Question

these figures are congruent. what is no? millimeters

Explanation:

Step1: Recall Congruent Figures Property

Congruent figures have corresponding sides equal. So, we need to find the corresponding side of \( NO \) in the first figure.

Step2: Identify Corresponding Sides

In the first figure (quadrilateral \( IJKH \)? Wait, no, the first is a quadrilateral \( IJKH \)? Wait, looking at angles: \( \angle I = 96^\circ \), \( \angle H = 47^\circ \), \( \angle K = 83^\circ \), \( \angle J \) (let's check sum: \( 96 + 47 + 83 + \angle J = 360 \)? Wait, no, wait the first figure: \( I, J, K, H \)? Wait, angles: \( \angle I = 96^\circ \), \( \angle H = 47^\circ \), \( \angle K = 83^\circ \), and the side \( IH = 44 \) mm. The second figure: \( N, O, P, Q \)? Wait, angles: \( \angle N = 47^\circ \), \( \angle O = 96^\circ \), \( \angle P = 134^\circ \), \( \angle Q \)? Wait, no, let's match angles. \( \angle I = 96^\circ \) corresponds to \( \angle O = 96^\circ \), \( \angle H = 47^\circ \) corresponds to \( \angle N = 47^\circ \), \( \angle K = 83^\circ \)? Wait, no, the second figure has angles \( 47^\circ \) (at N), \( 96^\circ \) (at O), \( 134^\circ \) (at P), and let's calculate the fourth angle: \( 360 - 47 - 96 - 134 = 83^\circ \), so that matches \( \angle K = 83^\circ \). Now, the side \( IH = 44 \) mm, and in the second figure, the side \( NQ = 48 \)? Wait, no, wait the side \( NO \): let's see the first figure, the side opposite or corresponding? Wait, the first figure: side \( IH = 44 \) mm, angle at \( I = 96^\circ \), angle at \( H = 47^\circ \). The second figure: side \( NO \), angle at \( N = 47^\circ \), angle at \( O = 96^\circ \). Wait, the side \( JK \)? No, wait the first figure has side \( IH = 44 \) mm, and the second figure: let's check the sides. Wait, the first figure: \( IH = 44 \) mm, and the second figure: \( NO \) should correspond to \( KH \)? Wait, no, wait the first figure: vertices \( I, J, K, H \), with \( IH = 44 \) mm, \( \angle I = 96^\circ \), \( \angle H = 47^\circ \), \( \angle K = 83^\circ \), \( \angle J \) (which we found is \( 83^\circ \)? Wait, no, sum of quadrilateral angles is \( 360^\circ \). So \( 96 + 47 + 83 + \angle J = 360 \), so \( \angle J = 360 - 96 - 47 - 83 = 134^\circ \), which matches \( \angle P = 134^\circ \) in the second figure. So now, let's map the vertices: \( I \) corresponds to \( O \) (both \( 96^\circ \)), \( H \) corresponds to \( N \) (both \( 47^\circ \)), \( K \) corresponds to \( Q \) (both \( 83^\circ \)), \( J \) corresponds to \( P \) (both \( 134^\circ \)). So the sides: \( IH \) (connecting \( I \) and \( H \)) corresponds to \( ON \) (connecting \( O \) and \( N \))? Wait, \( IH = 44 \) mm, so \( ON \) (which is \( NO \)) should be equal to \( KH \)? Wait, no, wait \( IH \) is between \( I \) (96°) and \( H \) (47°), and \( NO \) is between \( N \) (47°) and \( O \) (96°). So the side \( IH = 44 \) mm corresponds to side \( NO \)? Wait, no, wait the first figure: \( IH = 44 \) mm, angles at \( I = 96^\circ \) and \( H = 47^\circ \). The second figure: \( NO \) is between \( N = 47^\circ \) and \( O = 96^\circ \), so the side \( NO \) should correspond to \( IH \). Wait, but in the first figure, \( IH = 44 \) mm? Wait, no, wait the first figure: \( IH \) is 44 mm, and the second figure: \( NO \) – wait, maybe I mixed up. Wait, the first figure: vertices \( I, J, K, H \), with \( IH = 44 \) mm, \( \angle I = 96^\circ \), \( \angle H = 47^\circ \), \( \angle K = 83^\circ \), \( \angle J = 134^\circ \). The second figure: vertices \( N, O, P, Q \), with \( \angle N = 47^\circ \), \( \angle O = 96^\circ \), \( \angle P = 134^\circ \), \( \angle Q = 83^\circ \), and sides \( NQ = 48 \) mm, \( OP = 19 \) mm, \( PQ = 21 \) mm. Wait, no, the side \( IH = 44 \) mm: let's check the sides. The first figure has side \( IH = 44 \) mm, and the second figure: which side is between \( 47^\circ \) (N) and \( 96^\circ \) (O)? That's \( NO \). So \( NO \) corresponds to \( IH \), so \( NO = IH = 44 \) mm? Wait, no, wait the first figure: \( IH = 44 \) mm, and the second figure: \( NO \) – wait, maybe I made a mistake. Wait, the first figure: \( I \) to \( H \) is 44 mm, angle at \( I = 96^\circ \), angle at \( H = 47^\circ \). The second figure: \( N \) to \( O \) is between angle \( N = 47^\circ \) and angle \( O = 96^\circ \), so that's the same as \( I \) to \( H \) (between \( 96^\circ \) and \( 47^\circ \)). So corresponding sides, so \( NO = IH = 44 \) mm? Wait, but let's check the other sides. The first figure: \( JK \) – what's the length? The second figure: \( PQ = 21 \) mm, \( OP = 19 \) mm, \( NQ = 48 \) mm. Wait, no, the first figure: \( IH = 44 \) mm, and the second figure: \( NO \) should be equal to \( IH \). Wait, maybe I messed up the vertex order. Let's list the angles:

First quadrilateral (let's call it Quad1):
- Angles: \( 96^\circ \) (I), \( 83^\circ \) (K), \( 47^\circ \) (H), \( 134^\circ \) (J) [since \( 96 + 83 + 47 + 134 = 360 \)]
- Sides: \( IH = 44 \) mm (between I and H), \( JK \) (between J and K), \( KH \) (between K and H), \( IJ \) (between I and J)

Second quadrilateral (Quad2):
- Angles: \( 47^\circ \) (N), \( 96^\circ \) (O), \( 134^\circ \) (P), \( 83^\circ \) (Q) [since \( 47 + 96 + 134 + 83 = 360 \)]
- Sides: \( NQ = 48 \) mm (between N and Q), \( OP = 19 \) mm (between O and P), \( PQ = 21 \) mm (between P and Q), \( NO \) (between N and O)

Now, match the angles:
- \( 96^\circ \) in Quad1 (I) matches \( 96^\circ \) in Quad2 (O)
- \( 47^\circ \) in Quad1 (H) matches \( 47^\circ \) in Quad2 (N)
- \( 83^\circ \) in Quad1 (K) matches \( 83^\circ \) in Quad2 (Q)
- \( 134^\circ \) in Quad1 (J) matches \( 134^\circ \) in Quad2 (P)

So the sides between corresponding angles:
- Between \( 96^\circ \) (I) and \( 47^\circ \) (H) in Quad1: \( IH = 44 \) mm
- Between \( 96^\circ \) (O) and \( 47^\circ \) (N) in Quad2: \( NO \)

Therefore, \( NO \) corresponds to \( IH \), so \( NO = IH = 44 \) mm? Wait, no, wait \( IH \) is between I (96°) and H (47°), and \( NO \) is between N (47°) and O (96°), so the order is reversed, but in congruent figures, corresponding sides are equal regardless of order (since congruent means same shape and size, so the side between \( 96^\circ \) and \( 47^\circ \) in both should be equal). Wait, but maybe I got the side wrong. Wait, the first figure: \( IH = 44 \) mm, and the second figure: \( NO \) – let's check the other sides. Wait, the second figure has side \( OP = 19 \) mm, which is between \( 96^\circ \) (O) and \( 134^\circ \) (P), and in Quad1, the side between \( 96^\circ \) (I) and \( 134^\circ \) (J) would be \( IJ \), but we don't know that. Wait, no, maybe the side \( KH \) in Quad1: let's see, \( KH \) is between \( 83^\circ \) (K) and \( 47^\circ \) (H), and in Quad2, \( NQ \) is between \( 47^\circ \) (N) and \( 83^\circ \) (Q), so \( NQ = KH \), but \( NQ = 48 \) mm, which is not 44. Wait, I must have messed up the vertex correspondence. Let's try again.

Alternative approach: In congruent figures, corresponding sides are equal. So we need to find which side in the first figure corresponds to \( NO \) in the second.

First figure (Quad1):
- Vertices: I, J, K, H (in order)
- Angles: I=96°, J=?, K=83°, H=47°
- Sides: IH=44 mm (I to H), JK=?, KH=?, IJ=?

Second figure (Quad2):
- Vertices: N, O, P, Q (in order)
- Angles: N=47°, O=96°, P=134°, Q=83°
- Sides: NO=?, OP=19 mm (O to P), PQ=21 mm (P to Q), QN=48 mm (Q to N)

Now, let's list the angles in order:

Quad1 angles (in order): I=96°, J=134° (since 360-96-83-47=134), K=83°, H=47°

Quad2 angles (in order): N=47°, O=96°, P=134°, Q=83°

So the angle sequence of Quad1 is [96°, 134°, 83°, 47°], and Quad2 is [47°, 96°, 134°, 83°]. So to align the angle sequences, we can rotate Quad2. Let's shift Quad2's angle sequence: [47°, 96°, 134°, 83°] shifted by 1: [96°, 134°, 83°, 47°], which matches Quad1's sequence. So the vertices of Quad2, when shifted by 1, are O, P, Q, N. So Quad1 (I, J, K, H) corresponds to Quad2 (O, P, Q, N).

Therefore, the sides:

- I (Quad1) corresponds to O (Quad2)
- J (Quad1) corresponds to P (Quad2)
- K (Quad1) corresponds to Q (Quad2)
- H (Quad1) corresponds to N (Quad2)

So the side IH (I to H) in Quad1 corresponds to ON (O to N) in Quad2. Wait, IH is I to H, ON is O to N. Since I corresponds to O and H corresponds to N, then IH corresponds to ON, so IH = ON. But ON is the same as NO (since it's a side between O and N, order doesn't matter for length). So IH = 44 mm, so NO = 44 mm? Wait, but in Quad2, the side OP is 19 mm (O to P), which corresponds to IJ (I to J) in Quad1. PQ (P to Q) corresponds to JK (J to K) in Quad1. QN (Q to N) corresponds to KH (K to H) in Quad1. And ON (O to N) corresponds to IH (I to H) in Quad1. So yes, IH = 44 mm, so ON (NO) = 44 mm.

Wait, but let's check the side lengths in Quad2: OP=19, PQ=21, QN=48, and NO=?. In Quad1, the sides are IH=44, IJ=?, JK=?, KH=?. Wait, maybe I made a mistake in vertex correspondence. Alternatively, maybe the side \( NO \) corresponds to \( KH \)? No, let's use the angle-angle-side. The angle at N is 47°, angle at O is 96°, and the side between them: in Quad1, angle at H is 47°, angle at I is 96°, and the side between them is IH=44 mm. So by ASA (angle-side-angle), the side between 47° and 96° in both quadrilaterals should be equal. So in Quad2, the side between 47° (N) and 96° (O) is \( NO \), so \( NO = IH = 44 \) mm.

Answer:

44