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
Technical drawing wikipedia , lookup
History of trigonometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Integer triangle wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
Point What is it? An undefined term thought of as a location with no size or dimension. It is the most basic building block of geometry. What is it? How do you draw it? How do you write or name it? Draw point D. How do you draw it? A dot with a capital letter. How do you write or name it? A capital letter. (Ie. point D.) Draw point D. You need the dot and the capital letter near it. D Section 1.1 SJS Page 1 line What is it? An undefined term thought of as a straight, continuous arrangement of infinitely many points extending forever in two directions. A line has length, but no width or thickness, so it is one-dimensional. What is it? How do you draw it? How do you write or name it? Draw and name line NE. How do you draw it? A straight line with arrows at each end. How do you write or name it? Any two points on the line with a line symbol above it. or a lower case letter written near the arrow. Draw and name line NE. Check for arrows and the two points with labels near the point. N E HJJG NE Make sure a line (with arrows) is above the capital letters. Section 1.1 SJS Page 2 plane What is it? How do you draw it? How do you write or name it? Draw plane PLN. What is it? An undefined term thought of as a flat surface that extends infinitely along its edges. A plane has length and width but no thickness, so it is two-dimensional. How do you draw it? A 4-sided figure slanted to give it perspective. How do you write or name it? A capital script (or cursive) letter. Or with any three points in the plane that are not collinear. Draw plane PLN. Make sure the points are inside the parallelogram that you drew. (You could use a rectangle too.) L N P Section 1.1 SJS Page 3 Mixed up stuff Give all possible names for the line: Give all possible names for the line: C L t C A L t A HJJG HJJG HJJG HJG HJJG HJJG CL , LC , AL , LA , AC , CA or just t. Give all possible names for the plane. Give all possible names for the plane. R A Plane PAX, AXP, XPA, X XAP, PAX, APX or R . R A X P P Section 1.1 SJS Page 4 collinear Define it. Points on the same line. Draw collinear points D, O, and G. Make sure all three points are on a line (with arrows) or any part of a line (segment or ray). G Define it. O D Draw collinear points D, O, and G. Draw non-collinear points C, A, and T. Make sure any one of the three points are not in line with the other two points. Draw non-collinear points C, A, and T. C or T A T C A Section 1.1 SJS Page 5 coplanar Define it. Draw coplanar points M, A, and T. Then illustrate that points T, E, A, and M are non-coplanar. Draw coplanar lines n and m. Name the plane that contains them P. Define it. Points (or other figures) in the same plane. Draw coplanar points M, A, and T. Then illustrate that points T, E, A, and M are non-coplanar. E M A T Draw coplanar lines n and m. Name the plane that contains them P. P m n Section 1.1 SJS Page 6 ray Define it. A part of a line that starts at one point, called the endpoint, and extends infinitely through another point. Define it. Draw and name ray RA. Draw and name ray RA. JJJG RA Give all possible names for the ray: A (the only name) R C A B Give all possible names for the ray: JJJG JJJG BA or BC are the only 2 names. C endpoint must be first! A B Section 1.1 SJS Page 7 Line segment Define it. Use the word endpoint in your definition. Define it. Use the word endpoint in your definition. A part of a line that consists of two points called endpoints and all of the collinear points between them. Draw and give all possible names for segment ME. Draw and give all possible names for segment ME. Draw a segment AN congruent to segment ME. Show 3 different ways to write that the segments are equal in length, that they are congruent. M E ME , EM Draw a segment AN congruent to segment ME. Show 3 different ways to write that the segments are equal in length, that they are congruent. Make sure there are matching tic marks on the segments. E M A N mME = m AN , ME ≅ AN , ME = AN Section 1.1 SJS Page 8 Measure of line segments What type of units do you use to measure segments with? Inches, feet, millimeters, centimeters, etc ... What type of units do you use to measure segments with? Draw and label segment NA that measures 4.3 cm. (Check this with your ruler and label with 4.3 cm.) Draw and label segment NA that measures 4.3 cm. 4.3 cm N A Show 2 different ways to write that the segment you drew measures 4.3 cm. Show 2 different ways to write that the segment you drew measures 4.3 cm. NA = 4.3 cm , mNA = 4.3 cm Section 1.1 SJS Page 9 Midpoint [of a segment] Define it. Define it. A point on a segment that is the same distance from both endpoints. OR A point that divides the segment into two congruent segments. Draw and label A the midpoint of segment NP. Draw and label A the midpoint of segment NP. Make sure there are matching P tic marks on each side of A. A N Show 3 different ways to write that the midpoint gives you equal (or congruent) segments. Show 3 different ways to write that the midpoint gives you equal (or congruent) segments. mNA = m AP , NA ≅ AP , NA = AP Section 1.1 SJS Page 10 Bisect Define it. Means to cut into two equal parts Define it. P Draw “point A bisects segment PN”. Make sure there are matching A N Draw “point A bisects segment PN”. tic marks on each side of A. Draw “ray AR bisects angle BAN”. Then show 2 different ways to write the equal (or congruent) angles. Draw “ray AR bisects angle BAN”. Then show 2 different ways to write the equal (or congruent) angles. B Make sure there are matching arc marks and tic marks on each side of A. R A N m∠BAR = m∠RAN ∠BAR ≅ ∠RAN Draw “ray AR bisects segment NB”. R Draw “ray AR bisects segment NB”. N Section 1.1 SJS A Page 11 B Angle Define it (as well as side and vertex). Name all of the angles in the drawing along with their measures. Make sure you use the correct notation. Define it. An angle is a figure formed by two rays with a common endpoint. The two rays are the sides of the angle, and the common endpoint is the vertex. Name all of the angles in the drawing along with their measures. Make sure you use the correct notation. m∠LOM = 71° , m∠NOM = 69° , m∠LON = 140° Section 1.2 SJS Page 12 Angle bisector Define it. Name all of the angle bisectors in the drawing. Use the correct notation! Also explain why. Define it. A ray that contains the vertex of the angle and divides the angle into two congruent angles. Name all of the angle bisectors in the drawing. Use the correct notation! Also explain why. JJJG UR bisects ∠QUS because m∠QUR = m∠RUS JJJG US bisects ∠RUT because ∠TUS ≅ ∠RUS Note: there are various ways to write these statements correctly! Check to make sure your statements are correct. Section 1.2 SJS Page 13 Adjacent angles. Define it. Two angles that share a vertex, share a side, and they share no interior points (they don’t overlap). Define it. Name all of the pairs of adjacent angles in the drawing. Name all of the pairs of adjacent angles in the drawing. There are 4 pairs! D B R ∠DAB ∠BAR ∠DAR ∠BAN ∠BAR , is adjacent to ∠RAN , is adjacent to ∠RAN , is adjacent to ∠BAD D is adjacent to Note: If you use different names for the angles above, it is still correct. B R A N Check angles carefully! A N Do adjacent angles need to be congruent? Do adjacent angles need to be congruent? NO, but adjacent angles may be congruent, but don’t have to be. Section 1.2 SJS Page 14 Parallel lines Define it. Lines (or parts of lines) in the same plane that never intersect. Define it. D Draw line AB parallel to line CD. C Make sure you use matching arrow symbols A to indicate parallel. Draw line AB parallel to line CD. B How do you write it? HJJG HJJG AB & CD How do you write it? Show how to mark the parallelogram pictured to illustrate that the opposite sides are parallel. P L A R Show how to mark the parallelogram pictured to illustrate that the opposite sides are parallel. Make sure you use matching arrow symbols to indicate parallel. P L A R Section 1.3 SJS Page 15 Skew lines Define it. Lines that are not coplanar and never intersect. p Define it. Draw line p and line q are skew. q Draw line p and line q are skew. What is the difference between skew and parallel lines? Point out line segments around the room that are parallel and segments that are skew. What is the difference between skew and parallel lines? They both never intersect; however, parallel lines are coplanar where as skew lines are non-coplanar. Point out line segments around the room that are parallel and segments that are skew. Answers will vary. Section 1.3 SJS Page 16 Perpendicular lines Define it. Lines (or parts of lines) that intersect at a 90º angle. Define it. Draw segment BO is perpendicular to line OG. Make sure you indicate the 90º angle Draw segment BO is perpendicular to line OG. How do you write it? How do you write it? B with the little box in the corner. O HJJG BO ⊥ OG G Watch the notation on this!! (Segments don’t have arrows!) Point out line segments around the room that are perpendicular. Finish the statement with parallel, perpendicular and/or skew: If two lines are perpendicular to the same line, then they are _____ to each other. Section 1.3 SJS Point out line segments around the room that are perpendicular. Answers will vary. Finish the statement with parallel, perpendicular and/or skew: If two lines are perpendicular to the same line, then they are parallel or skew to each other. Page 17 Acute angle Define it. An angle that measures less than 90º (and more than 0º). Define it. Draw acute angle ANG. Use the correct notation. Draw acute angle ANG. A (Various: Make sure it is between 0 and 90 degrees.) 44 N G Write all possible names for your angle. Write all possible names for your angle. Measure your angle. Label your angle with the measure. How do you write the measure of your angle? ∠N , ∠ANG , ∠GNA Measure your angle. Label your angle with the measure. How do you write the measure of your angle? Various: Make sure you are reading the correct degree measure from the protractor! Make sure the measure is written inside the angle near the vertex, see above. Various ways to write the measure, one possible: m∠N = 44° Section 1.3 SJS Page 18 Obtuse angle Define it. An angle that measures more than 90º and less than 180º. Define it. Draw obtuse angle OBT. (Various: Make sure it is between 90 and 180 degrees.) Draw obtuse angle OBT. Write all possible names for your angle. Use the correct notation. T 136 B O Write all possible names for your angle. Use the correct notation. ∠B , ∠OBT , ∠TBO Measure your angle. Label your angle with the measure. How do you write the measure of your angle? Measure your angle. Label your angle with the measure. How do you write the measure of your angle? Various: Make sure you are reading the correct degree measure from the protractor! Make sure the measure is written inside the angle near the vertex, see above. Various ways to write the measure, one possible: m∠B = 136° Section 1.3 SJS Page 19 Right angle Define it. An angle that measures exactly 90º. Define it. Draw right angle RGT. (Various: Make sure it you indicate 90º with Draw right angle RGT. Write all possible names for your angle. the square in the corner.) R G T Write all possible names for your angle. ∠G , ∠RGT , ∠TGR The angle is right, so the sides are ___. (word) The angle is right, so the sides are [perpendicular]. (word) Show how to write the relationship between the sides of the angle. Show how to write the relationship between the sides. JJJG JJJG GR ⊥ GT (Both sides should be rays, not segments!) Section 1.3 SJS Page 20 Straight angle Define it. An angle that measures exactly 180º. Define it. Draw straight angle STR. S R T Draw straight angle STR. Write all possible names for your angle. Write all possible names for your angle. ∠T , ∠STR , ∠RTS How many straight angles are in the drawing? Name them. Is there a problem naming them? C O How many straight angles are in the drawing? Name them. Is there a problem naming them? B A C D You can name two: ∠COD, ∠BOA O B A D But actually there are 4 (one above and one below each line). The problem is without an arc mark in the drawing, you have no idea which straight angle you are talking about. Section 1.3 SJS Page 21 Complementary angles Define it. Any two angles that add to 90º. Define it. Draw non-congruent, adjacent complementary angles 1 and 2. Draw non-congruent, adjacent complementary angles 1 and 2. 1 2 (Various, must indicate 90º with the box. Make sure angle 1 and 2 do not look equal in measure. ) What would you need to know to be sure that the angles are complements? (Write a math sentence.) What would you need to know to be sure that the angles are complements? (Write a math sentence.) If three angles measure 20º, 30º and 40º are they complementary? Explain. Do two angles need to be adjacent to be complements? Explain. m∠1 + m∠2 = 90° If three angles measure 20º, 30º and 40º are they complementary? Explain. No, “complementary” only refers to two angles. Do two angles need to be adjacent to be complements? Explain. No, you could have a 10º angle on one wall and an 80º on another wall and they would still be considered complementary (2 angles that add to 90º). Section 1.3 SJS Page 22 Supplementary angles Define it. Any two angles that add to 180º. Define it. Draw non-congruent, non-adjacent supplementary angles 3 and 4. What would you need to know to be sure that the angles are supplements? (Write a math sentence.) Congruent, supplementary angles measure ____ each. Do two angles need to be adjacent to be supplements? Explain. If three angles measure 10º, 100º and 70º are they supplementary? Explain. Draw non-congruent, non-adjacent supplementary angles 3 and 4. (various) 3 4 What would you need to know to be sure that the angles are supplements? (Write a math sentence.) m∠3 + m∠4 = 180° Congruent, supplementary angles measure 90º each. Do two angles need to be adjacent to be supplements? Explain. No, you could have a 10º angle on one wall and an 170º on another wall and they would still be considered complementary (2 angles that add to 180º). If three angles measure 10º, 100º and 70º are they supplementary? Explain. No, complementary only refers to two angles. Section 1.3 SJS Page 23 Vertical angles Define it. Two non-adjacent angles formed by two intersecting lines. Draw intersecting lines AB and CD. Label the intersection O. (Make sure O is between A and B and also C and D. Name all pairs of vertical angles.) Define it. Draw intersecting lines AB and CD. Label the intersection O. (Make sure O is between A and B and also C and D.) Name all pairs of vertical angles. Name the pairs of vertical angles. 1 3 2 4 What terms can you use to describe the pair of angles ∠3 and ∠4 , above? C Vertical angles O ∠COA and ∠BOD . ∠COB and ∠AOD . Name the pairs of vertical angles: B A D 1 3 2 4 ∠1 and ∠4 , ∠3 and ∠2 What terms can you use to describe the pair of angles ∠3 and ∠4 above? They are adjacent and supplementary. Section 1.3 SJS Page 24 Linear pair of angles Define it. Two adjacent angles formed by a line and a ray. Define it. Draw intersecting lines AB and CD. Label the intersection O. (Make sure O is between A and B and also C and D.) Name all linear pairs of angles. Linear pairs of angles are always _____ (a word). Name all of the linear pairs of angles. 1 3 Draw intersecting lines AB and CD. Label the intersection O. (Make sure O is between A and B and also C and D.) Name all linear pairs of angles. ∠AOC ∠COB ∠BOD ∠DOA and ∠COB . C and ∠BOD . and ∠DOA . and ∠AOC . O A D Linear pairs of angles are always _____ (a word). Supplementary (You can’t say 90º or right. They may not be congruent!) 2 4 What terms can you use to describe the pair of angles ∠3 and ∠2 ? B Name all of the linear pairs of angles. 1 3 2 4 ∠1 and ∠2 , ∠2 and ∠4 , ∠4 and ∠3 , ∠3 and ∠1 What terms can you use to describe the pair of angles and ∠2 ? ∠3 Vertical or congruent both work here. Section 1.3 SJS Page 25 Reflex measure of an angle Define it. Draw angle LEX with measure 55 degrees. In the drawing find and label the reflex measure of the angle. Define it. The largest amount of rotation less than 360° between the two rays. Draw angle LEX with measure 55 degrees. In the drawing find and label the reflex measure of the angle. L 305 55 E X Various: But make sure they label the reflex measure as 305. Section 1.3 SJS Page 26 Polygon Define it (include side and vertex in your definition). Draw pentagon PENTA. Define it (include side and vertex). A closed figure in a plane, formed by connecting line segments endpoint to endpoint with each segment intersecting exactly 2 others. Each segment is a side. Each endpoint is a vertex. E Draw pentagon PENTA. P N A How do you name a polygon? Give another name for your pentagon above. How many diagonals does pentagon PENTA have? Name all of them. T How do you name a polygon? The name of it followed by a list of consecutive vertices. Give another name for your pentagon. (There are many!) pentagon NTAPE, pentagon NEPAT, etc… How many diagonals does this pentagon have? Name all of them. five diagonals: AE , ET , PN , PT , AN Section 1.4 SJS Page 27 Classifying Polygon Give the names of the different types of polygons (sides numbering 3 through 12). Give the names of the different types of polygons (sides numbering 3 through 12). How do you name a polygon if you have more than 12 sides? Sides 3 4 5 6 7 8 9 10 11 12 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon How do you name a polygon if you have more than 12 sides? Number of sides followed by “-gon”, 13-gon, 100-gon, etc… Section 1.4 SJS Page 28 Diagonal Define it. A line segment that connects two nonconsecutive vertices. Define it. How many diagonals does a triangle have? a quadrilateral? a pentagon? a hexagon? triangle: None, quadrilateral : 2, pentagon: 5, and hexagon: 9 How many diagonals does a triangle have? a quadrilateral? a pentagon? a hexagon? Name all of the diagonals in the quadrilateral QUDA. U Q D U A Name all of the diagonals in the quadrilateral QUDA. QD , UA Q D A Section 1.4 SJS Page 29 Define convex polygon vs. concave polygon. Convex [polygon] A polygon in which no diagonal is outside the polygon. Concave [polygon] A polygon in which at least one diagonal is outside the polygon. Define convex polygon vs. concave polygon. Draw a convex pentagon. Draw a concave pentagon. Show how each fits their definitions. E Draw a convex pentagon. Draw a concave pentagon. Show how each fits their definitions. In the second pentagon, diagonal PN is outside. P P A T convex Draw a convex triangle. Draw a concave triangle. Any problems? N E N A T concave Draw a convex triangle. Draw a concave triangle. Any problems? G Yup! In this case, there are no diagonals! R A triangle is always convex, never concave. T Section 1.4 SJS Page 30 Congruent Congruent polygons are polygons with the same [size] and same [shape]. (Use one word per blank.) Congruent polygons are polygons with the same ___ and same ___ . When naming congruent polygons, you must make sure the ___ vertices are written in the same order. Write the congruence shown below. Use the correct notation!! When naming congruent polygons, you must make sure the [corresponding] vertices are written in the same order. Write the congruence. Use the correct notation!! ΔABC ≅ ∠EFG or ΔCAB ≅ ΔGEF , etc... The corresponding vertices must match!! A → E, B → F and C → G. Also, make sure the congruence symbol is used. Section 1.4 SJS Page 31 Congruent The polygons are congruent. Copy the polygons and mark your diagram to show that C they are congruent. The polygons are congruent. Copy the polygons and mark your diagram to show that they are congruent. (Check that all of the corresponding parts are marked with the same number of tic marks.) E R T Z N E R P C A How many pairs of parts do you need to mark in order to show that the polygons are congruent? T Z N P A How many parts do you need to mark in order to show that the polygons are congruent? Eight, 4 pairs of sides and 4 pairs of angles. Section 1.4 SJS Page 32 Congruence If the quadrilaterals are congruent, show how to write the congruence statement. Ie. Quad RCTE ≅ Quad NZPA If the quadrilaterals are congruent, show how to write the congruence statement. Make sure the corresponding vertices match up. R → N, C → Z, T → P and E → A. E R C T Z N P A How do you write that the corresponding parts that are congruent? (Write the statements, remember there How do you write that the corresponding parts that are congruent? (Write the statements, remember there are eight.) ∠C ≅ ∠Z RE ≅ A N ∠R ≅ ∠N RC ≅ ZN ∠E ≅ ∠A CT ≅ ZP ∠T ≅ ∠P ET ≅ AP are eight.) Section 1.4 SJS Page 33 Equilateral [polygon] Define it. A polygon with all of its sides equal in length. Define it. If possible, draw an equilateral quadrilateral that is not equiangular. If possible, draw an equilateral pentagon that is not equiangular. (Make sure all figures are marked with tic marks correctly!) If possible, draw an equilateral quadrilateral that is not equiangular. If possible, draw an equilateral pentagon that is not equiangular. If possible, draw an equilateral hexagon that is not equiangular. If possible, draw an equilateral hexagon that is not equiangular. Section 1.4 SJS Page 34 Equiangular [polygon] Define it. A polygon with all of its angles equal in measure. (Make sure all figures are marked with tic marks correctly!) Define it. If possible, draw an equiangular quadrilateral that is not equilateral. If possible, draw an equiangular pentagon that is not equilateral. If possible, draw an equiangular hexagon that is not equilateral. If possible, draw an equiangular quadrilateral that is not equilateral. If possible, draw an equiangular pentagon that is not equilateral. If possible, draw an equiangular hexagon that is not equilateral. Section 1.4 SJS Page 35 Regular [polygon] Define it. A polygon that is both equilateral and equiangular. (Make sure all figures are marked with tic marks correctly!) Define it. If possible, draw a regular triangular. If possible, draw a regular triangular. If possible, draw a regular quadrilateral. If possible, draw a regular quadrilateral. If possible, draw a regular pentagon. If possible, draw a regular pentagon. If possible, draw a regular hexagon. If possible, draw a regular hexagon. Section 1.4 SJS Page 36 Perimeter [of a polygon] Define it. The sum of the lengths of the sides of a polygon. Define it. Find the perimeter of the pentagon. If the perimeter of a square is 40 inches, how much does each side measure? If one side of a regular hexagon measures 4.5 m, then its perimeter is? Find the perimeter of the pentagon. Perimeter = 6 + 5 + 5 + 4 + 8 = 28 cm If the perimeter of a square is 40 inches, how much does each side measure? Each side is equal in measure, so 40 ÷ 4 = 10 inches. If one side of a regular hexagon measures 4.5 m, then its perimeter is? Each side is equal in measure, so 4.5 * 6 = 27 m Section 1.4 SJS Page 37 Right triangle Define it. A triangle with one right angle. (Make sure all figures are marked with tic marks correctly!) Define it. C If possible, draw a right isosceles B ∆ABC with hypotenuse AC . If possible, draw a right isosceles ∆ABC with hypotenuse AC . If possible, draw a right scalene ∆ABC with right angle A. A If possible, draw a right scalene ∆ABC with right angle A. C A B If possible, draw an equilateral right ∆ABC with If possible, draw an equilateral right ∆ABC with legs AB and AC . legs AB and AC . Not possible! Equilateral is also equiangular. Can’t have three 90 degree angles. OR If you have a right triangle, the longest side is the hypotenuse so you can’t have it be equilateral. Section 1.5 SJS Page 38 Acute triangle Define it. A triangle with three acute angles. Define it. If possible, draw an acute scalene ∆ABC. C A B If possible, draw an acute scalene ∆ABC. If possible, draw an acute isosceles ∆ABC with vertex angle B. How many angles must be acute before you can classify a triangle as an acute triangle? C If possible, draw an acute isosceles ∆ABC with vertex angle B. A B How many angles must be acute before you can classify a triangle as an acute triangle? All three must be acute! Section 1.5 SJS Page 39 Obtuse triangle Define it. A triangle with only one obtuse angle. Define it. C If possible, draw an obtuse isosceles ∆ABC with legs AB and AC . A If possible, draw an obtuse isosceles ∆ABC with legs AB and AC . B If possible, draw an obtuse scalene ∆ABC with obtuse angle C. If possible, draw an obtuse scalene ∆ABC with obtuse angle C. How many obtuse angles can a triangle have? C B How many obtuse angles can a triangle have? Only one. Section 1.5 SJS Page 40 A Equilateral (or equiangular) triangle Define each. Define each. An equiangular triangle is a triangle with all angles congruent. An equilateral triangle is a triangle with all sides congruent. C If possible, draw an acute equilateral ∆ABC. If so, draw it! Is it possible to draw an equilateral triangle that is not equiangular? If so, draw it! Is it possible to draw an equilateral quadrilateral that is not equiangular? If so, draw it! If possible, draw an acute equilateral ∆ABC. B A Is it possible to draw an equilateral triangle that is not equiangular? No Is it possible to draw an equilateral quadrilateral that is not equiangular? Yes Section 1.5 SJS Page 41 Scalene triangle Define it. A triangle with no congruent sides. Define it. C If possible, draw a scalene acute ∆ABC. A If possible, draw a scalene acute ∆ABC. If possible, draw a scalene right ∆ABC, with right angle A. If possible, draw a scalene isosceles ∆ABC. B If possible, draw a scalene right ∆ABC, with right angle A. C A B If possible, draw a scalene isosceles ∆ABC. You can’t! Scalene says no sides equal and for isosceles, you must have at least two sides equal. Section 1.5 SJS Page 42 Isosceles triangle Define it. A triangle with at least two congruent sides. C Define it. If possible, draw an isosceles acute ∆ABC, with base AC. If possible, draw an isosceles acute ∆ABC, with base AC. If possible, draw a right isosceles ∆ABC, with right angle A. A B C A If possible, draw a right isosceles ∆ABC, with right angle A. B If possible, draw an isosceles equilateral ∆ABC. Isosceles says you must have at least two sides equal, and a triangle with 3 sides equal has at least two equal. (Note: an equilateral/equiangular triangle is a special case of an isosceles triangle.) C If possible, draw an isosceles equilateral ∆ABC. B A Section 1.5 SJS Page 43 Quadrilateral Define quadrilateral. A four-sided polygon. Define quadrilateral. A quadrilateral with one pair of opposite sides parallel is a _____. Draw it. A quadrilateral with one pair of opposite sides parallel is a [trapezoid]. A quadrilateral with both pairs of opposite sides parallel is a [parallelogram]. A quadrilateral with both pairs of opposite sides parallel is a _____. Draw it. A quadrilateral with exactly two distinct pairs of consecutive congruent sides is a _____. Draw it. A quadrilateral with exactly two distinct pairs of consecutive congruent sides is a [kite]. Section 1.6 SJS Page 44 Parallelogram Define it. A quadrilateral with both pairs of opposite sides parallel. Define it. An equiangular rhombus is a [square]. An equiangular rhombus is a _____. An equiangular parallelogram is a [rectangle]. An equiangular parallelogram is a _____. An equilateral rectangle is a _____. An equilateral rectangle is a [square]. An equilateral parallelogram is a [rhombus]. An equilateral parallelogram is a _____. Section 1.6 SJS Page 45 Quadrilateral Mix Up Answer with always, sometimes, or never. Fully explain. A parallelogram is [sometimes] a square. A parallelogram could be just a parallelogram, a rectangle, a rhombus or a square. A rectangle is [sometimes] a rhombus. If the rectangle has equal sides (a square), then it is a rhombus. A parallelogram is _____ a square. A rectangle is _____ a rhombus. A square is _____ a rhombus. A rectangle is _____ a parallelogram. A parallelogram that is not a rectangle is _____ a square. A square is _____ a rectangle. A square is [always] a rhombus. A square always has 4 equal sides, so it’s always a rhombus. A rectangle is [always] a parallelogram. In a rectangle, both pairs of opposite sides are parallel, so it’s always a parallelogram. A parallelogram that is not a rectangle is [never] a square. You have to have 4 congruent angles to have a square, so if it isn’t a rectangle, it can’t be a square. A square is [always] a rectangle. A square always has 4 congruent angles and it is a parallelogram, so it must be a type of rectangle. Section 1.6 SJS Page 46 Circle Define it (and center). Define it (and center). Circle is a set of points, a given distance from a given point, called the center. Draw and label circle A. Draw and label circle A. A Define congruent circles. Draw congruent circles X and Y. Define concentric circles. Draw two concentric circles with center Z. Define congruent circles. Draw congruent circles X and Y. Congruent circles are two or more circles with the same radius measure. (Specify the radii measure.) Section 1.7 Define concentric circles. Draw two concentric circles with center Z. Concentric circles are two or more circles with the same center point. SJS Page 47 Circles, segments & lines Define radius. A segment that goes from the center to any point on the circle. Define radius. Define chord. A segment connecting any two points on the circle. Define chord. Define diameter. Define tangent & point of tangency. Define diameter. A chord that goes through the center of a circle. The largest chord. Define tangent. A line that intersects a circle in only one point. The point of intersection is called the point of tangency. Section 1.7 SJS Page 48 Circles, segments & lines A Name all radii. AP , BP , RP Name all radii. Name all chords. R E C P B A R Name all chords. E C D CD , AB P B D Name all diameters. Name all diameters. AB Name all tangents & their points of tangency. Name all tangents & their points of tangency. HJJG HJJG BE with point of tangency, B. ER with point of tangency, R. Section 1.7 SJS Page 49 Circles & segment measures. Define radius (as a distance). The distance from the center to any point on the circle. Define radius (as a distance). Define diameter (in terms of length of radius). Give two formulas d = ? and r = ? If r = 25 inches, then d = ? Define diameter (as a length). The length of the diameter is two times the radius. d = 2r or ½ d = r. If r = 25 inches, then d = ? d = 2 * 25 = 50 inches If d = 13 cm, then r = ? r = 1/2 * 13 = 6.5 or 13 cm 2 If d = 13 cm, then r = ? Section 1.7 SJS Page 50 Arcs R ARB . Give another name for q Give another name ARB . for q Give another name q. for RBA R A C q BRA A C P B q. Give another name for RBA P B q RCA Name all semicircles. Name all semicircles. Name all minor arcs. Name all major arcs. Must use 3 letters! q ACB ARB , q Name all minor arcs. p , CA p , CR p p , BC AR , RB Must use 2 letters only! p Name all major arcs. Must use 3 letters! Make sure you are not counting the same q , so q is the same arc as RCA arc twice. In other words RBA only list one or the other, not both. q q , RBA q , CAB q q , BCR ARC , RBC Section 1.7 SJS Page 51 Arcs Define arc (include endpoints). Arc [of a circle] is formed by two points on a circle and a continuous part of the circle between them. The two points are called endpoints. Define arc (include endpoints). Define semicircle. An arc whose endpoints are the endpoints of the diameter. Define semicircle. Define minor arc. Define major arc. Define minor arc. An arc that is smaller than a semicircle. Define major arc. An arc that is larger than a semicircle. Section 1.7 SJS Page 52 Arcs & angles R A Define central angle. C P Define “the measure of an arc”. p= If m ∠ RPB = 43 ° , then m RB If m ∠ C PB = 152 °, then m p AC = Define central angle An angle whose vertex is the center of a circle. R A C P B B Define “the measure of an arc” Remember, the arc’s measure is equal to the measure of its central angle, so p = 43°. If m∠RPB = 43°, then mRB If m∠CPB = 152°, then m p AC = 28°. If m q ARC = 330°, then m∠APC = 30°. If m q ARC = 330 °, then m ∠ APC = Section 1.7 SJS Page 53 3-D Figures Sketch a cylinder. Sketch a cone. Sketch a sphere. Sketch a hemisphere. Sketch a cylinder. Sketch a cone. Sketch a sphere. Sketch a hemisphere. Section 1.8 SJS Page 54 3-D Figures Sketch a rectangular prism. Sketch a square pyramid. Sketch a triangular pyramid. Sketch a pentagonal prism. Sketch a rectangular prism. Sketch a square pyramid. Sketch a triangular pyramid. Sketch a pentagonal prism. Section 1.8 SJS Page 55