TOPIC D | LESSONS 12–16 HOW YOU CAN HELP AT HOME ▪ Ask your child to look around the house for objects that have lines of symmetry. Examples include the headboard of a bed, dressers, chairs, couches, and place mats. Ask him to show where the line of symmetry would be and what makes it a line of symmetry. Be careful of objects such as doors and windows. They could have a line of symmetry, but if there’s a knob or crank on just one side, then they are not symmetrical. ▪ Ask your child to name and draw all the quadrilaterals she can think of (e.g., square, rectangle, parallelogram, trapezoid, and rhombus).
    [Show full text]
  • Different Types of Loading on Right Angle Joints
    HIGH STRENGTH RIGHT ANGLE COMPOSITE JOINTS FOR CIVIL ENGINEERING APPLICATIONS Gerard M Van Erp, Fibre Composites Design and Development (FCDD), University of Southern Queensland (USQ), Toowoomba, Queensland, 4350, Australia, SUMMARY: This paper presents initial results of an ongoing research project that is concerned with the development of strong right angle composite joints for civil engineering applications. Experimental, analytical and numerical results for four different types of monocoque right angle joints loaded in pure bending will be presented. It will be shown that a significant increase in load carrying capacity can be obtained through small changes to the inside corner of the joints. The test results also demonstrate that there is reasonable correlation between the failure mode of the joints and the location and severity of the stress concentrations predicted by the FE method. KEYWORDS: Joining, Civil Application, Testing, Structures. INTRODUCTION Plane frames are common load carrying elements in buildings and civil engineering structures. These frames are combinations of beams and columns, rigidly jointed at right angles. The rigid joints transfer forces and moments between the horizontal and the vertical members (Fig. 1a) and play a major role in resisting lateral loads (Fig. 1b). Figure 1a: Frame subjected to vertical loading b: Frame subjected to horizontal loading Figure 1c: Different types of loading on right angle joints Depending on the type of loading and on the location of the joint in the frame, the joint will be subjected to a closing action (Fig. 1: case A and B) or an opening action (Fig. 1: case C). This paper will concentrate on joints with an opening action only.
    [Show full text]
  • Math 1330 - Section 4.1 Special Triangles
    Math 1330 - Section 4.1 Special Triangles In this section, we’ll work with some special triangles before moving on to defining the six trigonometric functions. Two special triangles are 30 − 60 − 90 triangles and 45 − 45 − 90 triangles. With little additional information, you should be able to find the lengths of all sides of one of these special triangles. First we’ll review some conventions when working with triangles. We label angles with capital letters and sides with lower case letters. We’ll use the same letter to refer to an angle and the side opposite it, although the angle will be a capital letter and the side will be lower case. In this section, we will work with right triangles. Right triangles have one angle which measures 90 degrees, and two other angles whose sum is 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs of the triangle. A labeled right triangle is shown below. B c a A C b Recall: When you are working with a right triangle, you can use the Pythagorean Theorem to help you find the length of an unknown side. In a right triangle ABC with right angle C, 2 2 2 a + b = c . Example: In right triangle ABC with right angle C, if AB = 12 and AC = 8, find BC. 1 Special Triangles In a 30 − 60 − 90 triangle, the lengths of the sides are proportional. If the shorter leg (the side opposite the 30 degree angle) has length a, then the longer leg has length 3a and the hypotenuse has length 2a.
    [Show full text]