Question

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these figures are congruent. what is vw?
kilometers
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Explanation:

Step1: Recall Congruent Figures Property

Congruent figures have corresponding sides equal. So we need to find the side in the first figure that corresponds to \( VW \) in the second figure.

Step2: Identify Corresponding Sides

In the first figure (quadrilateral \( IJKL \)) and the second figure (quadrilateral \( VWXU \)), we match the sides and angles. Looking at the side lengths: \( IJ = 43 \) km, \( IX \) (wait, \( IL \)? No, \( LI = 30 \) km, \( LK = 13 \) km, \( KJ = 55 \) km, \( IJ = 43 \) km, \( LI = 30 \) km, \( \angle I = 99^\circ \), \( \angle L = 126^\circ \). In the second figure, \( WX = 30 \) km, \( XU = 43 \) km, \( \angle V = 89^\circ \), \( \angle U = 46^\circ \). Wait, the side \( LK \) is 13 km, \( KJ = 55 \) km, \( IJ = 43 \) km, \( LI = 30 \) km. Wait, the angle sum in a quadrilateral is \( 360^\circ \). In first quadrilateral: \( 126 + 99 + \angle J + \angle K = 360 \). In second: \( 89 + 46 + \angle W + \angle X = 360 \). But more importantly, congruent figures have corresponding sides equal. So we need to find which side in the first figure corresponds to \( VW \). Let's list the sides:

First quadrilateral (let's order the vertices as \( I, L, K, J \)): \( IL = 30 \) km, \( LK = 13 \) km, \( KJ = 55 \) km, \( JI = 43 \) km.

Second quadrilateral (vertices \( V, W, X, U \)): \( VW = ? \), \( WX = 30 \) km, \( XU = 43 \) km, \( UV = ? \).

Wait, the side \( KJ \) in the first figure: \( KJ = 55 \) km? No, wait \( KJ = 55 \) km, \( JI = 43 \) km, \( IL = 30 \) km, \( LK = 13 \) km. Wait, the angle \( \angle L = 126^\circ \), \( \angle I = 99^\circ \). In the second figure, \( \angle V = 89^\circ \), \( \angle U = 46^\circ \). Wait, maybe the corresponding side to \( VW \) is \( KJ \)? No, wait no. Wait, let's check the side that is between the angles or the side that has the same corresponding length. Wait, the side \( LK \) is 13 km, \( KJ = 55 \) km, \( JI = 43 \) km, \( IL = 30 \) km. Wait, the side \( KJ \) is 55 km? No, wait the first figure: \( LK = 13 \), \( KJ = 55 \), \( JI = 43 \), \( IL = 30 \). The second figure: \( WX = 30 \), \( XU = 43 \), so \( JI = XU = 43 \), \( IL = WX = 30 \). Then the side \( KJ \) should correspond to \( UV \), and the side \( LK \) corresponds to \( VW \)? Wait no, wait the angle at \( L \) is \( 126^\circ \), angle at \( V \) is \( 89^\circ \), angle at \( I \) is \( 99^\circ \), angle at \( U \) is \( 46^\circ \). Wait, maybe I got the vertex order wrong. Let's try to match the angles. The angle \( \angle L = 126^\circ \), \( \angle I = 99^\circ \), so the remaining angles in first quadrilateral: \( 360 - 126 - 99 = 135^\circ \), split between \( \angle K \) and \( \angle J \). In second quadrilateral: \( 360 - 89 - 46 = 225^\circ \), split between \( \angle W \) and \( \angle X \). Wait, maybe the correct correspondence is \( LK \) with \( VW \)? No, wait the side \( LK \) is 13 km? No, wait the first figure: \( LK = 13 \) km, \( KJ = 55 \) km, \( JI = 43 \) km, \( IL = 30 \) km. The second figure: \( WX = 30 \) km, \( XU = 43 \) km, so \( JI = XU \), \( IL = WX \). Then the side \( KJ \) (55 km) would correspond to \( UV \), and the side \( LK \) (13 km) – no, wait no. Wait, maybe the side \( KJ \) is 55 km, but wait the question is \( VW \). Wait, maybe I made a mistake. Wait, the first figure: the side \( KJ \) is 55 km? No, wait the first figure's sides: \( LK = 13 \), \( KJ = 55 \), \( JI = 43 \), \( IL = 30 \). The second figure's sides: \( WX = 30 \), \( XU = 43 \), so \( JI = XU \), \( IL = WX \). Then the side \( LK \) (13 km) – no, wait the side \( KJ \) is 55 km, but that's not matching. Wait, no, maybe the corresponding side to \( VW \) is \( KJ \)? No, wait no. Wait, let's look at the side that is adjacent to the angles. Wait, in the first quadrilateral, the side \( KJ \) is 55 km? Wait, no, the first figure: \( KJ \) is 55 km, \( JI \) is 43 km, \( IL \) is 30 km, \( LK \) is 13 km. The second figure: \( WX \) is 30 km, \( XU \) is 43 km, so \( JI = XU \), \( IL = WX \). Then the side \( LK \) (13 km) – no, wait the side \( KJ \) is 55 km, but that's not. Wait, maybe I messed up the vertex order. Let's try to match the angles. The angle \( \angle L = 126^\circ \), \( \angle I = 99^\circ \), so the angle at \( K \) and \( J \). In the second figure, \( \angle V = 89^\circ \), \( \angle U = 46^\circ \), so angle at \( W \) and \( X \). Wait, maybe the correct correspondence is \( LK \) with \( UV \), \( KJ \) with \( VW \), \( JI \) with \( XU \), \( IL \) with \( WX \). Wait, if that's the case, then \( KJ = 55 \) km, but that can't be. Wait, no, the problem says the figures are congruent, so corresponding sides are equal. Let's list the sides:

First quadrilateral (vertices in order \( I, L, K, J \)):

- \( IL = 30 \) km

- \( LK = 13 \) km

- \( KJ = 55 \) km

- \( JI = 43 \) km

Second quadrilateral (vertices in order \( V, W, X, U \)):

- \( VW = ? \)

- \( WX = 30 \) km

- \( XU = 43 \) km

- \( UV = ? \)

Now, since they are congruent, the side \( LK \) (13 km) would correspond to \( UV \), \( KJ \) (55 km) to \( VW \), \( JI \) (43 km) to \( XU \) (43 km, which matches), \( IL \) (30 km) to \( WX \) (30 km, which matches). Wait, so \( KJ = 55 \) km, so \( VW = KJ = 55 \) km? Wait, no, wait the angle at \( L \) is 126°, angle at \( V \) is 89°, angle at \( I \) is 99°, angle at \( U \) is 46°. Wait, maybe the vertex order is different. Let's check the angle sum. In first quadrilateral: 126 + 99 + angle J + angle K = 360. So angle J + angle K = 360 - 126 - 99 = 135. In second quadrilateral: 89 + 46 + angle W + angle X = 360. So angle W + angle X = 360 - 89 - 46 = 225. That doesn't match, so maybe the quadrilaterals are not in the same vertex order. Wait, maybe the first figure is a quadrilateral with vertices \( L, I, J, K \)? No, the labels are \( I, L, K, J \). Wait, maybe the side \( KJ \) is 55 km, and in the second figure, \( VW \) corresponds to \( KJ \). So \( VW = 55 \) km? Wait, no, wait the first figure's side \( KJ \) is 55 km, so if \( VW \) corresponds to \( KJ \), then \( VW = 55 \) km. Wait, but let's check again. The first figure: \( KJ = 55 \) km, \( JI = 43 \) km, \( IL = 30 \) km, \( LK = 13 \) km. The second figure: \( WX = 30 \) km (matches \( IL \)), \( XU = 43 \) km (matches \( JI \)), so the next side \( UV \) should match \( LK \) (13 km), and \( VW \) should match \( KJ \) (55 km). So yes, \( VW = KJ = 55 \) km.

Answer:

55