<p>Interpret Drawings a. How many planes appear in this figure? There are four planes: plane M, plane XYZ, plane ZTY, and plane XTY b. Name three points that are collinear. Points Y, Q, and Z are collinear. c. Name the intersection of plane XYZ and plane M. Plane XYZ intersects plane M in XZ .</p><p> s s d. At what point do PR and TZ intersect? Explain. s s s s PR and TZ do not intersect. TZ lies in plane M but only point P of TZ lies in M. Find Measurements by Subtracting</p><p>Find FG. Assume that the figure is not drawn to scale. in. FG is the measure of . Point G is between F and H. Find F FG G H FG by subtracting GH from FH. FG + GH = FH Betweenness of points 15 in. 1 FG + 10 = 15 Substitution 4 1 1 1 1 FG +10  10 = 15  10 Subtract 10 from each side. 4 4 4 4 3 FG = 4 in. Simplify. 4</p><p>Write and Solve Equations to Find Measurements</p><p>Find the value of x and MN if N is between M and P, MP = 60, MN = 6x - 7, and NP = 2x + 3. Draw a figure to represent this information. Then write and solve an equation relating the given measures. MP = MN + NP Betweenness of points 60 60 = 6x - 7 + 2x + 3 Substitution 6x - 7 2x + 3 60 = 8x – 4 Simplify. M N P 64 = 8x Add 4 to each side. 8 = x Divide each side by 8. Now find MN. MN = 6x – 7 Given = 6(8) – 7 x = 8 MN = 41 Simplify.</p><p>Find Distance on a Number Line Use the number line to find QR.</p><p>The coordinates of Q and R are -2 and 3. QR = |x2 - x1| Distance Formula</p><p>= |-2 - 3| x1 = -2and x2 = -3 = |-5| or 5 Simplify.</p><p>Find Distance on a Coordinate Plane</p><p>Find the distance between A(4, 2) and B(-6, 4). 2 2 AB = x -x +y - y Distance Formula ( 2 1 ) ( 2 1 ) 2 2 = (-6- 4) +(4 - 2) (x1,y1) = (4, 2),(x2,y2) = (-6, 4) 2 2 = (-10) + ( 2) Simplify. = 104 Simplify. The distance from A to B is 104 units. You can use a calculator to find that 104 is approximately 10.20.</p><p>Check Graph the ordered pairs and apply the Pythagorean Theorem.</p><p>(AB)2  (BC)2  (AC)2 (AB)2  22  102 (AB)2  4  100 AB  104  Find the Midpoint in a Coordinate Plane</p><p>Find the coordinates of M, the midpoint of RS , for R(-3, -4) and S(5, 7).</p><p>骣x +x y + y 琪1 2 , 1 2 M = 琪 Midpoint Formula 桫 2 2 骣-3+ 5 -4 + 7 = 琪 , (x1, y1) = (-3, -4), (x2, y2) = (5, 7) 桫 2 2 骣2 3 骣 1 = 琪 , or 琪1,1 Simplify. 桫2 2 桫 2</p><p>Find the Coordinates of an Endpoint</p><p>Find the coordinates of P if M(3, 2) is the mid point of KP and K has the coordinates (1,-5). Step 1 Let P be (x1, y1) and K be (x2, y2) in the Midpoint Formula. 骣1+x - 5 + y M琪 2, 2 = M (3,2) 琪 (x1, y1) = (1, -5) 桫 2 2 Step 2 Write two equations to find the coordinates of P. 1 + x -5+ y 2 = 3 Midpoint Formula 2 = 2 Midpoint Formula 2 2 1 + x2 = 6 Multiply each side by 2. -5 + y2 = 4 Multiply each side by 2. x2 = 5 Subtract 1 from each side. y2 = 9 Add 5 to each side.</p><p>The coordinates of P are (5, 9).</p><p>Use Algebra to Find Measures Find the measure of QR if Q is the midpoint of PR . 4a + 27 R Because Q is the midpoint, you know that PQ = QR. Use this equation to find a value for a. -6a – 3 Q</p><p>PQ = QR Definition of midpoint P -6a - 3 = 4a + 27 PQ = -6a – 3, QR = 4a + 27 -3 = 10a + 27 Add 6a to each side. -30 = 10a Subtract 27 from each side. -3 = a Divide each side by 10. Now substitute -3 for a in the expression for PQ. QR = 4a + 27 Original measure. = 4(-3) + 27 a = -3 = -12 + 27 or 15 Simplify.</p><p>The measure of QR is 15. You can check your answer by substitution -3 for a in the expression for PQ. It should also have a length of 15. Angles and Their Parts</p><p>Use the map of a high school shown. a. Name all angles that have K as a vertex. F E Student 8, 6, HKJ, and JKC 1 Parking Science D b. Name the sides of 6. 2 Lab 5 3 KJ and or KJ and H K C G Office KC KG 8 6 4 E Gymnasium B c. What is another name for AJK? 9 10 7 J A 9, J, and KJA</p><p> d. Name a point in the interior of ECG. Point F. Measure and Classify Angles D M In the figure, ABD  FHG. If mABD = 3x + 6 and mFHG = x + 26, find the measures of ABD and FHG. B 蠤ABD FHG Given G m ABD = m FHG Definition of congruent angles A 3x + 6 = x + 26 Substitution 3x = x + 20 Subtract 6 from each side. H 2x = 20 Subtract x from each side. x = 10 Divide each side by 2. F J Use the value of x to find the measure of one angle. m ABD = 3x + 6 Given = 3(10) + 6 x = 10 = 30 + 6 or 36 Simplify.</p><p>The measures of ABD and FHG are 36.</p><p>You can check you solution by substituting 10 for x in the expression for FHG.</p><p>Identify Angles Pairs B H</p><p>Name an angle pair that satisfies each condition. a. two acute adjacent angles</p><p>C</p><p>A D G F BCH, ACD, DCG, and FCG are acute angles. ACD and  DCG are acute adjacent angles, and  FCG and  DCG are acute adjacent angles. b. two obtuse vertical angles BCD and HCG are obtuse vertical angles.</p><p>Perpendicular Lines   ALGEBRA Find x and y so that DG and BE are perpendicular.   If DG  BE , then mDFB = 90 and mGFE = 90. To find x, use BFC and DFC. m BFC + mD F C = mDFB Sum of parts = whole 2x + 4x = 90 Substitution 6x = 90 Combine like terms. x = 15 Divide each side by 6.</p><p>To find y, use GFE. m GFE = 90 Given 5y + 20 = 90 Substitution 5y = 70 Subtract 20 from each side. y = 14 Divide each side by 5.</p><p>Interpret Figures</p><p>Determine whether each statement can be assumed from the figure. Explain. a. BFC and AFG are complementary. No; they are congruent, but we do not know anything about their exact measurements. b. DFA and AFG are a linear pair. Yes; they are adjacent angles whose noncommon sides are opposite rays. c. DFC and BFC are complementary. Yes; there is a right angle symbol showing the adjacent angles form a right angle</p>