1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle?

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Arlene Harvey
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1 1 For all problems, NOTA stands for None of the Above. 1. If two angles of a triangle measure 40 and 80, what is the measure of the other angle of the triangle? (A) 40 (B) 60 (C) 80 (D) Cannot be determined 2. Lines l 1 and l 2 intersect. If the acute angle between the two lines measures 70, what is the measure of the obtuse angle between the two lines? l 1 70 l 2 (A) 110 (B) 140 (C) 160 (D) Lines l 1 and l 2 are coplanar and distinct, and both are perpendicular to line l 3. Which of the following must be true? (A) l 1 l 2 (B) l 1 l 2 (C) l 1 and l 2 intersect, but are not perpendicular (D) l 1 and l 2 are skew 4. Distinct points A, B, and C lie on a line in that order, so that AB : BC = 3 : 5. What is AB : AC? (A) 3 : 8 (B) 2 : 5 (C) 1 : 2 (D) 5 : 8 5. A store has an advertisement in its window stating If it s hot, it s here!. Assuming that this statement is true (and ignoring physical impossibilities), what can we correctly conclude? (A) All items in the store are hot. (B) Any item not in the store is not hot. (C) All items are hot. (D) No item is hot. 6. Which of the following is NOT a method used to prove two triangles are congruent? (A) AAS (B) ASA (C) SAS (D) SSS

2 2 7. Two angles of a triangle measure 35 and 110. The triangle can be described as (A) isosceles, acute (B) isosceles, obtuse (C) scalene, acute (D) scalene, obtuse 8. In triangle ABC, m A = 120. Suppose B measures x degrees. Which of the following describes all possible values of x? (A) 0 < x < 60 (B) 0 x < 60 (C) 0 < x 60 (D) 0 x Distinct points A, B, and C lie on a line in that order. Point D is not on the line. Which of the following conditions will NOT tell you that m DBA = 90? (A) DBA = DBC (B) DAB = DCB (C) DA = DC and BA = BC (D) DA = DC and DAB = DCB 10. In right triangle ABC, m B = 90, AB = , and BC = What is AC? (A) (B) (C) (D) A square has diagonal length d. What is the area of the square, in terms of d? (A) d2 4 (B) d2 2 (C) d2 2 2 (D) 2d The side lengths of a triangle are all even positive integers. The largest side has length 50. What is the smallest possible perimeter of the triangle? (A) 54 (B) 100 (C) 101 (D) If the side length of an equilateral triangle is 6, what is the length of one of its altitudes? (A) 3 (B) 3 2 (C) 3 3 (D) Let ABCD be a convex quadrilateral. Which of the following conditions will NOT tell you that ABCD is a parallelogram? (A) AB CD and AD BC (B) AB CD and AB = CD (C) AB CD and AD = BC (D) AB = CD and AD = BC 15. In convex quadrilateral ABCD, AC bisects BAD, and m ABC = m ACD. If AB = 4 and AC = 6, what is AD? (A) 4 (B) 2 6 (C) 6 (D) 9

3 3 16. Suppose we have a convex 20-gon. One of its diagonals is drawn, dividing it into two convex polygons, one of which has m sides, the other of which has n sides. What is m + n? (A) 20 (B) 21 (C) 22 (D) Cannot be determined 17. In trapezoid ABCD, AB CD, and AC and BD intersect at E. m AEB = 110, m BAC = 30, and m DBC = 60. What is the measure of BCD? (A) 70 (B) 80 (C) 90 (D) Turtle has a rectangular piece of paper with a clue on it. Before she eats it, she cuts off one of the corners as shown. What is the sum of the measures of the two marked angles in the second picture? (A) 180 (B) 210 (C) 240 (D) Two squares are in a plane. The sides of one square intersect the sides of the other square at a total of n points, where n is a finite number. What is the largest possible value of n? (A) 1 (B) 2 (C) 4 (D) ABCD is a convex quadrilateral with AD = BC. P is a point inside ABCD such that AP = BP = CP = DP. If m AP B = 20 and m CP D = 80, what is m BAD? (A) 95 (B) 100 (C) 105 (D) How many lines of symmetry does a regular hexagon have? (A) 3 (B) 6 (C) 12 (D) infinitely many 22. Let ABC be a triangle with area 12. Point D is inside triangle ABC. What is the sum of the areas of concave quadrilaterals ABCD, BCAD, and CABD? (A) 12 (B) 16 (C) 24 (D) For how many positive integer values of n, where n 3, does an exterior angle of a regular n-gon measure more than 10? (A) 15 (B) 17 (C) 33 (D) 35

4 4 Questions deal with the following theorem. Let ABC be a triangle, and let X, Y, and Z be points on sides BC, CA, and AB, respectively. A Z Y B X C Ceva s Theorem states that if AX, BY, and CZ intersect at one point, then AZ ZB BX XC CY Y A = 1. The converse of this statement is also true, and is often used to prove three lines intersect at one point. Three lines that intersect at one point are said to be concurrent. 24. In triangle ABC, points X, Y, and Z are on sides BC, CA, and AB, respectively, such that lines AX, BY, and CZ are concurrent. AZ = 1, BX = 3, XC = 2, CY = 4, and Y A = 3. What is ZB? (A) 1/2 (B) 8/9 (C) 9/8 (D) In triangle ABC, points X, Y, and Z are on sides BC, CA, and AB, respectively, such that AZ = ZB, BX = XC, and CY = Y A. Then AZ BX CY = = 1, so ZB XC Y A the converse of Ceva s theorem says that lines AX, BY, and CZ are concurrent. This proves that which of the following intersect at one point? (A) medians (B) angle bisectors (C) perpendicular bisectors (D) altitudes 26. In triangle ABC, points R and S are on sides AB and AC such that ARS ABC. Point P is on side BC such that lines AP, BS, and CR are concurrent. If AR = 3, RB = 2, and RS = 6, what is BP? (A) 2 (B) 3 (C) 4 (D) 5

5 5 27. In right triangle ABC with hypotenuse AC, AB = 7 and AC = 10. Which of the following angles is the smallest? (A) A (B) B (C) C (D) an angle measuring In the following diagram, ABCD is a square with area 16. What is the area of the shaded region? A 30 B 30 D C (A) (B) (C) 3 (D) 16/5 29. Let m and n be integers greater than or equal to 3, such that m > n. The interior angle measure of a regular m-gon is k/2 times the interior angle measure of a regular n-gon, where k is some positive integer. How many possible ordered pairs (m, n) are there? (A) 4 (B) 5 (C) 6 (D) In convex hexagon ABCDEF, m ABC = m CDE = m EF A = 100, AB = BC = 6, CD = DE = 3, and EF = F A = 3 3. What is the measure of F AB? (A) 110 (B) 130 (C) 140 (D) 170

2. In ABC, the measure of angle B is twice the measure of angle A. Angle C measures three times the measure of angle A. If AC = 26, find AB.

2. In ABC, the measure of angle B is twice the measure of angle A. Angle C measures three times the measure of angle A. If AC = 26, find AB. 2009 FGCU Mathematics Competition. Geometry Individual Test 1. You want to prove that the perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex. Which postulate/theorem