Answers: 21. Three noncollinear points determine 3 lines and 1 plane 22. If two quadrilaterals are similar, then they are squares 23. PR = 2.6 24. Midpoint is (3,-1) 25. Angle BXD = 108 degrees 26. The complement is 18 degrees 27. A) 30 28. Corresponding angles 29. False; Change "congruent" to "supplementary" 30. Neither 31. Equation is y = (-2/5)x + 9/5 32. Obtuse scalene triangle ---------------------------------------------------------------- Work Shown Problem 21) Three noncollinear points determine 3*2 = 6 pairings but half of those pairings are repeats, so we have 6/2 = 3 unique groups forming 3 lines (think of a triangle and its sides) The three noncollinear points form a single plane. This is simply an axiom. ----------------- Problem 22) Original Conditional is in the form If P, then Q The converse is the flip of that. So we go to If Q, then P. So we have Original Conditional: "If two quadrilaterals are squares, then they are similar" Converse: "If two quadrilaterals are similar, then they are squares" ----------------- Problem 23) P is between Q and R. By the segment addition postulate, we know QP+PR = QR We're given PQ or QP to be 10.2 and we know that QR = 12.8, so this means, QP+PR = QR 10.2+PR = 12.8 10.2+PR-10.2 = 12.8-10.2 PR = 2.6 ----------------- Problem 24) Add up the x coordinates and divide by 2: (x1+x2)/2 = (8+(-2))/2 = 6/2 = 3 Add up the y coordinates and divide by 2: (y1+y2)/2 = (-6+4)/2 = -2/2 = -1 Therefore the midpoint is (3,-1) ----------------- Problem 25) Angle DXE = 36 (given) Angle CXD = angle DXE (definition of bisection) Angle CXD = 36 Angle CXE = (angle CXD)+(angle DXE) Angle CXE = 36+36 Angle CXE = 72 Angle BXE = 2*(angle CXE) ... since XC bisects angle BXE Angle BXE = 2*72 Angle BXE = 144 Angle BXD = (angle BXE) - (angle DXE) Angle BXD = 144 - 36 Angle BXD = 108 ----------------- Problem 26) From problem 25, we found that Angle CXE = 72. Since XC cuts angle BXE in half, and the other angle is BXC, this means Angle BXE = angle CXE = 72 degrees Now subtract that from 90 90 - (angle BXE) = 90 - 72 = 18 The complement is 18 degrees ----------------- Problem 27) A+B+C = 180 x+x+120 = 180 2x+120 = 180 2x+120-120 = 180-120 2x = 60 2x/2 = 60/2 x = 30 ----------------- Problem 28) Angle 5 and angle 7 are corresponding angles. They are located on the same side of the transversal line. They both correspond to the same side of their respective parallel line counterparts. Both are on the right side of the parallel line they are attached to. ----------------- Problem 29) The statement in its current form is False. One way to fix it is to change the first underlined term from "congruent" to "supplementary". Angle 3 and angle 2 are same side interior angles which add up to 180 degrees. ----------------- Problem 30) Slope of AB = (y2-y1)/(x2-x1) Slope of AB = (-2-6)/(-2-10) Slope of AB = -8/(-12) Slope of AB = 2/3 Slope of CD = (y2-y1)/(x2-x1) Slope of CD = (2-6)/(6-(-6)) Slope of CD = -4/12 Slope of CD = -1/3 Multiply the slopes: (Slope of AB)*(Slope of BC) = (2/3)*(-1/3) = -2/9 The result is NOT equal to -1, so the lines are NOT perpendicular The two slopes are NOT equal, so the lines are NOT parallel So the answer is "neither" ----------------- Problem 31) Anything parallel to 2x+5y = 12 is of the form 2x+5y = C where C is some fixed number Plug in the given point (x,y) = (2,1) to find C 2x+5y = C 2*2+5*1 = C 4+5 = C 9 = C C = 9 So we go from 2x+5y = C to 2x+5y = 9. Now solve for y 2x+5y = 9 2x+5y-2x = 9-2x 5y = -2x+9 5y/5 = (-2x+9)/5 y = (-2/5)x + 9/5 ----------------- Problem 32) A+B+C = 180 16+B+64 = 180 B+80 = 180 B+80-80 = 180-80 B = 100 The angle B is 100 degrees, which is larger than 90 degrees. We have an obtuse triangle because of this fact. All three angles (16, 64, 80) are different, so the side lengths are different. The three different side lengths means we have a scalene triangle.
The length of an arc depends on the radius of a circle and the central angle Θ. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:
