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Name________________________ Basic Geometry Exam Date _________ Time _____ Final Exam Review Chapter 5 5-1 Bisectors, Medians, & Altitudes Fill in the blank and match the name of each special segment to the correct diagram. 1) perpendicular bisector – segment that is ____________________ to the opposite side of a triangle and also ____________ the opposite side. a) 2) altitude – segment that is ________________ to the opposite side of a triangle. b) 3) angle bisector – segment that _____________ an angle of a triangle. c) 4) median – segment that ______________ the opposite side of a triangle. d) Use the given diagram to solve for the variable and missing measures. 5) If MS is the altitude of D MNQ, and m and m 2 = 7x + 9, find the following: 1 = 3x + 11, x = _____ m 2 = ______ 1 6) If MS is the median of D MNQ, and QS = 3x – 14, SN = 2x + 1, and m MSQ = 7x + 1, find the following: x = _____ SN = _____ QN = _____ Is MS also an altitude of D MSQ? Explain. 7) If WP is an angle bisector of D WAH and m HWP = x + 12, m PAW = 3x – 2, and m HWA = 4x – 16, find the following: x = ______ m HWP = ______ m PAW = ______ 8) If WP is the perpendicular bisector of D WAH, and m WHA = 8x + 17, m HWP = 10 + x, AP = 6y + 4, and PH = 22 + 3y, find the following: x = ______ m HWP = ______ y = ______ AH = ______ 2 5-2 Inequalities & Triangles 9) Name the longest side of each triangle. a) ____________ b)_____________ A c)______________ X 78° M 104° C 42° Z Y N P B 10) Name the largest angle in each triangle. a) ____________ b) ____________ A J 15 c) _____________ 12 10 P K 5 B 18 C Q 13 12 14 L R 11) Find the value of x. List the sides of D PQR in order from shortest to largest for the given angle measures. (Hint: Draw a diagram) m P = 9x + 29, m Q = 93 – 5x, m R = 10x + 2 ________, _______, _______ 3 5-4 The Triangle Inequality 12) Can a triangle be formed from the given lengths? a) 12 in, 16 in, 20 in ____________ b) 2.4 ft, 5.7 ft, 8 ft ____________ c) 28¼ m, 43⅜ m, 71⅝ m ___________ 13) The measures of 2 sides of a triangle are given. The measure of the third side must be between what 2 measures? a) 16 ft, 17 ft ____________ b) 8¾ in, 8¾ in ____________ c) 9.2 cm, 17.8 cm ____________ 14) Determine whether the given coordinates are the vertices of a triangle. Explain. A(5, 8), B(2, -4), C(-3, -1) AB = _____ BC = _____ AC = _____ Using the Pythagorean Theorem 15) Tell if the triangle with the given lengths is acute, obtuse, or right. a) 6, 7, 10 ____________ b) 11, 11, 5 ____________ c) 1, 2, 5 ___________ 4 Chapter 6 6-1 Ratio & Proportions 1) Solve each proportion. a) 13 26 = 49 7 x b) 2y - 12 -4 = 28 7 c) k +1 5 = k -1 6 2) Solve each problem using a proportion or writing an equation. a) A 36 inch ribbon is cut into pieces so that the shorter piece and the longer piece have a 2:6 ratio. Find the length of the longer piece of ribbon. b) The lengths of a triangle are in a ratio of 2:3:5. If the perimeter of the triangle is 120 centimeters. Find the measure of each side of the triangle. c) The measures of the angles of a triangle have a ratio of 2:3:4. Find the measure of the largest angle. c) The ratio of the length of a rectangle to its width is 4:7. If the perimeter is 176, find the dimensions of the rectangle. 5 6-2 Similar Polygons 3) State the 2 conditions for 2 polygons to be similar. _______________________________________________________ _______________________________________________________ 4) Determine if the given figures are similar. State why or why not. If so, state the scale factor and state the similarity statement. a) b) 5) Rectangle ABCD is similar to rectangle ATSU. a) What is the scale factor of ABCD to ATSU? _______ T b) Find the ratio of the perimeter of rectangle ATSU to that of ABCD. 5.4 S C B 3 c) Find the ratio of the area of rectangle ATSU to that of ABCD. U A D 8 d) How is the scale factor related to the ratio of the perimeter of a polygon? e) How is the scale factor related to the ratio of the area of polygons.? 6 6) The two given polygons are similar. Find x and y. a) x = _____ y = _____ b) x = _____ c) x = _____ y = ___ __ d) x = _____ y = _____ y = _____ 7) Fill in the blank with sometimes, always, or never. a) Two squares are ______________________ similar. b) Two rectangles are _____________________ similar. c) A scalene triangle and an isosceles triangle are __________________ similar. d) Two regular polygons are ____________________ similar. 7 6-3 Similar Triangles 8) State the 3 triangle similarity theorems. __________, _________, __________ 9) State if each pair of triangles is similar. State the triangle similarity theorem and the triangle similarity statement. a) ________ b) ________ c) ________ 6-4 Proportions & Similarity in Triangles 10) Find the indicated variables. a) _________ b) __________ c) __________ 9 7 3 y k 8 Chapter 7 7-2 The Pythagorean Theorem & its Converse 1) Can the given numbers be the lengths of the sides of a right triangle? Why or why not? a) 7, 13, 18 b) 10, 24, 26 c) 12 , 13 , 5 2) Which of the following figures are rectangles? Why or why not? a) _______ 17 b) _______ 8 11 4 c) _______ 4 20 4 2 16 12 7-3 Special Right Triangles 3) Show how to represent the sides for the following special triangles. 45°, 45°, 90° triangle x 30°, 60°, 90° triangle 60 x 30 9 4) Find the value of the variables. b) x = _____ y = _____ a) x = _____ y = ______ 45 y 45 y 60 x 17 x 30 5 3 d) x = _____ y = _____ c) x = _____ y = _____ 30 x 60 26 y x 30 60 y 4 5 f) x = _____ e) x = _____ y = _____ y x y = _____ 60 x 11 30 y 9 g) x = _____ y = _____ 60 h) x = _____ y = _____ 30 9 y x y x 45 45 8 10 5) Draw a diagram for each problem, then answer the question. a) The perimeter of a square is 28 cm. What is the length of the diagonal? b) The diagonal of a rectangle divides the rectangle into two 30, 60, 90 triangles. Find the dimensions of the rectangle, if the length of the diagonal measures 18 inches? 7-4 Trigonometry 6) Find the following trig ratios and express each value as a decimal rounded to the nearest thousandth: B sin A = _____ = ______ sin B = _____ = ______ cos A = _____ = ______ cos B = _____ = ______ tan A = _____ = ______ tan B = _____ = ______ 5 A 1 2 C 7) Find the value of each angle. Round to the nearest degree. a) sin A = 7 11 m<A = _____ b) cos A = 12 23 m<A = _____ c) tan A = 1.483 m<A = _____ d) sin A = 0.995 m<A = _____ e) cos A = 0.743 m<A = _____ f) tanA = 0.194 m<A = _____ 11 8) Find the missing length or angle for each problem. (Round to tenths) a) x = _____ b) x = _____ 21 x 68° 9 x 32° d) x = _____ x° c) x = _____ x 54° 6 e) x = _____ 14 x f) x = _____ 75° 18 g) x = _____ h) x = _____ x° 4.6 6.8 i) x = _____ 29° 12.4 x 8 24 10.6 33.4 x° 83° x 12 7-5 Angles of Elevation & Depression 9) Write an equation to represent each problem and solve. a) The ground crew for a hot-air balloon can see the balloon in the sky at an angle of elevation of 11°. The pilot radios to the crew that the hot-air balloon in 950 feet above the ground. Estimate the horizontal distance of the hot-air balloon from the ground crew. b) Kierra is flying a kite. She has let out 55 feet of string. If the string makes a 35° angle with the ground, how high above the ground is the kite? Draw diagram here. c) A 12 foot ladder leans against a building and makes a 78º angle with the ground. How far from the building is the base of the ladder? Draw diagram here. d) At a certain time of the day, the angle of elevation of the sun is 44º. Find the length of the shadow cast by a building 30 meters high. Draw diagram here. 13 Chapter 8 8-1 Angles of Polygons 1) Find the sum of the interior angles of each convex polygon. a) heptagon _______ b) 37-gon _________ c) nonagon _______ 2) What is the measure of each interior angle for the given regular polygons? a) octagon _______ b) decagon ________ c) 90-gon _______ 3) The measure of each exterior angle of a regular polygon is given. Find the number of sides of the polygon. a) 72º ______ b) 45º _____ c) 14.4º _____ 4) What is the sum of the exterior angles of a decagon?_________ 5) What is the measure of each exterior angle in a regular 18-gon? _________ 6) How many sides does a regular polygon have if each interior angle measures 172.5º?______ 7) Find the value of x. a) x = ________ b) x = _______ c) x = ________ x° 88° 110° 87° x° 77° x° 93° x° 73° 102° 50° 14 d) x = ________ e) x = ________ x° 108° 2x° x° 123° f) x = _________ x° 55° 120° 135° g) x = ________ h) x = ________ i) x = _________ (regular octagon) x° x° 44º 115° 76° x° x° 8) True or False. a) Every quadrilateral is a parallelogram. ______ b) If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram. ______ c) The diagonals of a trapezoid are congruent. ______ d) Every square is a rhombus. ______ e) All rectangles are squares. ______ 15 Properties of Parallelograms and Other Quadrilaterals 9) Find each length or angle measure. a) ABCD is a rectangle A E b) PQRS is a parallelogram B P Q 28° 39° T 20 76° D 48 C S R AD = ______ m PSR = _______ AC = ______ m SRQ = _______ BE = ______ m PST = _______ m m PTQ= _______ ABC = ______ c) DEFG is a rhombus E D d) KITE is a kite I 12 S H 8 F K 6 G 16 13 FG = ______ E ET = _______ FD = ______ KI = _______ EH = ______ KS = _______ m DHG = _______ T m IST = _______ 16 10) Find the value of the variables for each parallelogram. a) x = _____ y = _____ b) x = _____ y = _____ c) u = _____ v = _____ d) b = _____ c = _____ d = ______ 11) Find the value of the given variable. a) x = ______ b) x = ______ x 17 Chapter 11 Area of Triangles, Quadrilaterals, Regular Polygons, & Circles 1) Find the area of each polygon. a) A = _________ b) A = __________ 5 17 c) A = __________ 6 8 25 11 d) A = ________ 23 25 e) A = __________ f) A = __________ 15.2 10 4.5 6 10 2) Find the area of each irregular figure. a) A = _______ b) A = ________ 6 5 6 8 5 21 5 8 6 8 18 14 18 3) Find the area of the shaded region. a) A = ________ b) A = ________ 5 8 5 7 5 5 16 Chapter 12 Surface Area & Volume 1) Find the base area, lateral area, surface area, and volume of each figure. a) B = _______ b) B = ________ c) B = ________ LA = _______ LA = ________ LA = ________ SA = _______ SA = ________ SA = ________ V = _______ V = ________ V = _______ 19 d) B = _______ e) B = ________ f) B = ________ LA = _______ LA = ________ LA = ________ SA = _______ SA = ________ SA = ________ V = _______ V = ________ V = _______ 20