Geometry Quiz #1 Answers & Comments

Math 310 € Spring 2006 Ë GEOquiz #1 ANSWERS & COM MENTS #1-8: Mark T for true statements, F for false. If the stat ement is false, give a count erexample, or explanat ion. 1. TF A square is a rectangle. True, a square has the required properties of a rectangle (quadrilat eral w it h all right angles). If you said f alse, you probably perc eive, incorrect ly, t hat one of t he properties of a rectangle is having t w o dif ferent length sides. 2. TF A rectangle is a parallelogram. True, a rectangle has the required properties of a parallelogram (quadrilateral w ith opposit e sides parallel). If you said f alse, you probably perc eive, incorrect ly, t hat one of the properties of a parallelogram is having t w o dif ferent size interior angles. 3. TF A parallelogram is a quadrilateral. True, a parallelogram has the required properties of a quadrilateral (polygon w it h f our sides). 4. TF A square is a rhombus. True, a square has the required properties of a rhombus (quadrilateral w ith four equal sides). It is NOT required t hat a rhombus have a “ diamond” shape, w it h one diagonal short er t han the ot her. In f act a square t urned 4 5° does indeed have a diamond-shaped appearance. .. 5. TF A quadrilat eral m ay have ex act ly one right angle* TF A quadrilat eral may have exact ly t hree right angles* These references are all t o int erior angles. Since the sum of all t he interior angles of any quadrilat eral is 360°, if three of those int erior angles measure 90 °, t he fourt h must also. The first st atement is t rue, example show n here: 6. TF A rhombus is a regular quadrilateral. No, a rhom bus may have tw o dif ferent sizes of interior angles ( º)... w hich violates the requirement t hat a regular polygon be equiangular. 7. TF The sum of t he measures of the int erior angles of any hexagon is 7 20°. True. Triangulation is probably the simplest w ay to demonstrate this fact. 8. TF The number of diagonals in a regular hexagon is 1 5. No, the number of w ays to connect 6 no-3-collinear point s w ith line segments is 1+ 2+ 3+ 4+ 5 = 5 C6/2 = 15, but siz of those connecting segments are sides of the hexagon, not diagonals. So the number of diagonals is 1 5 ! 6, w hich is 9. 9. Sketch a concave pentagon. 10. Sketch a concave quadrilateral. Can you sket ch a concave t riangle? 11. Sketc h a polygon w it h only tw o sides. This is impossible. If t he second segment com es back t o the same point w here the f irst segment began, then t he second segment is identical w ith t he first . (Tw o points determine ONE line segment.) So the second segment ret races all the point s covered by t he first. This is not even a curve, let alone a polygon. 12. Which of the f ollow ing is a curve? a b d f g h i & j. a. b. f. g. c. d. h. e. i. j. c. f ails because it is not connect ed (you have to lif t your pencil t o draw it ). e. f ails because the “ curv e” cannot be traversed (or draw n) w it hout retracing pat hs. (Those f our ODD vert ices give it aw ay as unt raversable.) 13. Which of the above figures is a simple closed curv e? (If it is not, w hy not?) Only b and d and h are simple closed curves. a. is not closed (does not end w here it began) and not sim ple. c. is not even a curve (see #1 2). e. is, lik ew ise, not even a curve. f. is not simple (it m eets it self, and not just by ending w here it began). g. is also not simple (crosses over it self). i & j are not sim ple. 14. Which of the f igures in #12, if any, is a polygon? Since a polygon is a simple closed plane curve consisting of straight segments, t he only ones to consider are b and d and h. Curve b does not consist entirely of straight segments, so it is not a polygon. The other tw o are polygons. 15. Identif y eac h of the f ollow ing triangles as A acute B obtuse C right D scalene E isosceles F equilateral a. A, E Acut e Isosceles b. A, D Acut e Scalene c. C, D Right Scalene d. B, E Obt use } Isosceles e. (all that apply!) f. A, D B, D Acut e Obtuse Scaleney Scalene g. B, D Obtuse Scalene } Use t he corner of your paper t o check t he angle! y Yes, it is close to being isosceles. But not ice how the t op “ leans right ” – the vert ex is not centered over the base; if isosceles, should be symmetric! (Problem should have been stat ed: Identif y t o w hich c lasses of t riangles each t he follow ing t riangles belongs. )