Answer Key. 1.1 The Three Dimensions. Chapter 1 Basics of Geometry. CK-12 Geometry Honors Concepts 1. Answers
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1 1.1 The Three Dimensions 1. Possible answer: You need only one number to describe the location of a point on a line. You need two numbers to describe the location of a point on a plane. 2. vary. Possible answer: a string. 3. vary. Possible answer: the surface of a desk. 4. vary. Possible answer: a person. 5. F, G, A, or D. 6. A or H. 7. A, B, C, D, E, or F. 8. AB, BC, AC, CD, DE, EF, AF, BE, DF 9. ABC, ABE, CBE, DEF, AFD 10. C, E, or F. 11. vary depending on answer to # D or B CK-12 Geometry Honors Concepts 1
2 1.2 Angles and Lines 1. A line segment has two endpoints, a ray has one endpoint, and a line has no endpoints. 2. vary. Angle should be vary. Angle should be between 90 and vary. Angle should be less than A straight angle is 180 so two angles that make a straight angle must have measures that add to 180. This is the definition of supplementary angles. 6. The measures of the angles are 40.4 and The measures of the angles are 76.4 and vary. Possible answer: ABC and FBG. 9. vary. Possible answer: CBF. 10. vary. Possible answer: ABC and ABF. 11. m ABF = m FBG = Perpendicular lines meet at right angles. Their slopes are opposite reciprocals. 14. K is obtuse. 15. Using a protractor, the angle is about 122. CK-12 Geometry Honors Concepts 2
3 1.3 Polygons vary. 4. octagon 5. decagon 6. quadrilateral 7. The hexagon and its interior angles have been divided into 4 triangles. The sum of the angles of each triangle is 180. Therefore, the sum of the interior angles of the hexagon is = The hexagon has been split into 6 triangles, but the angles in the triangles are not just interior angles. The 6 triangles are all of the interior angles from angles forming the central circle. Therefore, the sum of the interior angles is = sides 10. If the interior angle is 150 then the exterior angle is 30. A regular polygon with an exterior angle of 30 has 360 = 12 sides x = x = x 21.1 CK-12 Geometry Honors Concepts 3
4 1.4 Triangles 1. scalene, isosceles, equilateral 2. acute, right, obtuse, equiangular 3. No, because equiangular triangles have all angles 60. A right triangle must have one angle that is Isosceles and obtuse. 5. Scalene and obtuse. 6. x = m C = 41, m A = 94, m B = x = m F = 25.86, m D = , m E = Longest: ; EF Shortest: DE. 11. GH, FH, GF 12. K, L, J 13. The measure of one interior angle is 80 and the sum of the measures of the other interior angles is x = CK-12 Geometry Honors Concepts 4
5 1.5 Quadrilaterals 1. Always true. 2. Sometimes true. 3. Always true. 4. Always true. 5. Always true. 6. Never true. 7. Must be a rectangle (and therefore a parallelogram), could be a square. 8. Must be a quadrilateral. Could be a kite, parallelogram, rectangle, rhombus, or square depending on which sides are congruent and additional properties. 9. Possible conjectures: diagonals are perpendicular, one diagonal bisects the other, one diagonal bisects its angles. 10. Possible conjectures: diagonals are congruent, diagonals bisect each other. 11. Possible conjectures: diagonals are perpendicular, diagonals bisect each other. 12. Possible conjecture: one pair of opposite angles are congruent. 13. x = x = Adjacent angles are supplementary. This is due to the fact that the sum of the measures of the four interior angles is 360 and the opposite angles are congruent. CK-12 Geometry Honors Concepts 5
6 1.6 Area or Perimeter of Triangles and Quadrilaterals 1. Perimeter is 54 inches and area is 180 in Perimeter is 12 cm and area is 6 cm Area is 72 cm The length is 60 cm cm 6. 6 in, 8 in, 10 in. 7. Area is the number of square units it takes to cover a shape. Area is a two dimensional measurement. 8. Perimeter is a one dimensional measurement. It is the sum of the length of the sides. 9. b 1 and b 2 are the parallel sides, but it doesn't matter which is which in 2 (remember that a square is a rhombus) 11. A triangle is half a parallelogram with area bh. 12. A trapezoid is half a parallelogram with area (b 1 + b 2 )h 13. A = 90cm A square is a rectangle with base of "s" and height of "s". The Area = bh = s s = s Divide the kite into two triangles with base of d 1 and height of d 2. The area of each triangle is 2 d 1 ( d 2 2 ) = d 1d 2. The area of the kite is (d 1d 2 ) = d 1d CK-12 Geometry Honors Concepts 6
7 1.7 The Pythagorean Theorem 1. x x x x x = obtuse 7. acute 8. right 9. acute 10. d d d d vary. Common Pythagorean triples are 5, 12, 13 and 7, 24, (n 2 m 2 ) 2 + (2mn) 2 = n 4 2n 2 m 2 + m 4 + 4m 2 n 2 = n 4 + 2n 2 m 2 + m 4 = (n 2 + m 2 ) 2 CK-12 Geometry Honors Concepts 7
8 1.8 Circles 1. A = 9π in 2 ; C = 6πin 2. A = 36πin 2 ; C = 12πin 3. A = 56.25π cm 2 ; C = 15π cm 4. A = 121π in 2 ; C = 22π in 5. A = 256π cm 2 6. C = 8π 2 cm 7. A = 112.5π cm 2 8. P = π cm 9. A = 18.75π cm P = π cm 11. A = π in P = π in 13. A = 13.5πin P = 9 + 6π in 15. Cr = (2πr)r = 2πr2 = πr 2 = Area of a Circle CK-12 Geometry Honors Concepts 8
9 1.9 Composite Shapes P = π in 3. vary. 4. A = π in 2 5. vary P = 67 in 8. vary. CK-12 Geometry Honors Concepts 9
10 9. A = in vary P = 49.2 cm 13. vary. 14. A = 99.2 cm vary. CK-12 Geometry Honors Concepts 10
11 1.10 Volume of Solids π cm π in π cm in in π cm 3 7. vary. The area of the base tells you the volume of "one layer" of the prism. 8. A cylinder is like a prism with a circular base. 9. A pyramid is like a cone with a polygon base. 10. Both are the set of all points equidistant from a point. A circle is in two dimensions while a sphere is in three dimensions liters liters gallons inches in 2 CK-12 Geometry Honors Concepts 11
12 1.11 Surface Area and Nets 1. The surface area of a solid is the area of its net. 2. Surface area is a two dimensional measurement. It is the number of square units it would take to cover all the faces of a solid in in in in CK-12 Geometry Honors Concepts 12
13 in in in A triangular prism in CK-12 Geometry Honors Concepts 13
14 1.12 Cross Sections of Solids 1. circles 2. pentagons, rectangles 3. pentagons, triangles 4. Yes, as shown below. 5. The cross section will be the same shape as the base. 6. The cross section is a triangle. The shape of the base does not matter. 7. A cross section that is parallel to the base π in π in 2 CK-12 Geometry Honors Concepts 14
15 10. The cross section is a square. When the plane reaches the pyramid portion of the solid, the square gets smaller as the plane gets further from the base. 11. The cross section is a rectangle with a triangle on top in The cross section is a circle. When the plane reaches the cone portion of the solid, the circle gets smaller as the plane gets further from the base. 14. The cross section is a rectangle with a triangle on top in 2 CK-12 Geometry Honors Concepts 15
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