Sketches for Engineering and Design PDF

# SKETCHING ## Importance - The importance of freehand drawing or sketching in engineering and design cannot be overestimated - A person who possesses a complete knowledge of drawing as a language, gains the ability to execute quick, accurate and clear sketches of ideas - This constitutes a valuabl...

# SKETCHING ## Importance - The importance of freehand drawing or sketching in engineering and design cannot be overestimated - A person who possesses a complete knowledge of drawing as a language, gains the ability to execute quick, accurate and clear sketches of ideas - This constitutes a valuable means of expression ## Since most original ideas find their first expression through the medium of a freehand sketch, it is therefore an indispensable means of amplifying and clarifying, as well as recording verbal expressions. - Freehand sketches are of great assistance to the designer in organizing his thoughts and recording his ideas - They are an effective and economical means of formulating various solutions to a given problem so that a choice can be made between them on the onset - Often much time can be lost if the designer starts his scaled layout before adequate preliminary study with the aid of sketches ## The term freehand sketch is too often understood to mean a crude or sloppy freehand drawing in which no particular effort has been exerted. - On the contrary, a freehand sketch should be made with care and with attention to correct line widths ## Sketching Materials - Freehand sketching requires only paper, pencil and eraser which are items that anyone must have for ready use - For excellent sketching, one can use a plain white paper called sketch pad which is heavier than bond paper and may be purchased at low cost from local bookstore or school supply establishment - Beginning sketchers, especially those who cannot sketch reasonably well without guide lines, can use cross-section paper or graphing paper which are available in sheets or pads - These papers are also conveniently used for sketching to scale since values can be assigned to the squares and the squares counted to secure proportional distances ## Sharpening Pencils - Use a soft pencil, such as HB or F and sharpen it to a conical point for center lines, extension lines and dimension lines - For visible lines, hidden lines and cutting-plane lines, round off the point slightly to produce the desired thickness of line as illustrated - Except for the construction lines which are drawn very light, all lines should be made dark ## Note: Soft pencils, such as HB or F, should be used for freehand sketching and for carefully made sketches, soft erasers are recommended for use. - Sharpen it to a conical point as shown - Use sharp point for center lines, dimension lines and extension lines - For visible lines, hidden lines and cutting-plane lines, round off the point slightly to produce the desired thickness of line and make all lines dark except for construction lines which should be drawn very light ## Technique of Lines - The chief difference between a mechanical drawing and a freehand sketch lies in the technique of the lines - A good instrumental line is drawn rigidly straight and exactly uniform while that of freehand line is drawn with freedom and variety - The freehand construction line is a very light rough line wherein some of the strokes may overlap and all other lines should be dark and clean-cut - Ends of all dashes are accented and sharp contrasts between line thicknesses are maintained - All visible lines are drawn heavy so the outline will stand out clearly and make hidden lines, center lines, dimension lines and extension lines thin ## Straight Lines - Since almost all lines on the average sketch are straight lines, it is necessary to learn on how to make them well - Figures are pictorial demonstrations of drawing straight lines ## Circles and Arcs - Small circles and arcs can be easily sketched in one or two strokes as in stroking the circular portions of letters - Figures are the preferred methods as presented ## Ellipses are circles viewed obliquely. - With a little practice, you can learn to sketch small ellipses with a free hand movement and large ellipses with a free arm movement - Figures are for sketching ellipses ## Proportions - The most important rule in freehand sketching is to keep the sketch in proportion - No matter how brilliant the technique or how well the small details are drawn, if the proportions particularly the large over-all proportions are bad, the sketch will be bad - In sketching from an actual object, various distances can be easily compared on the object by using your pencil to compare measurements as shown - It should be remembered that while doing this, you should always hold your pencil at arm's length and stay at your position until you are through comparing measurements - The length sighted can then be compared in similar manner with any other dimension of the object ## In establishing proportions, especially for irregular shapes, the blocking-in method is recommended. - The steps in blocking-in and completing the sketch of a Shaft Hanger are shown - As always, first give attention to the main proportions, next to the general sizes and direction of flow of curved shapes and finally to the snappy lines of the completed sketch ## Pictorial Sketching - The foregoing are simple methods of preparing pictorial sketches. - Although prepared in freehand, they must be carefully sketched applying the rules in estimating dimensions and in establishing proportions. - Let us now examine several methods of preparing pictorial sketches that will be of great assistance in learning the principles of multiview projection. ## Isometric Sketching - The steps in sketching isometric are: - Sketch the enclosing box lightly, closely resembling the tilted object as shown - Block in the recess and the projecting block - Dim all construction lines with soft eraser and heavy in all final lines ## Isometric Ellipses - Shown are objects having cylindrical or conical shapes and are placed in the isometric position; hence, circles viewed at an angle will appear as ellipses ## Another simple method for sketching pictorially is oblique sketching. - Hold the object as shown - Block in the front face of the Bearing, as if you were sketching a front view - Sketch receding lines parallel to each other and at any convenient angle, normally at 30° or 45° approximately with horizontal - To obtain a more natural appearance, cut the estimated depth to about three-quarters or one-half size approximately - Dim all construction lines with soft eraser and heavy in the final lines ## Perspective Sketching - The Bearing sketched in oblique can easily be drawn in one-point perspective (one vanishing point), as shown - Sketch the true front face of the object, the same as in oblique sketching. - Select the vanishing point (VP) for the receding lines. In most cases, it is desirable to place VP above and to the right of the drawing, as shown. - Sketch the receding lines toward VP. - Estimate the depth to look well, and sketch in the back portion of the object. Note that the back circle and arc will be slightly smaller than the front circle and arc - Dim all construction with a soft eraser, and heavy in all final lines # LETTERING ## History and Development - The designs of modern alphabets had their origin in Egyptian hieroglyphics which were developed into a cursive hieroglyphics or hieratic writing - This was adopted by the Phoenicians and was developed by them into an alphabet of twenty-two (22) letters. - The Greeks later adopted the Phoenician alphabet but evolved into two distinct types in different sections of Greece namely: the Eastern Greek type used also in Asia Minor and the Western Greek type which was used in the Greek colonies in and near Italy. - In this manner, the Western Greek alphabet became the Latin alphabet about 700 B. C. - The Latin alphabet came into general use throughout the old world. - Originally the Roman capital alphabet consisted of twenty-two characters - These characters may still be seen on Trajan's Column and other Roman monuments - The letter V was used for both U and V until the 10th century. - Letter J, j, was adapted at the end of the 14th century as a modification of the letter I, i, The dot over the lower-case j still indicates its kinship to letter i. - In Old English, these two letters are very much similar. - The numerous modern styles of letters were derived from the design of the original Roman capitals. ## Importance - Lettering being the written language of industry is integral in communicating the complete description of the size, shape and function of the various components of the object under construction. - It records ideas and transmits the same from one culture and generation to another. - Since lettering can either make or break an otherwise carefully prepared drawing, it is therefore essential to every student to learn to letter properly if it is to function as an effective means of graphic communication. ## Styles of Letters - **Single-Stroke Gothic Letters/Gothic Letters** - Generally, the lettering used in engineering and technical drawing consists of single-strokes which are uniform in width. - It is the plainest and most legible of all the styles of letter and is comparatively easy to make, - **Roman Letters** - The Roman Letters are distinctively divided into two: the Old Roman and the Modern Roman. - The Old Roman which is the basis of all our letters and considered to be the most beautiful are still prepared to be used on buildings and for inscription on metals or stones while the letters used in most modern newspapers, magazines, books and maps especially in the names of countries and cities are of Modern Roman. - The text used in this book is set in Modern Roman Letters or simply Roman Letters. - **Italic Letters** - All inclined letters regardless of style used is classified as Italics Letters. - Because of its beauty and versatility, Italic is widely recognized as the most popular calligraphic hand. - It is used for designing book jackets, menus and greeting cards to addressing envelopes, lettering poems, invitations and announcements. - **Text Letters** - This letter style includes all styles of Old English, German Text, Bradley Text and other trade names. - Oftentimes used in diplomas and certificates, this style is too illegible to read and cumbersome to prepare for commercial purposes. ## General Division of Letters - Letters are generally divided according to: - **Usage** - **Built-up Letters** - These are letters often used on big advertisements. - They are first outlined and then filled up for more emphasis. - **Single-Stroke Letters** - These are letters made one stroke of lettering pencils or pens. - **Proportion** - To meet the design or space requirements, letters may be made normal, narrower or wider. - **Normal Letters** - These are letters drawn in its usual proportion and are neither too narrow nor too wide. - **Compressed/Condensed Letters** - These letters are spaced closed together and are narrower in width in proportion to the height and are used when space is limited. - **Extended Letters** - These letters are drawn wider than the normal letters and are used when the space is wider. - **Thickness of the Stem** - **Lightface** - Letters having very thin stems are called LIGHTFACE letters. - **Boldface** - Letters having very thick stems or heavy strokes are classified as BOLDFACE letters. ## Uniformity in Lettering - Uniformity in lettering is essential in any style of lettering. - Letters that are uniform in height, proportion, inclination, strength of lines and spacing insure a pleasing appearance. ## Stability - Letters appear top heavy when their upper and lower portions are drawn equal in width. - To correct this so as to produce the effect of stability and appear more pleasing to the reader, the upper portions of letters are reduced in size. ## Technique of Lettering - While it is true that practice makes perfect, it must be understood that practice alone is not enough - It must be accompanied by continuous effort to improve - Lettering as what normally regarded is not writing but freehand drawing - Therefore freehand drawing's six fundamental strokes and their directions are basic to lettering - The horizontal strokes are drawn to the right, and all vertical, inclined and curved strokes are drawn downward. ## There are three necessary steps in lettering: - Knowledge of the proportions and forms of the letters and the order of the strokes - Knowledge of composition - the spacing of letters and words. Rules governing composition should be thoroughly mastered - Persistent practice with continuous effort to improve. ## Pencil lettering should be executed with a fairly soft pencil, such as an F or H for ordinary paper and the strokes should be dark and sharp, not gray and blurred. - The correct position of the hand in lettering is shown - In general, draw vertical strokes downward or toward you with a finger movement, and draw horizontal strokes from left to right with a wrist movement without turning the paper. ## Guide Lines in Lettering - Guide lines are absolutely essential for good lettering - The better draftsman always uses guide lines and believes that good lettering can be acquired through the use of guide lines. - Guide lines are fine light lines that are used to regulate the height of letters and keep letters uniformly vertical or inclined. ## Spacing of Letters - Uniformity in spacing of letters is a matter of equalizing spaces by eye. - The background areas between letters and not the distances between them should be made approximately equal. ## Spacing of Words - space between words is equal to capital letter "o" capital letter "O"after comma and period - space between words for lower-case letters is equal to capital letter "o". - Lower-case words also should be kept well apart. # INSTRUMENTAL DRAWING ## Basic Drawing Equipment - The essential items of equipment needed by students in technical schools are shown. - To produce the most satisfactory results, the drawing equipment should be of high quality - A complete list of these items is enumerated below and is shown pictorially ## Objectives in Drawing - The following are the important objectives the student in technical drawing should strive to attain: - **Accuracy** - No drawing is of maximum usefulness if it is not accurate. - The student must learn from the beginning that success in a college career or later in professional employment cannot be achieved if the habit of accuracy is not acquired. - **Legibility** - The student in technical drawing should remember that the drawing is a means of communication to others, and that it must be clear and legible in order to serve its purpose well, Care should be given to details, especially to lettering. - **Neatness** - If a drawing is to be accurate and legible, it must also be clean; therefore the student should constantly strive to acquire the habit of neatness. - Untidy drawings are the result of sloppy and careless methods and will not be accepted by the instructor. - **Speed** - Speed in any undertaking comes with study and practice and the student who draws faster is usually more mentally alert. - Speed is not attained by hurrying; it is an unsought by-product of intelligent and continuous work. ## Drawing at School - Technical drawing requires headwork - The efficient drawing student must see to it that the correct equipment is available and refrains from borrowing which is annoying to others. - While the student is drawing, the Basic Drawing 1 which is our chief source of information should be available and in a convenient position. - When questions arise, first refer to the book and endeavor to find the answer for yourself. - In doing so, self-reliance and initiative are developed. - On the other hand, if you really need help, you may ask your instructor. - The student who goes about his work intelligently, with a minimum waste of time, first studies the assignment carefully to be sure that he understands the principles involved; second, makes sure that the correct equipment is in proper condition (such as sharp pencils, etc.); and third, makes an effort to dig out answers for yourself (the only true education). ## Proper Care and Use of Drawing Instruments - On the succeeding pages are discussions of the correct use and care of the drawing instruments. - The student should learn and practice correct manipulation of the drawing instruments so that correct habits may be formed and even maintained. - Eventually, he should draw correctly by habit so that his full attention may be given to the problems at hand. - The instructor will insist upon absolutely correct form at all times, making exceptions only in cases of physical disability. ## Drawing Boards/Tables - Drawing boards traditionally are available in soft woods, hardwoods, masonite and linoleum. - For student's use, the recommended size of the board is 20" x 24". - Drawing tables with wooden seat pan stools are used as standard drawing furniture. - The left-hand of the board/table which must be straight, is called the working edge, because the head of the T-square slides against it. ## T-square - The T-square is composed of a long strip called a blade fastened rigidly at right angles to a shorter piece called the head. - The upper edge of the blade and the inner edge of the head are working edges of the T-square and must be straight. - The working edge of the head must not be convex or the T-square will rock when the head is placed against the edge of the drawing board or table. - The blade should not be warped or the Triangles will slide underneath it. - The blade should have transparent plastic edges and should be free of nicks along the working edge. - Transparent edges are recommended, since they permit the user to see drawing in the vicinity of the lines being drawn. - T-square is used in constructing horizontal lines. - Do not use the T-square for any rough purpose. - Never cut paper along its working edge, as the plastic is easily cut and even a slight nick will ruin the T-square. ## The Triangles - The triangles are available in transparent plastic so that lines of the drawing can be seen through them. - A good combination of triangles is the 30° x 60° triangle with a long side of 18 cm and the 45° triangle with 14cm sides. - Triangles are used with the aid of the T-square in constructing vertical and inclined lines. - Since most inclined lines in mechanical drawing are drawn at 30°, 45° and 60°, these angles can be easily drawn using your triangles. - Besides the standard angles, using these triangles in combination, other angles such as 15º and 75° can also be drawn. ## Drawing Horizontal Line Using the T-square - In drawing a horizontal line, the T-square should be in its working position (the head should be pressed firmly against the working edge of the drawing board), then slide the left hand to the blade of the T-square as shown, pressing the blade tightly against the drawing paper. - Draw horizontal line from left to right with the pencil at an angle of approximately 60° with the drawing paper. - While drawing the horizontal line, rotate the pencil slowly between the thumb and forefinger in clockwise direction so as to distribute the wear uniformly on the lead, maintain a symmetrical point and keep the pencil in continuous contact with the working edge of the T-square to keep the line perfectly horizontal. ## Drawing Vertical Line Using the Triangle - To draw vertical lines, it should be remembered that the T-square should be in its working position and the triangle should be resting upon the working edge of the T-square. - With the left hand holding both the blade and the triangle firmly in position, draw the line upward rotating the pencil slowly between the thumb and forefinger. - Lean the pencil in the direction of the line at an angle of approximately 60° with the paper and in a vertical plane. - Twist the upper part of the body to the right as shown ## Drawing Inclined Lines - Drawing inclined lines with the aid of triangles are shown. - Note carefully that all lines in the left half of the circle are drawn toward the center while those in the right half are drawn away from the center. ## Set of Drawing Instruments - Drawing instruments are usually sold in sets or cases but they may be purchased separately. - Good quality drawing instruments are usually made of nickel silver, which has a silvery luster and is corrosive resistant. - Most common of these instruments are the compass. - With pencil and inking attachments, they are used primarily for drawing circles. - Most compass needle points have a plain end for use as a compass. - Adjust the needle point with the shoulder end out so that the small point extends slightly farther than the pencil lead or pen point. ## To draw a circle - place the needle point at the exact intersection of the center line, then adjust the compass to the required radius, lean the compass forward and draw the circle clockwise while rotating the handle between the thumb and forefinger. - It must be remembered that any error in radius will result in a doubled error in diameter; hence, it is best to draw a trial circle first on scrap paper and then check the diameter with the scale. ## On drawings having circular arcs and tangent straight lines, draw the arcs first, whether in pencil or in ink, as it is much easier to construct a straight line to an arc than an arc to a straight line. ## Triangular Metric Scale - Since almost all of the dimensions in the illustrations and the problems in this book are given in metric units, and the international language of measurement used in the industry is the metric system, the triangular metric scale with variety of scale graduations will be used. - A scale is used to reproduce the drawing into full size, reduced size or enlarged size. - **Full Size** - The 1:1 scale is full size and each division is actually 1 mm in width with the numbering of the calibrations at 10mm intervals This same scale is convenient also for the ratios of 1:10, 1:100, etc. - **Reduced Size** - The 1:2 scale is one-half size and each division equals 2 mm with the calibration numbering at 20-unit intervals. - This scale is convenient for ratios of 1:20, 1:200, etc. - **Enlarged Size** - The enlarged size is the exact opposite of the reduced size and is given by the scale ratios of 2:1, 3:1, etc. ## Irregular Curve - Irregular curves or French curves are made of transparent plastic and are used to draw mechanical curves other then circles or circular arcs. - The curves are composed largely of successive segments of the geometric curves, such as the ellipse, parabola, hyperbola and involute. - To draw a mechanical curve over the freehand curve with the aid of the irregular curve, it is important to match the various segments of the irregular curve with successive portions of the freehand curve and to draw the line with pencil or technical pen. ## Protractor - Protractor is used for measuring or setting off angles other than those obtainable with the triangles. - The best protractors are made of nickel silver and are capable of most accurate work. - For ordinary work, the plastic protractor is satisfactory and is much cheaper than the former. ## Drawing Pencil - Drawing pencil is the most important tool of a student in any drawing subjects. - Drawing pencils come in various grades of softness or hardness. - With HB, 2B and H pencils, the student can easily make variations in the weight of lines even if the pressure of the pencil applied on the paper is the same for each grade pencil. - Pencil points may be sharpened into three different shapes: the conical, chisel and the elliptical but the most common and the pencil point prepared by the College especially for sketching purposes is the conical point. ## Eraser - Erasers are available in various degrees of hardness or abrasiveness. - For general drawing purposes, a soft rubber eraser is suggested for removing pencil lines and cleaning smudges and for ink lines, a hard rubber eraser is recommended. - Either eraser is recommended for general cleaning of large areas of a drawing or for removing unnecessary pencil lines from an inked drawing. - It is likewise suggested that the erasing shield which is made of thin flat metal with cut outs of various shapes and sizes, must be used to protect the lines of the drawing not intended to be erased. ## Technical Pens - Technical pens are precision inking tools that come with needle points of varying sizes and ink colors depending on the line widths and colors required. - Nowadays, technical pens come with different accessories like the cleaning unit with cleaning fluid and cleaner concentrate for proper care and maintenance. - Because of their fixed line widths, they are suitable for accurate line work, freehand and mechanical lettering. - For uniform line work as well as to protect the needle point from accidental bending or unnecessary wearing, the pen should be used perpendicular to the paper. ## Drawing Paper - For working drawings and for general use, the preferred paper is the white drawing paper (printed borders and title blocks are optional) and come in cut sheet of 36 x 45 cm. - In fastening the drawing paper to the drawing board, the paper should be placed close enough to the upper edge of the board to permit adequate space at the bottom of the sheet for using the T-square and supporting the arm while drawing as well as close enough to the working edge of the drawing board to reduce to a minimum any error resulting from the bending of the blade of the T-square. - To fasten the paper, place the T-square in working position with the drawing board, then press the former with the left hand while the paper is adjusted with the right hand until the upper or lower horizontal border line coincides with the working edge of the T-square. - Fasten the upper-left corner or lower-left corner then the opposite corner and the remaining two corners of the drawing paper with the masking tape as shown. # GEOMETRICAL CONSTRUCTION ## Introduction - Geometrical construction presents definitions and illustrations of geometrical terms and figures considered important in technical drawing. - The succeeding pages discuss practical applications of simplified methods of geometrical constructions that are based on plane geometry, likewise afford opportunities for excellent practice in accurate instrumental drawings and produce drawings according to industry standards. - Since accuracy is most important in instrumental drawing, the use of prescribed tools, instruments and materials should always be observed. - Make all construction lines thin and light and all object lines thick and dark. - Except for construction lines and guide lines, all unnecessary lines should be erased. ## Geometrical Terms and Figures - **Point** - it represents a location in space or on a drawing - It has no width, height or even depth. - It is represented by: - the intersection of two lines - a short cross bar on a line - a small cross or dash - **Line** - it is the shortest distance between two points. - It can be: - **Horizontal line** - is a line drawn from left to right and is parallel to the horizon. - **Vertical line** - is a line drawn from bottom to top exactly perpendicular to the horizon - **Perpendicular line** - is a line drawn exactly 90° with reference to an existing line - **Inclined line** - is a line which is neither vertical nor horizontal in nature - **Angle** - It is a figure formed by the intersection of lines and is described by the number of degrees it contains. - The point produced by this intersection is called the vertex. - Angle can be: - **Acute angle** - is an angle that measures less than 90°. - **Obtuse angle** - is an angle that is greater than 90°. - **Right angle** - is an angle which measures at exactly 90° and can be called complementary angles when angles formed together have a sum of 90°. - **Straight angle** - is an angle formed by two lines which in effect is a straight line with a measure of 180° and can be called supplementary angles when the sum of angles formed is exactly 180°. - **Triangle** - It is a plane figure bounded by three straight sides and the sum of its interior angles is always 180°. - It can be: - **Right triangle** - it is a triangle with a right angle and the square of the hypotenuse is equal to the sum of the squares of the two sides. - **Isosceles triangle** - it is a triangle with two equal sides and two equal angles. - **Equilateral triangle** - it is a triangle with all its three sides and angles equal and sometimes referred to as equiangular triangle. - **Scalene triangle** - it is a triangle with no sides and angles equal. - **Quadrilaterals** - It is a plane figure bounded by four straight sides. - It is called a parallelogram if the opposite sides are parallel. - **Polygons** - it is any plane figure bounded by straight lines and can be called regular polygon if it has equal sides and equal angles, hence, can be inscribed in or circumscribed around a circle. - **Circle** - It is a closed curve bounded by points which are equidistant from a fixed point called the center. - **Arc** - it is a part or a subset of a circle. - **Semicircle** - it is an arc of a circle whose degree measure is equal to 180°. - **Circumference** - it is the distance around a circle. - **Chord** - it is a segment or straight line joining the two points of the center. - **Diameter** - it is a chord of a circle that passes through the center of a circle. - **Radius** - it is a segment joining the center and any point of the circle. - **Sector** - it is a segment of a circle bounded by two radii and the arc between them. - **Tangent line** - it is a line which intersects a circle at exactly one point. - The point where the line and circle meet is called the point of tangency. - **Solid** - it is a figure bounded by plane surfaces and these surfaces are called the faces, and if these faces are equal regular polygons, then they are classified as regular polyhedra. - **Prism** - it is a solid with two bases, which are parallel equal polygons with three or more lateral faces that are parallelograms. - **Pyramid** - it is a solid having a polygon as a base and triangular lateral faces intersecting at a common point called the vertex. - **Cylinder** - it is a solid generated by a straight line called the generatrix moving in contact with a curved line and always remaining parallel to its previous position or to its axis. - Each position of the generatrix or the straight line of the cylinder is called the element. - **Cone** - it is a solid generated by a straight line moving in contact with a curved line and passing through a fixed point called the vertex of the cone. - **Sphere** - it is a solid generated by a circle revolving about one of its diameters. ## Basic Techniques of Drawing Geometrical Figures - **Bisecting a Line using a Perpendicular Bisector** - Given line AB, and using points A and B as centers, strike two arcs which intersect at points C and D. - Connect the intersections C and D with a straight line producing line CD. - This line bisects line AB which is also perpendicular to the given line AB. - **Bisecting an Arc** - Using the ends of the arc A and B as centers, strike two arcs with the same radius which intersect at points C and D. - Connect the intersections C and D with a straight line. - This line bisects the given arc. - **Drawing a Perpendicular Line passing through a given Point on a given Line** - Given line AB and point O where the perpendicular line should pass, strike an arc which intersects line AB at point C and D. - Using points C and D as centers, strike two arcs having the same radius such that these intersect at point E. - Connect points O and E with straight line which is perpendicular to line AB and passes through point O on the given line. - **Drawing a Perpendicular Line passing through a Point outside the given Line** - Given line AB and point O where the perpendicular line should pass, strike an arc using point O as center. - The arc should intersect line AB at points C and D. - With points C and D as centers, strike two arcs having the same radius such that they intersect at point E. - Connect points O and E with a straight line making line OE perpendicular to line AB. - **Bisecting an Angle** - Given angle AOB, strike an arc which intersects line OB at point D and line OA at point C. - With points C and D as centers, strike two arcs of equal radius which intersect at point E. - Connect points E and O with a straight line. - This line is the bisector of the given angle AOB. - **Drawing a 45° Angle** - Given line OA, erect a perpendicular line passing through point O. - Bisect right angle AOC, producing angle AOG which is 45° or one half of the given angle. - **Drawing a 60° Angle** - Given line OA and using points O and A as centers, strike two arcs with the radius equal to the given line. - The two arcs intersect at point B. - Connect points O and B with a straight line producing angle AOB which is 60°. - **Drawing a 30° Angle** - Draw a 60° angle. - Using points A and B as centers, strike arcs having the same radius and mark their intersection at point C. - Connect points O and C producing angles AOC and BOC each with a measure of 30° which is one half measure of angle AOB. - **Drawing a 15° Angle** - Draw a 30° angle - Bisect the given angle following step 2 and step 3 of number VIII producing angles AOC and COB each with a measure of 15° which is one half measure of angle AOB. - **Drawing a Line Parallel to a given Line** - Using points A and B as centers of the given line AB, strike at least two arcs of the same radius. - Draw a straight line on the top of the arcs. - This line is parallel to line AB. - **Striking an Arc Tangent to two Lines at an Angle** - Given two lines OA and OB forming an angle to each other, draw lines parallel to these lines as shown (see Fig. 3.1). - Draw straight line parallel to line OB and marked it line ED, and draw straight line parallel to line OA and marked it line EC. - Lines ED and EC intersect at point E which is the center of the required arc. - **Locating the Center of a Circle** - Given is a circle. - Draw two chords AB and AC. - Erect perpendicular bisectors to the chords which intersect at point D, hence, the center of the circle. - **Drawing a Tangent Arc in a Right Angle** - Given a right angle, with radius R, strike arc intersecting the lines at tangent points T. - Using point T as radius, strike arcs intersecting at C - With C as center and given radius R, draw required tangent arc. - **Drawing a Line Tangent to a Circle on a given point of contact** - Given a circle with O as its center and point D as the point of contact, draw radius OD and extend it to point A. - Piercing the compass' needle leg at point D, strike an arc from point O intersecting line OA at point A. - From points O and A, strike arcs of equal radius intersecting each other at points B and C. - Connect points B and C passing through the given point of contact D (also called the point of tangency), hence, line BC is tangent to the given circle at point D. - **Drawing an Arc Tangent to Two Arcs** - Given are arcs with centers A and B and radius R, using A and B as centers, draw arcs parallel to the given arcs and a distance R from them. - Draw lines of centers AB and BC to locate points of tangency T, and draw required tangent arc between the points of tangency, as shown. - **Drawing Tangents to Two Circles** - For Figure 3.7 (A) and (B), move the T-square and triangle as one unit until one side of the triangle is tangent by inspection to any of the given circles; then slide the triangle until the other side passes through the center of the said circle and lightly mark the point of tangency. - Do similar activity to the other circle and draw the required lines between the points of tangency. - **Drawing an Equilateral Triangle** - **Inscribing an Equilateral Triangle in a given circle using the compass method.** - Given a circle, draw diameter AB. - With B as the center and the distance BO as radius, strike arcs intersecting the circle at points C and D. - Connect points A to C, C to D and D to A to produce equilateral triangle which is also called equiangular triangle. - **Drawing an Equilateral Triangle using the compass method.** - With A and B as centers and AB as radius, strike arcs to intersect at point C. - Draw lines AC and BC to complete the triangle. - **Drawing an Equilateral Triangle using the triangle method.** - Given line AB. -