Question

these figures are congruent. what is m∠q?

(there are two triangles: triangle def with sides de=45 mi, df=50 mi, ef=31 mi, angles ∠d=37°, ∠f=63°, ∠e=80°; triangle pqr with sides pr=50 mi, rq=45 mi, angle ∠r=37°.)

Explanation:

Step1: Recall Congruent Triangles

Congruent triangles have corresponding angles equal. So, we need to find the corresponding angle to ∠Q in triangle DEF.

Step2: Identify Corresponding Angles

In triangle DEF, sides: \( DE = 45 \) mi, \( DF = 50 \) mi, \( EF = 31 \) mi. Angles: \( \angle D = 37^\circ \), \( \angle F = 63^\circ \), \( \angle E = 80^\circ \). In triangle PQR, sides: \( PR = 50 \) mi, \( RQ = 45 \) mi, \( PQ \) (unknown). Angles: \( \angle R = 37^\circ \). By matching sides ( \( DE = RQ = 45 \) mi, \( DF = PR = 50 \) mi ), the corresponding angle to ∠Q is ∠F? Wait, no, let's check angles. Wait, triangle DEF: angles sum to \( 180^\circ \), \( 37 + 63 + 80 = 180 \). Triangle PQR: angles sum to \( 180 \), \( 37 + \angle P + \angle Q = 180 \). But since they are congruent, corresponding angles: let's match the sides. \( DE = 45 \) (side between ∠D (37°) and ∠E (80°)), \( DF = 50 \) (side between ∠D (37°) and ∠F (63°)). In triangle PQR, \( RQ = 45 \) (side between ∠R (37°) and ∠Q), \( PR = 50 \) (side between ∠R (37°) and ∠P). So the angle opposite to the side of length 31 in DEF (which is ∠D? No, wait EF is 31, which is opposite ∠D (37°). Wait, maybe better: in triangle DEF, angle at F is 63°, angle at E is 80°, angle at D is 37°. In triangle PQR, angle at R is 37°, so the other angles: let's see the sides. DE = 45 (DFE: side DE is 45, between D (37) and E (80)). RQ = 45 (PQR: side RQ is 45, between R (37) and Q. So ∠Q corresponds to ∠E? Wait no, maybe I mixed. Wait, let's list the sides:

Triangle DEF:
- Side DE: 45 mi (between D (37°) and E (80°))
- Side DF: 50 mi (between D (37°) and F (63°))
- Side EF: 31 mi (between E (80°) and F (63°))

Triangle PQR:
- Side RQ: 45 mi (between R (37°) and Q)
- Side PR: 50 mi (between R (37°) and P)
- Side PQ:? (between P and Q)

So the angle at F in DEF is 63°, angle at E is 80°, angle at D is 37°. In PQR, angle at R is 37°, so the angle at Q should correspond to angle at F? Wait no, maybe the correspondence is D <-> R, E <-> Q, F <-> P? Wait, DE = 45, RQ = 45; DF = 50, PR = 50. So vertex D corresponds to R, E corresponds to Q, F corresponds to P. So ∠E (80°) corresponds to ∠Q? Wait no, ∠D (37°) corresponds to ∠R (37°), ∠E (80°) corresponds to ∠Q, ∠F (63°) corresponds to ∠P. Wait, but in DEF, angle at F is 63°, angle at E is 80°, angle at D is 37°. So if D <-> R, E <-> Q, F <-> P, then ∠Q = ∠E? No, that can't be. Wait, maybe I made a mistake. Wait, let's calculate the angle in DEF: 37 + 63 + 80 = 180. In PQR, 37 + ∠P + ∠Q = 180. So ∠P + ∠Q = 143. In DEF, ∠F = 63, ∠E = 80. So if they are congruent, the angles should match. Wait, maybe the correspondence is D <-> R, F <-> Q, E <-> P. Let's check sides: DF = 50 (D to F), RQ = 45 (R to Q) – no. Wait, DE = 45 (D to E), RQ = 45 (R to Q); DF = 50 (D to F), PR = 50 (P to R). So the sides: DE = RQ (45), DF = PR (50), so the included angle? Wait, DE and DF meet at D (37°), RQ and PR meet at R (37°). So the triangle DEF and PQR are congruent by SAS? Wait, DE = RQ (45), DF = PR (50), and the included angle at D and R is 37°, so SAS congruence. Therefore, the corresponding angle to ∠Q is ∠F? Wait, no, in SAS, the angle is between the two sides. So in DEF, sides DE (45) and DF (50) meet at D (37°), so the angle between DE and DF is ∠D (37°). In PQR, sides RQ (45) and PR (50) meet at R (37°), so the angle between RQ and PR is ∠R (37°). Therefore, the correspondence is D <-> R, E <-> Q, F <-> P. So ∠E (80°) corresponds to ∠Q? No, wait, DE is from D to E, RQ is from R to Q. DF is from D to F, PR is from P to R. So the side opposite to ∠D (37°) in DEF is EF (31 mi), and in PQR, the side opposite to ∠R (37°) is PQ. Then, the side opposite to ∠F (63°) in DEF is DE (45 mi), and in PQR, the side opposite to ∠Q is PR (50 mi)? No, this is confusing. Wait, maybe a better approach: since the triangles are congruent, their corresponding angles are equal. Let's list the angles in DEF: ∠D = 37°, ∠F = 63°, ∠E = 80°. In PQR, ∠R = 37°, so the other two angles must be 63° and 80°. Now, looking at the sides: in DEF, side DE is 45 (between ∠D and ∠E), side DF is 50 (between ∠D and ∠F), side EF is 31 (between ∠E and ∠F). In PQR, side RQ is 45 (between ∠R and ∠Q), side PR is 50 (between ∠R and ∠P), side PQ is? (between ∠P and ∠Q). So the angle between RQ (45) and PR (50) is ∠R (37°), same as between DE (45) and DF (50) is ∠D (37°). Therefore, the angle opposite to DF (50) in DEF is ∠E (80°), and in PQR, the angle opposite to PR (50) is ∠Q. Wait, no, DF is 50, opposite ∠E (80°). PR is 50, opposite ∠Q. So ∠Q = ∠E = 80°? Wait, no, that can't be. Wait, no, in DEF, side DF is 50, which is opposite ∠E (80°). In PQR, side PR is 50, which is opposite ∠Q. So ∠Q = ∠E = 80°? But wait, in DEF, angle at F is 63°, which is opposite DE (45°). In PQR, angle at Q is opposite PR (50°)? No, I think I messed up. Wait, let's calculate the angle in DEF: angles are 37, 63, 80. So 37 + 63 = 100, 180 - 100 = 80. In PQR, angle at R is 37, so the other two angles sum to 143. If the triangles are congruent, the angles must match. So if ∠D (37) = ∠R (37), then ∠F (63) = ∠P, and ∠E (80) = ∠Q. Wait, but let's check the sides. DE is 45 (from D to E), RQ is 45 (from R to Q). DF is 50 (from D to F), PR is 50 (from P to R). So the side DE (45) is between ∠D (37) and ∠E (80), and side RQ (45) is between ∠R (37) and ∠Q. Therefore, ∠E (80) corresponds to ∠Q. Wait, but in DEF, angle at F is 63, which is between DF (50) and EF (31). In PQR, angle at P is between PR (50) and PQ. So yes, ∠Q should be equal to ∠E, which is 80°? Wait, no, wait the problem says the figures are congruent. Let's check the angles again. In triangle DEF: angles are 37° (D), 63° (F), 80° (E). In triangle PQR: angle at R is 37°, sides PR=50, RQ=45. So the sides: PR=50 (same as DF=50), RQ=45 (same as DE=45). So the triangle DEF has sides DE=45, DF=50, EF=31. Triangle PQR has sides PR=50, RQ=45, PQ=?. So by SSS? Wait, DE=RQ=45, DF=PR=50, EF=PQ=31? Wait, EF is 31, so PQ should be 31. Then, the angles: in DEF, angle at E is 80°, which is opposite DF (50). In PQR, angle at Q is opposite PR (50), so ∠Q = ∠E = 80°? Wait, no, opposite angles: in triangle, the angle opposite a side is equal to the angle opposite the corresponding side in congruent triangle. So side DF (50) is opposite ∠E (80°), side PR (50) is opposite ∠Q, so ∠Q = ∠E = 80°? Wait, but let's check the angle sum. In PQR: 37 + ∠Q + ∠P = 180. If ∠Q is 80, then ∠P is 63, which matches ∠F (63) in DEF. Yes, that makes sense. So ∠Q = 80°.

Wait, but wait, maybe I made a mistake. Let's re-express:

Triangle DEF:
- Vertex D: 37°, sides DE=45, DF=50
- Vertex F: 63°, sides DF=50, EF=31
- Vertex E: 80°, sides DE=45, EF=31

Triangle PQR:
- Vertex R: 37°, sides PR=50, RQ=45
- Vertex P:?, sides PR=50, PQ=?
- Vertex Q:?, sides RQ=45, PQ=?

Since they are congruent, the corresponding vertices are D <-> R, E <-> Q, F <-> P. Therefore, angle at E (80°) corresponds to angle at Q, so m∠Q = 80°.

Step3: Confirm Congruence

Since the triangles are congruent, corresponding angles are equal. So ∠Q corresponds to ∠E, which is 80°.

Answer:

80