Question

what are the corresponding parts of the figures below? select all that apply.

$overline{jk} \cong \overline{ut}$ $overline{kj} \cong \overline{us}$ $\angle k \cong \angle s$ $\angle l \cong \angle s$

Explanation:

Step1: Analyze triangle \( \triangle JKL \) and \( \triangle UTS \)

• \( \triangle JKL \) has a right angle at \( K \), \( \angle L = 58^\circ \), \( \angle J = 32^\circ \), \( LK = 13 \) km, \( JK = 20 \) km, \( JL = 24 \) km.
• \( \triangle UTS \) has a right angle at \( T \), \( \angle S = 58^\circ \), \( \angle U = 32^\circ \), \( TS = 13 \) km, \( UT = 20 \) km, \( US = 24 \) km.

Step2: Check each option

• \( \overline{JK} \cong \overline{UT} \): \( JK = 20 \) km, \( UT = 20 \) km, so this is true.
• \( \overline{KJ} \cong \overline{US} \): Wait, \( KJ = 20 \) km, \( US = 24 \) km? Wait, no, re - check. Wait, \( KJ \) is 20 km, \( US \) is 24 km? Wait, no, in \( \triangle JKL \), \( JL = 24 \) km, \( JK = 20 \) km, \( LK = 13 \) km. In \( \triangle UTS \), \( US = 24 \) km, \( UT = 20 \) km, \( TS = 13 \) km. So \( \overline{JK} \cong \overline{UT} \) (both 20 km), \( \overline{LK} \cong \overline{TS} \) (both 13 km), \( \overline{JL} \cong \overline{US} \) (both 24 km). And \( \angle K \) (right angle) \( \cong \angle T \) (right angle), \( \angle L = 58^\circ \cong \angle S = 58^\circ \), \( \angle J = 32^\circ \cong \angle U = 32^\circ \).

Wait, the original options:

• \( \overline{JK} \cong \overline{UT} \): Correct, as \( JK = UT = 20 \) km.
• \( \overline{KJ} \cong \overline{US} \): \( KJ = 20 \) km, \( US = 24 \) km? No, this is wrong. Wait, maybe a mislabeling. Wait, \( KJ \) is 20 km, \( UT \) is 20 km, \( JL = 24 \) km, \( US = 24 \) km. So \( \overline{JL} \cong \overline{US} \), \( \overline{JK} \cong \overline{UT} \), \( \overline{LK} \cong \overline{TS} \). And \( \angle L \cong \angle S \) (both \( 58^\circ \)), \( \angle K \cong \angle T \) (both right angles), \( \angle J \cong \angle U \) (both \( 32^\circ \)).

So the correct corresponding parts:

• \( \overline{JK} \cong \overline{UT} \): Correct.
• \( \angle L \cong \angle S \): Correct, as both are \( 58^\circ \).
• The option \( \overline{KJ} \cong \overline{US} \) is wrong ( \( KJ = 20 \), \( US = 24 \) ), and \( \angle K \cong \angle S \) is wrong ( \( \angle K \) is right angle, \( \angle S = 58^\circ \) ).

So the correct options are \( \overline{JK} \cong \overline{UT} \) and \( \angle L \cong \angle S \), and also \( \overline{LK} \cong \overline{TS} \) (not in options) and \( \overline{JL} \cong \overline{US} \) (not in options) and \( \angle J \cong \angle U \) (not in options) and \( \angle K \cong \angle T \) (not in options).

From the given options, the correct ones are:

• \( \overline{JK} \cong \overline{UT} \) (because \( JK = UT = 20 \) km)
• \( \angle L \cong \angle S \) (because \( \angle L=\angle S = 58^\circ \))

The option \( \overline{KJ} \cong \overline{US} \) is incorrect ( \( KJ = 20 \), \( US = 24 \) ), and \( \angle K \cong \angle S \) is incorrect ( \( \angle K \) is \( 90^\circ \), \( \angle S \) is \( 58^\circ \) ).

Answer:

The correct options are:

• \( \overline{JK} \cong \overline{UT} \) (since \( JK = UT = 20 \) km)
• \( \angle L \cong \angle S \) (since \( \angle L=\angle S = 58^\circ \))