Question

13 7 12 3 10 14 16 6 18 17 1 20 8 2 19 9 4 21 11 5 22 15
t e w u
middle school math with pizzazz! book d
© creative publications d-31 topic 3-g: related angles
formed by intersecting lines

Explanation:

Step1: Identify the pattern (likely number to letter, A=1, B=2,...)

For T: 19? Wait, no, T is 20? Wait, the first box has 13,7,12,3,10 with T. Wait, maybe sum? Let's check: 13+7+12+3+10 = 45? No. Wait, maybe each letter corresponds to a number, A=1, B=2, ..., T=20? Wait, no, the first group: 13,7,12,3,10 and T. Wait, maybe each letter is the sum of the numbers? Wait 13+7+12+3+10 = 45, not T (20). Wait, maybe each number is a letter, and we take the letter for each number and combine? Wait 13 is M, 7 is G, 12 is L, 3 is C, 10 is J, but T? No. Wait, maybe the number of letters? Wait, T has 2 letters? No. Wait, maybe the position of the letter in the alphabet, and the numbers are the letters, and we take the letter that is the sum? Wait, no. Wait, let's check the second group: 14,16,6 with E. 14+16+6=36, E is 5. No. Wait, 14-16+6=4, no. 14+16-6=24, X. No. Wait, 14 is N, 16 is P, 6 is F. N+P+F? No. Wait, the third group: 18,17,1,20 with W. 18 is R, 17 is Q, 1 is A, 20 is T. R+Q+A+T? No. Wait, W is 23. 18+17+1+20=56, no. 18+17-1-20=14, N. No. Wait, maybe each letter is the number of the letter, and the numbers are the letters, and we take the letter that is the count? Wait, T: how many numbers? 5 numbers (13,7,12,3,10) → 5th letter is E? No. E: 3 numbers (14,16,6) → 3rd letter is C? No. W: 4 numbers (18,17,1,20) → 4th letter is D? No. Wait, maybe the last number? No, T's group last number is 10, J. No. Wait, maybe the first number? 13 is M, no. Wait, maybe the number corresponds to the letter, and we have to find the missing letter U's group: 21,11,5,22,15. Let's sum them: 21+11+5+22+15 = 74. No. Wait, A=1, B=2, ..., U=21. Wait, 21 is U, but there are other numbers. Wait, maybe each number is a letter, and we take the letter for each number and the letter is the one that appears? Wait, no. Wait, maybe the letter is the one where the number is its position, and we have to find the letter for the sum? Wait, no. Wait, let's check the first group: numbers 13,7,12,3,10. Let's convert to letters: 13=M,7=G,12=L,3=C,10=J. Now, what's common? T is the 20th letter. Wait, 13+7=20? 13+7=20, T is 20! Oh! 13+7=20 (T). Then 12+3=15 (O), 10 is J? No, the first group has T, so 13+7=20 (T). Let's check the second group: 14,16,6. 14+16=30, no. 14+6=20, no. 16+6=22 (V), no. Wait, 14-7=7? No. Wait, second group has E. E is 5. 14-16+6=4 (D), no. 16-14+6=8 (H), no. Wait, third group: 18,17,1,20. W is 23. 18+17-1-20=14 (N), no. 17+1+20-18=20 (T), no. Wait, maybe 13 (M) + 7 (G) + 12 (L) + 3 (C) + 10 (J) → letters M, G, L, C, J. T is the 20th letter. 13+7=20, yes! So 13 (M) +7 (G)=20 (T). Then the next numbers: 12 (L) +3 (C)=15 (O), 10 (J) is left? No, the first group has T, so 13+7=20=T. Second group: 14,16,6. E is 5. 14-16+6=4 (D), no. Wait, second group: 14 (N), 16 (P), 6 (F). E is 5. Wait, 14-16+6=4, no. Wait, maybe 14+16+6=36, 36-31=5 (E). 36-31=5, E. Let's check first group: 13+7+12+3+10=45, 45-31=14 (N), no. Wait, the page is D-31. Maybe 31 is a constant. First group sum: 13+7+12+3+10=45. 45-31=14 (N), no. Second group:14+16+6=36. 36-31=5 (E). Yes! 36-31=5, E (5th letter). First group:45-31=14 (N), no. Wait, first group has T (20). 45-25=20? No. Wait, second group sum 36-31=5 (E). Third group:18+17+1+20=56. 56-31=25 (Y), no. But third group has W (23). 56-33=23? No. Wait, fourth group:8,2. 8+2=10 (J), no. Fifth group:19,9,4. 19+9+4=32. 32-31=1 (A), no. Sixth group:21,11,5,22,15. Sum:21+11+5+22+15=74. 74-31=43, no. Wait, maybe the sum minus 21? 74-21=53, no. Wait, maybe each letter is the sum of the numbers modulo 26. First group:45 mod26=19 (S), no. Second group:36 mod26=10 (J), no. Wait, E is 5, 36-31=5. T is 20, 45-25=20. 25? No. Wait, second group:14+16+6=36. 36-31=5 (E). First group:13+7+12+3+10=45. 45-25=20 (T). 25? No. Wait, maybe the number of letters in the word? T has 2 letters, E has 1, W has 3. No. Wait, T (20), E (5), W (23). Let's see the numbers:

First group numbers:13,7,12,3,10. Let's take the largest number:13, no. Smallest:3, no.

Wait, maybe each letter is the number of the letter, and we have to find the letter for the sum of the numbers. Wait, E is 5, sum of second group is 36, 36-31=5. T is 20, 45-25=20. 25? No. Wait, the page is Middle School Math with Pizzazz, so it's a math puzzle, likely number to letter (A=1, B=2,...Z=26). Let's list the letters:

T (20), E (5), W (23), (next is 8,2: 8+2=10 (J)? No), (19,9,4:19+9+4=32→32-26=6 (F)? No), (21,11,5,22,15: sum=74→74-2*26=22 (V)? No, but we need U's group. Wait, U is 21. The group has 21,11,5,22,15. Let's sum:21+11=32, 5+22=27, 15. 32+27+15=74. 74-53=21 (U). 53? No. Wait, maybe each letter is the sum of the numbers minus a multiple of 26 to get the letter. Let's try:

First group:13+7+12+3+10=45. 45-25=20 (T). 25? No. Second group:14+16+6=36. 36-31=5 (E). 31 is the page number (D-31). Ah! Page D-31, so 31. So 36-31=5 (E). First group:45-31=14 (N), no. Wait, first group has T, which is 20. 45-25=20, 25=31-6? No. Wait, second group:36-31=5 (E). Third group:18+17+1+20=56. 56-31=25 (Y), no. But third group has W (23). 56-33=23, 33=31+2. No. Fourth group:8+2=10. 10-31= negative, no. Fifth group:19+9+4=32. 32-31=1 (A), no. Sixth group:21+11+5+22+15=74. 74-31=43. 43-26=17 (Q), no. Wait, maybe the sum of the numbers minus the number of numbers. First group:5 numbers, 45-5=40, no. Second group:3 numbers, 36-3=33, no. Wait, E is 5, 36/7.2=5, no. Wait, maybe each letter is the median? First group:3,7,10,12,13. Median 10 (J), no. Second group:6,14,16. Median 14 (N), no. Third group:1,17,18,20. Median (17+18)/2=17.5, no. Fourth group:2,8. Median 5 (E), no. Fifth group:4,9,19. Median 9 (I), no. Sixth group:5,11,15,21,22. Median 15 (O), no. Wait, maybe the mode? No. Wait, the letters are T, E, W, (next), (next), U. Wait, T, E, W: maybe a word? TEW... No. Wait, U is in the last group. Let's check the numbers in the last group:21 (U), 11 (K), 5 (E), 22 (V), 15 (O). Wait, U is 21, so maybe the first number is 21, which is U. But why other numbers? Wait, maybe each group's letter is the letter corresponding to the sum of the numbers, and we made a mistake earlier. Let's try again:

Group 1: 13,7,12,3,10. Sum=45. 45-25=20 (T). 25=45-20. Group 2:14,16,6. Sum=36. 36-31=5 (E). 31=36-5. Group 3:18,17,1,20. Sum=56. 56-33=23 (W). 33=56-23. Group 4:8,2. Sum=10. 10-?=? No. Group 5:19,9,4. Sum=32. 32-?=? No. Group 6:21,11,5,22,15. Sum=74. 74-?=21 (U). 74-53=21. 53=74-21. So 74-53=21 (U). But why 53? Wait, maybe the sum minus (sum of previous letters' positions). T=20, E=5, W=23. Sum of positions:20+5+23=48. 74-48=26, no. Wait, this is getting too complicated. Let's try the number to letter (A=1) for each number in the last group and see if U is formed. 21=U, 11=K, 5=E, 22=V, 15=O. Wait, U, K, E, V, O. No. Wait, maybe the letter is the sum of the numbers modulo 26. 74 mod26=74-2*26=22 (V), no. 74-3*26=74-78=-4, no. Wait, 21 is U, so maybe the answer is U, but we need to confirm. Wait, the last group has U, and the first number is 21, which is U. Maybe the letter is the first number? No, group 1 first number 13 (M), not T. Group 2 first number 14 (N), not E. Group 3 first number 18 (R), not W. So no. Wait, maybe the sum of the numbers minus 21 (U's position) is the sum of the other numbers. 74-21=53. 11+5+22+15=53. Yes! 11+5+22+15=53, and 74-53=21 (U). So the sum of the other numbers is 53, and 74-53=21 (U). So the letter is U, which is already given, but maybe we need to find the missing letter? Wait, no, the last group has U, so maybe the answer is U, and the numbers sum to 74, and 74-53=21 (U).

Step2: Confirm the pattern (sum of numbers except one equals the position of the letter)

In the last group: numbers 21 (U), 11,5,22,15. Sum of 11+5+22+15=53. 21+53=74 (total sum). So the letter is U (21), which matches. So the missing letter (if any) is U, but since U is already there, maybe we confirm the pattern.

Answer:

U (since 21 corresponds to U in A=1, B=2, ..., U=21, and the sum of the other numbers in the group is 53, 21+53=74, confirming the pattern)