Question

these figures are congruent. what is ( mangle t )?

Explanation:

Step1: Match congruent triangle parts

First, identify corresponding parts: $\angle U$ corresponds to $\angle B$ (both $80^\circ$). Side $AB=74$ km, $CB=50$ km, so side $AC = \sqrt{AB^2 + CB^2 - 2\cdot AB\cdot CB\cdot\cos(80^\circ)}$ (or use triangle sum to find side, but easier: match angles to sides. The side opposite $\angle C$ ($63^\circ$) is $AB=74$ km, side opposite $\angle A$ ($37^\circ$) is $CB=50$ km, side opposite $\angle B$ ($80^\circ$) is $AC$. In $\triangle TUV$, side $TV=82$ km is opposite $\angle U$ ($80^\circ$), so $TV$ corresponds to $AC$. Then $\angle T$ corresponds to $\angle A$? No, wait: $\angle U = \angle B=80^\circ$, so side opposite $\angle T$ is $UV$, which corresponds to $CB=50$ km, and side opposite $\angle V$ is $TU$, which corresponds to $AB=74$ km. So $\angle T$ corresponds to $\angle C$? No, wait triangle sum: in $\triangle ABC$, angles are $37^\circ, 63^\circ, 80^\circ$. Congruent triangles have matching angles. $\angle U = 80^\circ = \angle B$, so $\angle T$ must match $\angle A$ or $\angle C$. The side opposite $\angle T$ is $UV$, which in $\triangle ABC$ the side opposite $\angle C$ is $AB=74$ km, side opposite $\angle A$ is $CB=50$ km. Since $\triangle TUV$ is congruent, the side opposite $\angle U$ ($80^\circ$) is $TV=82$ km, which is the longest side, matching the longest side in $\triangle ABC$, which is $AC$ (opposite $80^\circ$). So $AC=82$ km, $AB=74$ km (opposite $63^\circ$), $CB=50$ km (opposite $37^\circ$). In $\triangle TUV$, $TU$ corresponds to $AB=74$ km (opposite $\angle V$), $UV$ corresponds to $CB=50$ km (opposite $\angle T$). So $\angle T$ is opposite the 50 km side, which in $\triangle ABC$ is $\angle A=37^\circ$? No, wait: $\angle A$ is opposite $CB=50$ km, yes. Wait no: $\angle A$ is at vertex $A$, so opposite side is $CB$, which is 50 km. $\angle C$ is opposite $AB=74$ km. $\angle B$ is opposite $AC=82$ km. In $\triangle TUV$, $\angle U$ is $80^\circ$ (opposite $TV=82$ km, which is $AC$). $\angle T$ is opposite $UV$, which is equal to $CB=50$ km, so $\angle T = \angle A=37^\circ$? No, wait no: $\angle A$ is opposite $CB=50$ km, so yes, $\angle T$ is opposite $UV=50$ km, so $\angle T=37^\circ$? Wait no, wait triangle sum: $37+63+80=180$, correct. In $\triangle TUV$, angles must sum to 180. $\angle U=80$, so $\angle T + \angle V=100$. The sides: $TV=82$ (longest, opposite $80$), $TU$ and $UV$ are the other two sides, 74 and 50. So $\angle T$ is opposite $UV$, which is 50, so $\angle T=37^\circ$, since 37 is opposite 50 in $\triangle ABC$.

Step2: Confirm congruent angle match

Since congruent triangles have corresponding angles equal, and $\angle U \cong \angle B (80^\circ)$, the side opposite $\angle T$ is congruent to the side opposite $\angle A$ (50 km), so $\angle T = \angle A = 37^\circ$.

Answer:

$37$