Math 310 ! Self-test " Geo me try Basics This is NOT a comprehensive study guide. The follow ing may be used to TEST YOURSELF after review of the assignedexercises and class work. Memorization of t hese questions and answers is NOT advised.1. Find the word or phrase or number w hich best completes each statement.a. The intersection of i. tw o parallel planes is .... ii. tw o perpendicular planes is ....iii. tw o planes which form a dihedral angle is .... iv. three distinct planes is ....b. The intersection ofi. two parallel lines is .... ii. a set of concurrent lines is .... iii. two non-parallel lines is ....c. The number of planes passing through (i.e. containing)i. a point is .... ii. a line is ....iii. tw o distinct points is .... iv. three non-collinear points is ....v. a line and a separate point is .... vi. a pair of intersecting lines is ....d. The number of circles that can be drawn t hrough (passing through)i. tw o points is .... ii. three noncollinear points is ....e. An equilateral five-pointed "star" (fig. A) can be folded up into a convex polyhedron called a .... fig. Af. A closed simple plane curve consisting of line segments is a ....g. A hexagonal hatbox most closely resembles the geometric surface ... Ch. A quadrilateral with all sides equal is called a .... C Ci. The number of segments connecting eight points, no three of which are collinear, is .... C CThe number of diagonals in a convex octagon is .... C Cj. Coplanar lines which do not intersect are .... Ck. Lines which cannot be contained in a plane are ....l. The measure of each of the interior angles of a regular pentagon is ....m. The union of four coplanar segments, joined so each endpoint is shared with exactly one other segment is a ....n. The number of planes containing i. t w o parallel lines ii. t w o skew lines iii. f our points iv. three concurrent lineso. Given points A and B, how many rays pass through the tw o, w ith one as an endpoint?p. The set of all point s in a pl ane a dist ance " r: f rom a giv en poi nt " P" is a .... [replace “ plane” w ith “ space” ...]2. Stat e the Jordan Curve Theorem as clearly as you can. Given a point and a simple closed curv e, can you easilydetermine whether the point is on the interior or exterior of t he curve? How ?3. Name the five regular convex polyhedra (Platonic solids) & describe each o ne briefly (number of f aces, type of faces,w hat f orms a vertex of each). Explain w hy t here are no regular c onvex poly hedra w ith oct agonal faces.4. Find the indicated sums of the measures of angles: a. The sum of t he tw o acute angles in a right t riangle.b. The su m of t he int erior angle and ext erior angle at any v ertex of an equilateral t riangle.c. A straight angle and two angles w hich are complementary to each other.5. Sketc h each of the f ollow ing; f ind t he number of faces, vert ices and edges. Show Euler's f ormula holds.a. prism w ith a pentagonal base b. py ramid w ith hexagon al base c. octahedron6. Given AB||CD and EF a transversal of th e tw o: Sel ect one ang le, mark it """ ; then find all angles congruent t o theselect ed angle, and mark them " "" also. How many did you f ind?7. The figure at right could be folded up into a polyhedron in the shape of a ....8. For a 70-sided convex polygon find: a. sum of t he int erior angles b. sum of t he exterior angles c. number of diagonals.9. Tru e-Fa lse (where false, how do you know ?):a. A cone is a type of polyhedron. b. A rectangle is a square. c. Every rect angle is a parallelogram. d. An equilateral triangle is a polygon.e. A py ramid is a type of prism. f. Every r ight tr iangl e is acute.g. A t riang le is an ex amp le of a simp le closed cu rve. h. A figure-8 is an examp le of a simp le closed cu rve.i. If t w o angles of a tr iangl e are com plem ent ary, th en t he triangle is a rig ht triangle.j. There is only one quadrilateral composed of segments of length 2cm, 4cm, 2 cm, 4 cm in that order.10. a. Is it possible for a line & a plane to intersect in: i. just 1 poi nt ? ii. exact ly 2 point s? iii. exactly 3 point s? iv. infinit ely man y p oin ts? v. no point s?Same questions for: b. t w o dist inct l ines c. tw o dist inct planes d. t w o dist inct t riang les11. Given the curve shown* at right is a simple closed curve in the planeof th e paper, are poi nt s A and B on t he same side of the curve?How do you know ? (The censor's strip is not part of the curve.)*(Well, partially shown. The censor thought t he curve too curvy and decided to cov er part of it. )13. Why is the triangle so pervasive in geometry?CENS ORED12. Can the line segments form th e sides of a trian gle? . .. a dif ferent tr iang le? (How man y d if feren t tr iang les c an be fo rm ed us ing th ose 3 leng ths?) Answers (we hope): 1. a-i. N a-ii. a line a-iii. a line a-iv. unpredictable: may be empty, a line, or a point. (Can you d escribe scenarios?)b-i. N b-ii. a point b-iii. unpr edictab le: m ay be N if lines are skew; a point, if coplanar; since the lines are non-parallel,they cannot be one line, so we can eliminate the possibility of intersection being the whole line. c-i. in fi nit e c-ii. ditto c-iii. same! (essentially same as a line) c-iv. only one c-v. only one c-vi. only one.d-i. inf inite (imagine hanging dif ferent size hoops on tw o nails) d-ii. only one. (picture a third nail on th e w all...)e. rhombus f. polygon g. (hexagonal) prism h. (pentagonal) pyramidi. 7+ 6+ 5+ 4+ 3+ 2+ 1 = 8C7/2 = 28 ; 20 j. parallel k. skew l. 3(180°)/5 = 108° m. quadrilateral n-i. 1 n-ii. 0 n-iii. unpredictable (if the four points are collinear, infinitely many; if three are noncollinear, and the 'fourth' n-iv. point happens to lie in the plane of the first three, one, and if fourth point happens to lie outside the plane, none! o. t w o: AB an d BA p. circle. [if w e replace th e w ord “ plane” w ith “ space” , answ er is sphere..] 2. A simple cl osed c urv e in …