Geometry Problem Solving Drill 07: Properties of Triangles Question No. 1 of 10 Question 1. In ABC, m A = 44 and m C = 71. Find m B. Question #01 (A) 46 (B) 65 (C) 19 (D) 75 You thought the sum of A and B was 90. B. Correct! You used the Triangle Sum Theorem. You thought the sum of C and B was 90. Using the triangle sum theorem, we have m A + m B + m C = 180 44 + m B + 71 = 180 m B = 65
Question No. 2 of 10 Question 2. GHK has a median GS drawn from G to side HK, and R is the centroid. If GS = 36, what is the length of GR? Question #02 (A) 32 (B) 9 (C) 12 (D) 24 When simplifying (2/3)(36), you incorrectly found 36/3 = 16. You used GR = (1/4)(GS). You used GR = (1/3)(GS). D. Correct! You used GR = (2/3)(GS). Using the equivalences for medians and a centroid: GR = (2/3)(36) = 24
Question No. 3 of 10 Question 3. In the figure, ABC is an equilateral triangle and AC is a median in ABD. If AB = 24, what is the length of CD? Question #03 (A) 24 (B) 12 (C) 30 (D) 16 A. Correct! You used the definitions of equilateral triangle and median. You thought the length of CD was one-half the length of BC. You found the measure of a base angle in ACD as the length of CD. You divided the sum of AB and BC by 3. All the sides in an equilateral triangle are the same, therefore: AB = AC = BC = 24 On the other hand, BC = CD Therefore, CD = 24
Question No. 4 of 10 Question 4. In PQR, the median PT is drawn from P to the side QR, and N is the centroid. If the length of PT is 102, what is the length of NT? Question #04 (A) 32 (B) 36 (C) 34 (D) 52 Check you arithmetic and try again. C. Correct! You used the equivalences for medians and a centroid. You divided 102 by 2 instead of dividing by 3. Using the equivalences for medians and a centroid: NT = (1/3)(PT) = (1/3)(102) = 34
Question No. 5 of 10 Question 5. In the right triangle below, m 1 = 59. What is the value of m 1 m 2? Question #05 (A) 31 (B) 28 (C) 38 (D) 48 You found the value of m 2. B. Correct! You first solved for m 2 and then found the difference between the two angles. We are given m 1 = 59 m 1 + m 2 = 90 Solving these equation, we obtain m 2 = 31 Then m 1 m 2 = 59 31 = 28
Question No. 6 of 10 Question 6. In the following isosceles triangle m 1 = 110. What is m 2? Question #06 (A) 70 (B) 30 (C) 55 (D) 35 You found the value of the sum of the base angles. You found the value of half of the vertex angle. D. Correct! You used the Triangle Sum Theorem and the properties of isosceles triangles to find m 2. We have 110 + m 2 + m 3 = 180 m 2 = m 3 Therefore, 110 + 2(m 2) = 180 2(m 2) = 70 m 2 = 35
Question No. 7 of 10 Question 7. In the isosceles triangle below, m 4 = 124. What is the value of m 1 m 2? Question #07 (A) 12 (B) 14 (C) 6 (D) 68 A. Correct! You used the Exterior Angle Theorem along with the properties of isosceles triangles to solve. m 3 = 180 124 = 56 Since this is an isosceles triangle: m 3 = m 2 By the Exterior Angle Theorem: m 4 = m 1 + m 2 124 = m 1 + 56 68 = m 1 Therefore, m 1 m 2 = 68 56 = 12
Question No. 8 of 10 Question 8. In ABC, m A = 90 and m B m C = 20. What is m B? Question #08 (A) 45 (B) 55 (C) 35 (D) 65 B. Correct! You correctly solved the system of equations composed of the given equation along with m B + m C = 90. m B m C = 20 m B + m C = 90 Solve this system of equations to get m B = 55 and m C = 35
Question No. 9 of 10 Question 9. Given equilateral ABC and isosceles MNP in which M is 120, which statement is true? Question #09 (A) A = M (B) A = N (C) B = (1/2) N (D) B = 2( N) The measure of an interior angle of an equilateral triangle is not 120. Review the definition of an equilateral triangle and try again. The measure of an interior angle of an equilateral triangle is not 30. Review the definition of an equilateral triangle and try again. You divided N by 2 instead of multiplying by 2. D. Correct! You used the properties of isosceles and equilateral triangles to find the true statement. Since ABC is an equilateral triangle, then A = 60 B = 60 C = 60 We are given M = 120 and P = m Q Using the Triangle Sum Theorem, P = 30 Q = 30 So, B = 2( N) = 2(30) = 60
Question No. 10 of 10 Question 10. In ABC, segment CD is an angle bisector and segment AE is an altitude. If m ACB = 38, what is ANC? Question #10 (A) 109 (B) 119 (C) 136 (D) 92 A. Correct! You correctly applied properties of altitude, angle bisector, and the Triangle Sum Theorem. Check your arithmetic in the final step and try again. You forgot to divide m C by 2. While using the Triangle Sum Theorem, you divided m A by 2. In AEC: EAC = 90 38 = 52 On the other hand, the angle bisector CD gives ACN = 38/2 = 19 Applying the Triangle Sum Theorem to ANC, we conclude ANC = 180 (19 + 52) = 109