Math 310  Spring 2 009 Ë GEOquiz #2 > NAMEÚ #1-8: Mark T for true statements, F for false. If the statement is false, give a counterexample, or explanation.1. TF A square is a rhombus.2. TF A rhombus is a parallelogram.3. TF A parallelogram is a quadrilateral.4. TF A square is a rectangle.5. TF A quadrilat eral may have exactly one right int erior angle.TF A quadrilateral may have exactly three right interior angles.6. TF A rhombus is a regular quadrilateral.7. TF The sum of the measures of the interior angles of any hexagon is 720E.8. TF The number of diagonals in a regular hexagon is 15.9. If possible, sketch a concave pentagon.If possible, sketch a concave quadrilateral.10. Sketch a polygon w ith only tw o sides.11. Which of the f ollow ing is a curve?2 a. b. c. d. e. f. g. h. i. j. k. 1 6 5 3 413 Which of t he above f igures is a simple closed plane curve? (If it is not , w hy not ?)Note about f igure k: The f igure is comprised of only t he numbered edges of the cube. The cube itself is not part of the figure, but is show n to make the figure understood.14. Which of the figures in #11, if any, is a polygon?15. Ident if y each of t he f ollow ing triangles asA acute B obtuse C right D scalene E isosceles F equilateral (ALL that apply!)a. b. c. d. e. f. g.On t he reverse:16. Prove that vertical angles are congruent. 17. Prove that the sum of the measures of the interior angles of a triangle is 180E.Math 310  Spring 2 00 9 Ë GEOquiz #2 ANSWERS & COMMENTS #1-8: T indicates statement is true; F, false. False statements are accompanied by a reason or counterexample.1. T A square is a rhombus.True; a square has the required properties (of a rhombus– quadrilateral w ith 4 sides ).It is NOT required that a rhombus have a “ diamond” shape, w it h one diagonal short er t hanthe other. In f act a square t urned 45E does indeed have a “ diamond-shaped” appearance...2. T A rhombus is a parallelogram.True; a rhombus has the required properties (of a parallelogram– quadrilateral w ith opposite sides parallel).If you said false, you probably perceive, incorrectly, that one of the properties of aparallelogram is having tw o different size sides. Not so!3. T A parallelogram is a quadrilateral.True; a parallelogram has the required properties (of a quadrilateral– four-sided polygon).4. T A square is a rect angle.True, a square has the required properties of a rectangle (quadrilateral w ith all right angles).If you said false, you probably perceive, incorrectly, that one of the properties of a rectangleis having t w o diff erent length sides. A rectangle MAY have 2 different length sides (or NOT).5. T A quadrilateral may have exact ly one right interior angle. True, just draw one! ...like this:F A quadrilateral may have exactly three right angles.NEVER! Since the sum of all the interior angles of any quadrilateral is 360E, if three of t hose int erior angles measure 90E, t he f ourt h must also be 90E. 6. F A rhombus is a regular quadrilateral.No, a rhombus may have tw o different sizes of interior angles ( º)...w hich violates the requirement that a regular polygon be equiangular (as w ell as equilateral).A rhombus may be a square, but might not be, so w e cannot claim it is a regular polygon.7. T The sum of the measures of the interior angles of any hexagon is 720E.True. Triangulation is probably the simplest w ay to demonstrate this fact.8. F The number of diagonals in a regular hexagon is 15.No, the number of ways to connect 6 no-3-collinear points with line segments is1+ 2+ 3+ 4+ 5 = 5C6/2 = 15, but six of those connect ing segment s are sides of thehexagon, not diagonals. So the number of diagonals is 15 ! 6, w hich is 9.9. Sketch a concave pentagon.Sketch a concave quadrilateral. Can you sket ch a concave t riangle?10. Sketch a polygon w ith only tw o sides.This is impossible. A polygon is a CLOSED curve. If the second segment comes back to thesame point where the first segment began, then the second segment is identical with thefirst. (Tw o points determine ONE line segment.) So the second segment retraces all thepoints covered by t he first. This is not even a curve.11. Which of the following is a curve? a b d f g h i j & k.a. b. c. d. e. f . g. h. i. j. k. c. fails because it is not connected (you have to lift your pencil to draw it).e. fails because the “ curve” cannot be traversed (or draw n) w ithout retracing paths.(Those f our ODD vert ices give it aw ay as unt raversable.)Note about f: if you took this to be a cylinder, then your answ er w ould be that it is not a curve, becausea cylinder is a 2-dimensional SURFACE, not the 1-dimensional set of points that is required to be a curve.13 . Which of t he above f igures is a simple closed plane curve? (If it is not , w hy not ?)Only b and d and h are simple closed plane curves.a is not closed (does not end w here it began) and not simple.c & e are not even curves (this w as explained in #11).f is not simple (it meets itself, and not just by ending w here it began).g & I & j are also not simple .k is a simple closed curve, but it is not a plane curve.14. Which of the figures in #12, if any, is a polygon?Since a polygon is a simple closed plane curve consisting of straight segments, the onlyones to consider are the SCPC: b and d and h. Curve b does not consist entirely ofstraight segments, so it is not a polygon. The other tw o, d & h, are polygons.15. Ident if y each of t he f ollow ing triangles asA acute B obtuse C right D scalene E isosceles F equilateral (all that apply!)a. b. c. d. e. f. g. A, E A, D C, D C, E A, D B, E B, D Acute Acute Right Right } Acute Obtuse } ObtuseIsosceles Scalene Scalene Isosceles Scaleney Isosceles Scalene } Cheap t rick : Use t he corner of your paper to check the angle!y Yes, it is close to being isosceles. But notice how t he t op “ leans right ” – t he vertex isnot centered over t he base; if isosceles, should be symmetric– top centered, base ps !BTW: “ Acute” means ALL the angles of t he triangle are acut e. EV ERY triangle contains atleast TWO acute angles! 16. Suppose tw o lines, one through A & B, t he other through C & D, intersect at