Question

these figures are congruent. what is sq?
\boxed{} miles

Explanation:

Step1: Recall Congruent Triangles

Congruent triangles have corresponding sides equal. So, we need to find the corresponding side of SQ in triangle JIH.

Step2: Identify Corresponding Angles/Sides

In triangle JIH, angle at I is 80°, side JI is 50 mi. In triangle RSQ, angle at S is 80°, so side SQ should correspond to side JI (since angles and sides correspond in congruent triangles). Wait, wait, let's check angles again. Triangle JIH: angles - at I is 80°, sides: JH=56, JI=50, IH=35. Triangle RSQ: angles - at S is 80°, at R is 38°, at Q is 62°. Let's find the sum of angles in triangle RSQ: 38+62+80=180, correct. In triangle JIH: let's find angle at J. Sum of angles: angle J + 80 + angle H = 180. Wait, maybe better to match angles. Angle at I (80°) in JIH corresponds to angle at S (80°) in RSQ. Then, side JI (50 mi) in JIH: let's see sides. Wait, SQ: let's see the sides. Wait, in triangle JIH, side JI is 50 mi, and in triangle RSQ, side SQ should correspond to JI? Wait, no, let's check the sides. Wait, triangle JIH has sides: JH=56, JI=50, IH=35. Triangle RSQ has sides: RQ=56, RS=?, SQ=?. Wait, angle at Q is 62°, angle at R is 38°, angle at S is 80°. In triangle JIH, angle at I is 80°, angle at J: let's calculate. 180 - 80 - angle H. Wait, maybe angle at J: in triangle JIH, sides JH=56, JI=50, IH=35. Let's check angles. Using Law of Sines? Wait, but since they are congruent, corresponding sides are equal. So, angle at I (80°) in JIH: side opposite is JH (56). Angle at S (80°) in RSQ: side opposite is RQ (56), which matches. Then angle at Q (62°) in RSQ: side opposite is RS. Angle at H in JIH: let's calculate angle H. 180 - 80 - angle J. Wait, angle J: using Law of Sines in JIH: sin(angle J)/35 = sin(80°)/56. sin(angle J) = (35 sin80°)/56 ≈ (35*0.9848)/56 ≈ 34.468/56 ≈ 0.6155, so angle J ≈ 38°, which matches angle at R (38°) in RSQ. So angle J (38°) in JIH corresponds to angle R (38°) in RSQ. Then angle H in JIH corresponds to angle Q (62°) in RSQ. So, side opposite angle J (38°) in JIH is IH (35), which should correspond to side opposite angle R (38°) in RSQ, which is SQ. Wait, angle R is 38°, side opposite is SQ. In JIH, angle J is 38°, side opposite is IH (35). Wait, no, angle J is 38°, side opposite is IH (35). So SQ should be equal to IH? Wait, IH is 35? No, wait IH is 35 mi? Wait, no, in triangle JIH, IH is 35 mi? Wait, the diagram: J to I is 50, I to H is 35, J to H is 56. So angle at J is 38°, angle at I is 80°, angle at H is 62° (since 38+80+62=180). Ah! So angle at H is 62°, which corresponds to angle at Q (62°) in RSQ. So side opposite angle H (62°) in JIH is JI (50 mi). Side opposite angle Q (62°) in RSQ is RS. Wait, no, angle at Q is 62°, side opposite is RS. Angle at H is 62°, side opposite is JI (50 mi). Then angle at R (38°) in RSQ: side opposite is SQ. Angle at J (38°) in JIH: side opposite is IH (35 mi). Wait, that can't be. Wait, maybe I mixed up. Wait, let's list the angles:

Triangle JIH:
- Angle at J: let's calculate. Using Law of Sines: JH/sin(angle I) = JI/sin(angle H) = IH/sin(angle J)
JH=56, angle I=80°, so 56/sin80° = 56/0.9848 ≈ 56.86
JI=50, so 50/sin(angle H) = 56.86 → sin(angle H)=50/56.86≈0.879 → angle H≈62°, correct. Then angle J=180-80-62=38°, correct.

Triangle RSQ:
- Angle at R=38°, angle at Q=62°, angle at S=80°, correct.

So corresponding angles:
- Angle J (38°) ↔ Angle R (38°)
- Angle I (80°) ↔ Angle S (80°)
- Angle H (62°) ↔ Angle Q (62°)

Therefore, corresponding sides:
- Side opposite angle J (IH) ↔ Side opposite angle R (SQ)
- Side opposite angle I (JH) ↔ Side opposite angle S (RQ)
- Side opposite angle H (JI) ↔ Side opposite angle Q (RS)

Side opposite angle J (38°) in JIH is IH=35 mi? Wait, no: in triangle JIH, angle J is at vertex J, so side opposite is IH (from I to H). Yes, IH is 35 mi. So side opposite angle R (38°) in RSQ is SQ, so SQ should be equal to IH=35 mi? Wait, no, wait IH is 35 mi? Wait the diagram: I to H is 35 mi, J to I is 50 mi, J to H is 56 mi. So IH=35, JI=50, JH=56. Then in triangle RSQ, RQ=56 (matches JH=56), angle at S=80° (matches angle at I=80°), angle at R=38° (matches angle at J=38°), angle at Q=62° (matches angle at H=62°). So side SQ: which side in JIH corresponds to SQ? Let's see the vertices. Triangle JIH: J, I, H. Triangle RSQ: R, S, Q. So J ↔ R, I ↔ S, H ↔ Q. So vertex J corresponds to R, I to S, H to Q. Therefore, side JI (J to I) corresponds to side RS (R to S), side IH (I to H) corresponds to side SQ (S to Q), side JH (J to H) corresponds to side RQ (R to Q). Ah! That's the key. So I ↔ S, H ↔ Q, so IH (I to H) corresponds to SQ (S to Q). Therefore, SQ = IH = 35 mi? Wait, no, IH is 35 mi? Wait the diagram: I to H is 35 mi, yes. Wait but JI is 50 mi, JH is 56 mi. So if I ↔ S, H ↔ Q, then IH (I-H) corresponds to SQ (S-Q). So SQ = IH = 35? Wait, but let's check again. Wait, the problem says "These figures are congruent". So congruent triangles have corresponding sides equal. So the correspondence is J→R, I→S, H→Q. So:

- JH (56) ↔ RQ (56) – matches.
- JI (50) ↔ RS (50) –?
- IH (35) ↔ SQ (35) –?

Wait, but in triangle RSQ, RQ is 56, angle at Q is 62°, angle at R is 38°, angle at S is 80°. In triangle JIH, JH is 56, angle at H is 62°, angle at J is 38°, angle at I is 80°. So yes, the correspondence is J↔R, I↔S, H↔Q. Therefore, side IH (from I to H) corresponds to side SQ (from S to Q). Therefore, SQ = IH = 35 mi? Wait, no, wait IH is 35 mi? Wait the diagram: I to H is labeled 35 mi, yes. So SQ should be 35? Wait, but wait, maybe I made a mistake. Wait, let's check the sides again. Wait, JI is 50 mi (J to I), and in triangle RSQ, SQ: if I↔S, H↔Q, then IH (I to H) is 35, so SQ (S to Q) is 35. Wait, but let's confirm with angles. Angle at I (80°) in JIH: side opposite is JH (56), which is opposite angle I? Wait no, in triangle JIH, angle at I is between JI (50) and IH (35), so side opposite angle I is JH (56). In triangle RSQ, angle at S is between RS and SQ, so side opposite angle S is RQ (56), which matches JH (56). Then angle at J (38°) in JIH: side opposite is IH (35), which is between I and H. In triangle RSQ, angle at R (38°) is between RQ (56) and RS, so side opposite angle R is SQ. Therefore, SQ = IH = 35 mi. Wait, but wait, the problem is asking for SQ. So according to the correspondence, SQ corresponds to IH, which is 35 mi? Wait, no, wait IH is 35 mi, yes. Wait, but let's check the diagram again. The first triangle: J to I is 50, I to H is 35, J to H is 56. The second triangle: R to Q is 56, R to S is?, S to Q is?. So if J↔R, I↔S, H↔Q, then:

- JH (J-H) = 56 ↔ RQ (R-Q) = 56 ✔️
- JI (J-I) = 50 ↔ RS (R-S) = 50 ✔️
- IH (I-H) = 35 ↔ SQ (S-Q) = 35 ✔️

Yes, that makes sense. So SQ is equal to IH, which is 35 miles. Wait, but wait, maybe I mixed up the correspondence. Let's check angles again. Angle at H in JIH: 62°, angle at Q in RSQ: 62°, so H↔Q. Angle at I: 80°, angle at S: 80°, so I↔S. Angle at J: 38°, angle at R: 38°, so J↔R. Therefore, sides:

- J (R) to H (Q): 56 ↔ 56 ✔️
- I (S) to H (Q): 35 ↔ SQ ✔️
- J (R) to I (S): 50 ↔ RS ✔️

Therefore, SQ = 35 miles. Wait, but wait, the problem says "What is SQ?" So the answer should be 35? Wait, but let me check again. Wait, maybe I made a mistake in the correspondence. Let's list the sides:

Triangle JIH:
- Vertices: J, I, H
- Sides: JH=56, JI=50, IH=35
- Angles: ∠J=38°, ∠I=80°, ∠H=62°

Triangle RSQ:
- Vertices: R, S, Q
- Sides: RQ=56, RS=?, SQ=?
- Angles: ∠R=38°, ∠S=80°, ∠Q=62°

So the correspondence is J↔R, I↔S, H↔Q. Therefore, side IH (I to H) is between I (S) and H (Q), so SQ is the side between S and Q, which is IH. Therefore, SQ = IH = 35 miles.

Wait, but wait, the first triangle's side JI is 50, and in the second triangle, is SQ equal to JI? No, because JI is J to I, which is R to S. So SQ is I to H, which is 35. So the answer is 35.

Answer:

35