(Current Re nweb Grade)x.90 + ( finalexam grade) x.10 = semester grade

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1 2//2 5:7 PM Name ate Period This is your semester exam which is worth 0% of your semester grade. You can determine grade what-ifs by using the equation below. (urrent Re nweb Grade)x.90 + ( finalexam grade) x.0 = semester grade The first semester exam will cover geometry content from chapters to 5. calculator will be allowed. Most of the major formulas will be provided. ========================================================================. hapter : Points, Lines, and Planes. Terms: Point, line, plane, segment, ray, angle, angles, vertex, slope of a line y 2 y x 2 x, space, collinear, noncollinear, coplanar, noncoplanar, distance x + x y + y (, ) a + b or (x 2 x ) 2 + ( y 2 y ) 2, midpoint 2 2, congruent segments (are =in length), Segment addition propery. sum of x ' s sum of y ' s (, ) or Practice Problems Refer to the figure to the right to answer problems - 0. X. The line intersecting plane P. 2. The intersection of and X. 3. re points,, and X collinear?. re points,, and X coplanar? P 5. re points,, and X contained in Plane P? 6. Identify 3 non-collinear points 7. Identify non-coplanar points. j, 8. Identify 2 angles that have as their vertex. 9. Name a line. 0. Name a ray. 3. Segment and midpoint definitions 2 segments having the same length are. The midpoint of a segment is a point half way between the endpoints of the segment. Thus, if X is the midpoint of, then X = X, also 2X =, or 2X =.

2 Segment bisector is any segment, line, ray, or plane that intersects a segment at its midpoint. The ddition Postulate says that if point is between the endpoints and of a segment, then + =.. Practice problems concerning segments and midpoints: 2. If is the midpoint of and = 2x 3 and = 5x 2, find x,, and. x =, =, = 3. If X = and X = 20, find. =. If is the midpoint of X and X = x + and = 5x, find x and X. x =, X = 5. If = 3x, = x + 2, and = 38, find x and. x =, = 6. If the x coordinate of G is 8 and the x coordinate of H is 9, find GH. GH = 7. ind the midpoint of the segment having the given endpoints: a. (-2, -), (3, 8) b. ( 3, -), ( -3, -) c. ( 2, ), (5, ) 8. ind the distance between the given endpoints: a. (-2, -), (3, 8) b. ( 3, -), ( -3, -) c. ( 2, ), (5, ) d. If the length of PQ is twice the length of, then find PQ. e. If the length of RS is one third the length of, then find RS. 9. How many sides does a pentagon have? 20. What does it mean for a polygon to regular? 2. ind the perimeter of a regular hexagon with a side equal to 5. ===========================================================================================. hapter 2 Topics: Logic & Reasoning. Terms: eductive reasoning, inductive reasoning, conjecture, conditional, hypothesis, conclusion, converse, contrapositive, counterexample 2. Practice Problems. Restate each of the following given statement into an if-then statement.. Underline the hypothesis and circle the conclusion.. Is the statement true or false? ircle your answer.. Write the converse of the conditional and determine whether it is true or false.. Write the inverse of the conditional and determine whether it is true or false.. Write the contrapositive of the conditional and determine whether it is true or false. G. If possible, write the bi-conditional statement in if and only if form. If not, write a counter example demonstrating why not. 2

3 . Tardy students receive detention.. &.. T or. T or. T or. T or G. 2. ll right angles are congruent.. &.. T or. T or. T or. T or G. 3. triangle is a polygon that has three sides.. &.. T or. T or. T or. T or G.. Supplementary angles are two angles whose sum is 80.. &.. T or. T or. T or. T or G. ============================================================================. hapter 3 Topics: ngle Relationships. Terms: The vertex angle & sides of an angle, naming an angle ( G means point is the vertex of the angle), right (= 90º), acute (< 90º ) and obtuse angles (> 90º ), congruent angles( ) adjacent angles, vertical angles, complementary angles(sum is 90º), supplementary angles (sum is 80º), linear pair(sum is 80º), perpendicular lines (Lines that form 90º angles), slope of parallel lines are equal, slope of 2 3 perpendicular lines are opposite reciprocals ( If m =, then m = ) Practice Problems Refer to igure 2. Matching, you may use more than one letter to describe the angle(s).. and 2 2. and and. and O 5. and 6 6. O and O 7. O and O 8. 2 and 5 9. and O a. acute angles b. right angles c. obtuse angles d. adjacent angles e. linear pair f. complementary angles g. supplementary angles h. vertical angles i. congruent angles G O 5 igure 2 3

4 Refer to figure 2 to solve problems If m 3 = 27, then m =. m + m O = m. 2. If m = 6 and m = 59, then m O =. 3. If O bisects O, then m =.. If O, then m + m 5 =. 5. If O and m = 65, then m =, m 2 =, m 6 =, m O =. 6. If O, name all the pairs of complementary angles. 7. If O is the bisector of, which segments are congruent? Refer to figure 3 to solve problems O 6 G 5 igure 2 8. Given: m 2 = 9x +28 and m 3 = 7 2x, x =, m 2 = 9. Given: m = 3x + 5 and m 3 = 65, x = 20. Given: m 2 = 9x +2 and m = 7x + 36, x =, m 2 = 2. Given: m = x-9 and m 2 = 2x, x =, m = 2 3 igure 3. hapter 3 Topics: parallel Lines & Their Relationships. Terms: Parallel (//) lines, transversal, corresponding angles ( ), alternate interior angles ( ), alternate exterior angles ( ), same side (or consecutive) interior angles (sum of 80), (supplementary angles still occur), parallel lines never intersect, parallel lines have the same slope 2. Practice Problems or problems -, refer to figure to determine which lines if any are parallel.. Given: 5 2. Given: Given: 7 3. Given: 5. Given: 6 6. Given: Given: 3 and 3 are supplementary Given a b, l m. (Refer to figure ) 8. If m 2 = 67 o, then m 3 = 9. If m 6 = 08 o, then m 6 = 0. If m = 23 o, then m 0 =. If m = 7 o, then m 0 = 2. m = 2x + 7 and m 6 = x + 30, x =, m =, m 6 = 3. m = 3x + 6 and m 3 = x + 26, x =, m =, m 3 =. m 2 = x - 6 and m 7 = 7x + 28, x =, m 2 =, m 7 = ind the slope of the line through the given points. 5. a. (-3,8), (,2) b. What is the slope of any line parallel to the line through points and? c. What is the slope of any line perpendicular to the line through points and? a b 2 igure l 9 0 m

5 6. a. (,-3), (9,-9) b. What is the slope of any line parallel to the line through points and? c. What is the slope of any line perpendicular to the line through points and? 7. a. (-2,-3), (-6,-5) b. What is the slope of any line parallel to the line through points and? c. What is the slope of any line perpendicular to the line through points and? =============================================================================. hapter Triangle Relationships. Term: {classified by angles} right ( right Δ ), acute (all acute Δ s), obtuse( obtuse Δ ), equiangular triangles (all 60 angles). {lassified by sides}, Scalene (no sides are =), isosceles (at least 2 sides are =), equilateral triangles (all sides are =).sum of the interior angles is 80, sum of the remote interior angles is = to the exterior angle of the triangle, 2. Practice Problems ind the value of x.. x = 2. x = 3. x = x x x In Δ, find x and m, then classify the type of triangle according to sides and angels.. m = 6x 2, m = 2x 7, and m = x + x =, m = lassification:_by sides: by angles: 5. m = 8x + 9, m = 3x, m = 9x + 5 x =, m = lassification:_by sides: by angles: Using the given information, classify each triangle according to its sides and angles. 6. Δ Z, < Z and m = Δ MNO, 7. Δ WV, W = V and m < 90.. Δ LJR, m M = 27 and m L = 35 and m O = 82. m R =0. 8. Δ PON, PO = 5, ON = 5, PN = Δ KMN, m M >90, MN = MK. 9. Δ LJI, m L = 5 and m I = Δ SYX, m S = 60 and m Y = 60. Use the distance formula to classify the triangle by the measure of its sides.. (, 0) (3, 3) (2, ) = = = lassification: 5

6 5. (, -6) (-2, 5) V(0, 7) = V = V = lassification:. hapter Topic: ongruent Triangles. Term: constructions of congruent triangles, 2 sides and the included angle are (SS), 2 angles and the included side are (S), three congruent sides are (SSS), 2 angles and the non-included side are (S). 2. Practice Problems Identify the congruent triangles and justify your answer. If congruency can not be proven write n p in both blanks.. Given:,, and Δ Δ by. M 2. Given: SM MT, MP MP, andmp bisects SMT ΔMPS Δ by. S P T 3. Given: OM MN, PR PQ, MO PR, and ON RQ ΔMNO Δ by.. Given: G JK, GH HK ΔHJK Δ by. G H O Q P M N R 5. Given: is the Midpoint of Δ Δ by J K 6. Given: bisects YXW, YZX is a right angle. ΔXYZ Δ by Y Z W or the following problems, Δ Δ. 7. Given: = 3y + 2, = 5y 8, find. X 8. Given: m = y 23, m = 2y 5, find the m. 9. Use the distance formula to determine whether the triangles with the given vertices are congruent. Given: PQR : P(,2), Q(3,6), R(6,5) KLM : K(-2,), L(-6,3), M(-5,6) PQ = KL = QR = LM = PR = KM = re they ongruent? Why? 6

7 Proofs:. Given : a // b and m // n Prove: 0 Statements Reasons a m n b Given : // ; Prove:. 2. Statements Reasons escribe the location of point. Point. Point. 2. Where does to triangle formed by points,, and lie? 3. Where does the line containing points and lie?. oes a line containing point have to intersect the plane? M 5. If ΔGHK is a right triangle, name another right triangle. 7. Name a pair of alternate interior angles. 8. Name a pair of corresponding angles. H G J I K L O M P 7

8 9. Use the graph to the right and use the Pythagorean Theorem to determine the length of the longest segment. Round to the nearest hundredth. e sure to indicate the segment. List the segments is order from least to greatest. 0. Use the graph to the right to answer the following questions. State the coordinates for an endpoint of the segment with point as one endpoint and point as a midpoint.. Given: is the midpoint of, = (2x 3) cm and = (5x 2)cm. ind the length of. 2. ind the value of x in the figure. 3x + 2x + 3. If m = 7 and m = 3x - 8 and m 2 = 5x + 26, find x and m 3.( points) 2 3. If m =, and m O = 87, what is m? 5. If m 3 = 8x 2 and m = x + 6, and m = 3x 9, 2 3 find m. 6. If 3, then O is a(n). O If G //, their slopes are. G 8. If point and point are equidistant from, what conclusion can be made about and? igure 2 9. What is the sum of, 2, 3,, 5, and 6? If the equation of line is y+ 3 = ( x+ 5), state a possible equation which would describe line Δ KNG is an isosceles triangle with K as the vertex angle, and KN = 5x 2, and GK = 2x +. a. raw a diagram and label the angles and the sides with their lengths in algebraic form. b. What is the length of KN? Of KG? c. or what range of values for GN will the lengths still form a triangle? d. Make a table of lengths possible for NG. (Use only integers) e. Using the range of values above, find value that will form an UT triangle. Justify using the Pythagorean theorem. f. Using the range of values above, find value that will form an OTUS triangle. Justify using the Pythagorean theorem. 8

9 Why are the following triangles are congruent? Justify your reasoning! e sure to use the phrase two sides and the included angle are congruent instead of SS! 2. Given : ; 22. Given: ;, is the midpoint of Prove : Prove: 23. etermine which postulate can be used to prove the triangles are congruent. If the triangles cannot be proven congruent write NON. e sure to write out the postulate (X: 2 sides and the included angle are congruent instead of SS) =============================================================================================== G. hapter 5 Triangle Relationships. Terms: altitude, centroid, circumcenter, concurrent lines, incenter, median, orthocenter, perpendicular bisector, point of concurrency, perpendicular bisector, angle bisector. 2. hapter 5 Theorems: 5. If a point on a bisector of a segment, then it is equidistant from the endpoints of the segment. 5.2 onverse of the bisector theorem is also true ircumcenter theorem: The bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle. 5. ngle isector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. 5.5 The converse of the angle bisector theorem is also true. 9

10 5.6 Incenter Theorem The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. The point of concurrency of the angle bisectors is called an incenter. 5.7 entroid Theorem The medians of a triangle intersect at a point called the centroid that is 2/3 of the distance from each vertex to the midpoint of the opposite side. The point of concurrency of the medians is called the centroid. n altitude of a triangle is a segment from a vertex to the line containing the opposite side and the line containing that side. The lines containing the altitudes of a triangle are concurrent, intersecting at a point called the orthocenter. oncurrent lines are the 3 bisectors, or the 3 angle bisectors, or the 3 medians, or the 3 altitudes drawn in a triangle. The altitude, median, bisector, or the angle bisector 2. Practice Problems: a. Make a drawing of each of the theorems above, without looking at your book! b. 0

11 omplete the following. Show all work.. Given: is the bisector of m = 50 m = ind: m = m 2 = m 3 = 2. Given: is the bisector of m = 0 m = 70 m = ind: m = m = m = 3. Given: is the bisector of m = 35 m = 80 m = ind: m = m 2 = m 3 = m = 3. Inequalities in One Triangle efinition of inequality: or any real numbers a and b, a > b, iff, there is a positive number c such that a = b + c. 5.8 xterior ngle Inequality Theorem The measure of an exterior angle of a triangle > the measure of either of its corresponding remote interior angles. ngle side relationships in triangles theorems: 5.9 If one side of a triangle is longer than another side, then the angle opposite the longer side has greater measure than the angle opposite the shorter side. 5.0 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.. In ΔQRT, the angles listed from largest to smallest are: Q a) Q, R, T b) R, Q, T 25 9 c) T, R, Q d) Q, T, R T 30 R

12 . The triangle Inequality 5. Triangle Inequality Theorem The sum of the lengths of any 2 sides of a triangle must be > than the length of the 3 rd side. Practice problems: Is it possible to form a triangle with the given lengths? If not, explain why not. ) 5, 6, 9 2) 3,, 8 ind the range for the measure of the 3 rd side of a triangle given the measure of the 2 sides: 3) 5 ft., 7 ft. ) 0.5 cm, cm 2