Geometry Problem Solving Drill 08: Congruent Triangles

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1 Geometry Problem Solving Drill 08: Congruent Triangles Question No. 1 of 10 Question 1. The following triangles are congruent. What is the value of x? Question #01 (A) (B) 10 (C) 31 (D) 18 You set 3x + 1 = 41. B. Correct! You set 3x + 1 = 31. You set 3x + 1 = 94. You set 3x + 1 = 55. First find the value of unknown angle in the top triangle: 180 ( ) = 31 So, 3x + 1 = 31 3x = 31 1 x = 10

2 Question No. 2 of 10 Question 2. What postulate can be used to prove that ABC and DEF are congruent given m B = 90, m E = 90, AB = 12, DE = 12, m A = 38, and m D = 38? Question #02 (A) SAS (B) ASA (C) HL (D) HA You thought the given information stated that two sides and the included angles from one triangle are equal to their corresponding elements from the other. B. Correct! You matched two pairs of corresponding congruent angles and a pair of sides common to these angles. Since one of the angles in each triangle equals 90, you thought that HL could be the postulate. Since one of the angles in each triangle equals 90, you thought that HA could be the postulate. We have m B = m E = 90 AB = DE = 12 m A = m D = 38 So, the triangles are congruent by ASA.

3 Question No. 3 of 10 Question 3. In the figure, A is the midpoint of segments BE and CD. To be congruent, which condition is needed in ADE and ABC? Question #03 (A) m B = m E (B) m D = m C (C) m B = m D (D) m DAE = m BAC D. Correct! This condition makes the two triangles congruent by the SAS postulate. Using the definition of midpoint AD = AC AB = AE So, we need only m DAE = m BAC to prove that the triangles are congruent by SAS.

4 Question No. 4 of 10 Question 4. In the figure below, AC = DE, BC = DN. In order to be congruent, what condition is needed for the given triangles? Question #04 (A) m A = m E (B) m B = m N (C) m C = m N (D) m D = m C D. Correct! This condition makes the triangles congruent by SAS. The condition m D = m C along with the given equal sides makes the triangles congruent by SAS.

5 Question No. 5 of 10 Question 5. The triangles below are congruent. Which statement is true? Question #05 (A) a b = 24 (B) a + b = 24 (C) a b = 0 (D) a = b A. Correct! You found the values of a and b by equating the measures of congruent angles. Instead of adding a and b, you subtracted their values. Set up the following equations and solve: 2a + 1 = 61 5b + 2 = 32 2a = 60 a = 30 5b = 30 b = 6

6 Question No. 6 of 10 Question 6. In equilateral ABC, the median AM divides side BC into two parts, each measuring (4x + 11). Each side of equilateral RST measures 126. If one side of ABC is equal to one side of RST, what is the value of x? Question #06 (A) 13 (B) (C) (D) A. Correct! You solved 2(4x + 11) = 126 to find the answer. You did not multiply the expression by 2 before solving. You divided the expression by 2 instead of multiplying. You switched the operation from addition to subtraction in the expression. The triangles are congruent. So, 2(4x + 11) = 126 8x + 22 = 126 8x = 104 x = 13

7 Question No. 7 of 10 Question 7. In the figure, BE = CD, AE = AD, and m E = m D. Which statement is true? Question #07 (A) m C = m BAD (B) m B = m CAE (C) m BAE = m DAC (D) m E = m C ABE is congruent to ACD, but these are not corresponding angles. ABE is congruent to ACD, but these are not corresponding angles. C. Correct! Corresponding angles in two congruent triangles are equal. ABE is congruent to ACD, but these are not corresponding angles. We are given two pairs of corresponding sides and the included angles equal, so we conclude that the triangles are congruent by SAS. Thus, m BAE = m DAC.

8 Question No. 8 of 10 Question 8. In rectangle ABCD, what is the value of m? Question #08 (A) m = -15 (B) m = 5 (C) m = 15 (D) m = -5 C. Correct! You solved 3m 4 = 2m + 11 to find the value of m. If we have m ABD = m CBD, then we can conclude that the triangles are congruent by SAS. Let m ABD = m CBD and solve for m. 3m 4 = 2m + 11 m = 15

9 Question No. 9 of 10 Question 9. The horizontal sides of the equilateral triangles below are equal. Which statement is true? Question #09 (A) 3x = 4a (B) 4x = 3a (C) 3x = 4a 16 (D) 3x = 4a + 16 A. Correct! You set the side length expressions equal to each other and solved. When solving 3x 8 = 4(a 2), you exchanged the coefficients of a and x. Having a pair of equal sides between two equilateral triangles, we conclude they are congruent. So, 3x 8 = 4(a 2) 3x 8 = 4a 8 3x = 4a

10 Question No. 10 of 10 Question 10. In equilateral ABC, m ACD = m BCD = m ACB, and AC = BD. Which statement is true? Question #10 (A) Only ACD and BCD are congruent. (B) Only ACD and ABC are congruent. (C) Only BCD and ABC are congruent. (D) ABC, BCD, and ACD are all congruent. You applied the given information to only two triangles. You applied the given information to only two triangles. You applied the given information to only two triangles. D. Correct! You applied the definition of an equilateral triangle and the given information to find the true statement. We are given m ACD = m BCD = m ACB AC = BD These data imply that ABC, ACD, and BCD are all isosceles and congruent.

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