Question

8.3 puzzle time
what speaks every language?
circle the letter of each correct answer in the boxes below. the circled letters will spell out
the answer to the riddle.
complete the sentence.

1. if the corresponding side lengths of two triangles are _______, then the triangles

are similar.

1. if an angle of one triangle is congruent to an angle of a second triangle and the

lengths of the sides including these angles are proportional, then the triangles
are _________.

1. if two nonvertical lines are _______, then they have the same slope.
2. if two _______ lines are perpendicular, then the product of their slopes is −1.

name the triangle that is not similar to the other two triangles.

1. triangles abc, jkl, rst with side lengths: abc: 35, 28, 49; jkl: 24, 18, 32; rst: 15, 12, 21
2. triangles abf, rkl, vst with angle 93° and side lengths: abf: 80, 120; rkl: 42, 84; vst: 21, 42

t: △abc, a: proportional, h: concurrent, i: △vst, n: parallel, k: △rkl, e: △abf
r: congruent, s: perpendicular, c: similar, o: equal, h: △jkl, o: nonvertical, b: △rst

Explanation:

Step1: Match Q1 to key term

Triangles with proportional corresponding sides are similar. This matches "proportional" (letter A).

Step2: Match Q2 to key term

SAS similarity means triangles are similar. This matches "similar" (letter C).

Step3: Match Q3 to key term

Parallel nonvertical lines have equal slopes. This matches "parallel" (letter N).

Step4: Match Q4 to key term

Perpendicular nonvertical lines have slope product -1. This matches "nonvertical" (letter O).

Step5: Analyze Q5 triangle ratios

- $\triangle ABC$: $\frac{28}{35}=\frac{4}{5}$, $\frac{35}{49}=\frac{5}{7}$
- $\triangle JKL$: $\frac{18}{24}=\frac{3}{4}$, $\frac{24}{32}=\frac{3}{4}$
- $\triangle RST$: $\frac{12}{15}=\frac{4}{5}$, $\frac{15}{21}=\frac{5}{7}$
$\triangle JKL$ is not similar (letter H).

Step6: Analyze Q6 triangle similarity

- $\triangle ABF$: sides 80,120; included angle $93^\circ$
- $\triangle RKL$: sides 42,84; included angle $93^\circ$, $\frac{42}{80}
eq\frac{84}{120}$
- $\triangle VST$: sides 21,42; included angle $93^\circ$, $\frac{21}{80}
eq\frac{42}{120}$
$\triangle VST$ is not similar (letter I).

Step7: Spell riddle answer

Combine letters: A, C, N, O, H, I → **A C N O H I** (rearranged to spell "AN CH OI" → correct riddle answer: **A NOISE**? No, correct ordered letters from answers: 1.A, 2.C, 3.N, 4.O, 5.H, 6.I → **ANCHOI** → rearranged to **A CHOIN**? No, correct mapping:
1. A (proportional)
2. C (similar)
3. N (parallel)
4. O (nonvertical)
5. H ($\triangle JKL$)
6. I ($\triangle VST$)
The correct spelled answer is **ANCHOI** → adjusted to the riddle answer: **A NOISE** (no, correct: the letters spell **ANCHOR**? No, correct letter mapping:
Wait, correct step for 5: $\triangle ABC$: 28,35,49; $\triangle JKL$:18,24,32; $\triangle RST$:12,15,21.
$\frac{28}{12}=\frac{7}{3}$, $\frac{35}{15}=\frac{7}{3}$, $\frac{49}{21}=\frac{7}{3}$ → $\triangle ABC \sim \triangle RST$, so $\triangle JKL$ is different (letter H).
6. $\triangle ABF$: 80,120, $93^\circ$; $\triangle RKL$:42,84, $93^\circ$ → $\frac{42}{80}=\frac{21}{40}$, $\frac{84}{120}=\frac{7}{10}$ → not similar; $\triangle VST$:21,42, $93^\circ$ → $\frac{21}{80}=\frac{21}{80}$, $\frac{42}{120}=\frac{7}{10}$ → not similar. Wait, $\frac{80}{21}
eq\frac{120}{42}$, $\frac{80}{42}=\frac{40}{21}$, $\frac{120}{84}=\frac{10}{7}$ → $\triangle VST$ is not similar (letter I).
Final letters: A, C, N, O, H, I → **ANCHOR**? No, correct riddle answer is **A NOISE**? No, the correct spelled answer from the letters is **ANCHOI** → the actual riddle answer is **AN ECHO**? No, correct mapping:
1. A (proportional)
2. C (similar)
3. N (parallel)
4. O (nonvertical)
5. H ($\triangle JKL$)
6. I ($\triangle VST$)
Wait, no, 4: "nonvertical" is letter O, correct. 5: $\triangle JKL$ is letter H, correct. 6: $\triangle VST$ is letter I, correct.
The correct riddle answer is **ANCHOR**? No, the letters are A, C, N, O, H, I → **ANCHOI** → the actual answer is **A CHOIR**? No, the correct answer to "What Speaks Every Language?" is **AN ECHO**? No, the letters from the correct answers:
1. A
2. C
3. N
4. O
5. H
6. I
Rearranged: **A NOCHI** → no, the correct answer is **ANCHOR** (missing R? No, 4 is O, not R. Wait, 4: "nonvertical" is O, correct. 1: "proportional" is A, correct. 2: "similar" is C, correct. 3: "parallel" is N, correct. 5: $\triangle JKL$ is H, correct. 6: $\triangle VST$ is I, correct.
Wait, the correct riddle answer is **A NOISE**? No, the letters are A, C, N, O, H, I → **ANCHOI** → the correct answer is **AN ECHO** (swap C and E? No, E is $\triangle ABF$, which is not correct.
Wait, recheck Q2: "If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are ______." The answer is "similar" (letter C), correct.
Q4: "If two ______ lines are perpendicular, then the product of their slopes is -1." The answer is "nonvertical" (letter O), correct.
Q5: $\triangle ABC$: 28,35,49; $\triangle JKL$:18,24,32; $\triangle RST$:12,15,21. $\frac{28}{12}=\frac{7}{3}$, $\frac{35}{15}=\frac{7}{3}$, $\frac{49}{21}=\frac{7}{3}$ → $\triangle ABC \sim \triangle RST$, so $\triangle JKL$ is different (letter H), correct.
Q6: $\triangle ABF$: 80,120, $93^\circ$; $\triangle RKL$:42,84, $93^\circ$ → $\frac{42}{80}=\frac{21}{40}$, $\frac{84}{120}=\frac{7}{10}$ → not similar; $\triangle VST$:21,42, $93^\circ$ → $\frac{21}{80}=\frac{21}{80}$, $\frac{42}{120}=\frac{7}{10}$ → not similar. Wait, $\frac{80}{42}=\frac{40}{21}$, $\frac{120}{84}=\frac{10}{7}$ → $\triangle RKL$ is not similar? No, $\frac{42}{80}=\frac{21}{40}$, $\frac{84}{120}=\frac{7}{10}$ → not proportional. $\triangle VST$: $\frac{21}{80}=\frac{21}{80}$, $\frac{42}{120}=\frac{7}{10}$ → not proportional. $\triangle ABF$ is not similar to either? No, the question says "Name the triangle that is not similar to the other two." So $\triangle ABF$ and $\triangle RKL$: $\frac{80}{42}=\frac{40}{21}$, $\frac{120}{84}=\frac{10}{7}$ → not similar. $\triangle ABF$ and $\triangle VST$: $\frac{80}{21}$, $\frac{120}{42}=\frac{10}{3}$ → not similar. $\triangle RKL$ and $\triangle VST$: $\frac{42}{21}=2$, $\frac{84}{42}=2$ → $\triangle RKL \sim \triangle VST$ (SAS similarity, included angle $93^\circ$). So $\triangle ABF$ is not similar to the other two (letter E). Oh! I made a mistake here. $\triangle RKL$ and $\triangle VST$ have sides 42,84 and 21,42, so $\frac{42}{21}=2$, $\frac{84}{42}=2$, included angle $93^\circ$ → they are similar. So $\triangle ABF$ is not similar (letter E).
So Q6 answer is E ($\triangle ABF$).
Now letters: 1.A, 2.C, 3.N, 4.O, 5.H, 6.E → **ANCHOR**! Yes! That makes sense. The riddle answer is **ANCHOR**.

Answer:

Completed Sentences:

1. proportional
2. similar
3. parallel
4. nonvertical

Non-Similar Triangles:

1. $\triangle JKL$
2. $\triangle ABF$

Riddle Answer:

ANCHOR