Question

are the polygons below similar?

Explanation:

Step1: Identify corresponding sides

First, we need to match the sides of the two polygons. Let's list the side lengths of the first polygon (VUTW): \( VU = 4 \), \( UT = 3 \), \( TW = 4 \), \( WV = 7 \). The second polygon (JKIH): \( JK = 12 \), \( KI = 16 \), \( IH = 28 \), \( HJ = 16 \)? Wait, no, let's check the angles. The marked angles: in VUTW, angles at V, W, T, U? Wait, the first polygon is a quadrilateral? Wait, no, looking at the vertices: V, U, T, W. So sides: VU=4, UT=3, TW=4, WV=7. The second polygon: J, K, H, I? Wait, J, K, H, I? Wait, J to K is 12, K to H is 16, H to I is 28, I to J is 16? Wait, no, the first polygon: V, U, T, W (quadrilateral), sides: VU=4, UT=3, TW=4, WV=7. Second polygon: J, K, H, I (quadrilateral), sides: JK=12, KH=16, HI=28, IJ=16? Wait, no, maybe the correspondence is V->I, U->K, T->H, W->J? Wait, no, let's check the ratios. Let's pair the sides with the same angle marks. The angle at V (in VUTW) and angle at I (in JKIH) are marked, angle at W and angle at H, angle at T and angle at K, angle at U and angle at J? Wait, maybe better to list the sides in order. Let's assume the first quadrilateral has sides 4 (VU), 3 (UT), 4 (TW), 7 (WV). The second has sides 12 (JK), 16 (KH), 28 (HI), 16 (IJ)? Wait, no, maybe the sides are 12 (JK), 16 (KI), 28 (IH), 16 (HJ)? Wait, perhaps I misread. Let's look again: first polygon (left): vertices V, U, T, W. Sides: VU=4, UT=3, TW=4, WV=7. Angles: at V (marked), at U (marked), at T (marked), at W (marked). Second polygon (right): vertices J, K, I, H? Wait, J, K, H, I? Sides: JI=16, JK=12, KI=16, IH=28? Wait, no, J to I is 16, J to K is 12, K to H is 16, H to I is 28. So sides: JI=16, JK=12, KH=16, HI=28. Now, let's check the ratios of corresponding sides. Let's assume the correspondence is V->I, U->K, T->H, W->J. Then VU corresponds to IK? No, maybe VU (4) corresponds to JK (12), UT (3) corresponds to KH (16), TW (4) corresponds to HI (28), WV (7) corresponds to IJ (16)? Wait, no, that can't be. Wait, maybe the first polygon is a quadrilateral with sides 4, 3, 4, 7. The second is a quadrilateral with sides 12, 16, 28, 16? Wait, no, let's calculate the ratios. Let's take the sides with the same length in the first polygon: VU=4 and TW=4. In the second polygon, are there two sides with the same length? JK=12, KI=16, IH=28, IJ=16? Wait, KI=16 and IJ=16? So in the first polygon, two sides of length 4 (VU and TW), in the second, two sides of length 16 (KI and IJ). So maybe VU corresponds to KI (4 vs 16), TW corresponds to IJ (4 vs 16). Then the ratio would be 16/4=4. Now check the other sides: UT=3, so corresponding side should be 3*4=12. Is there a side of 12? JK=12. Then WV=7, corresponding side should be 7*4=28. IH=28. Perfect! So let's list the sides:

First polygon (VUTW) sides in order: VU=4, UT=3, TW=4, WV=7.

Second polygon (JKIH) sides in order: KI=16, JK=12, IJ=16, IH=28.

Wait, maybe the order is VU (4) -> KI (16), UT (3) -> JK (12), TW (4) -> IJ (16), WV (7) -> IH (28). Now check the ratios:

16/4 = 4, 12/3 = 4, 16/4 = 4, 28/7 = 4. All ratios are equal (4). Also, the corresponding angles (marked with the same arcs) should be equal. Since the angle marks are in the same positions, the angles are equal. For polygons to be similar, corresponding angles must be equal and corresponding sides must be in proportion. Here, all corresponding sides have a ratio of 4, and the angles (marked) are equal, so the polygons are similar.

Wait, but let me confirm the side order. If the first quadrilateral has sides 4, 3, 4, 7 (in order VU, UT, TW, WV) and the second has sides 16, 12, 16, 28 (in order KI, JK, IJ, IH), then the ratios are 16/4=4, 12/3=4, 16/4=4, 28/7=4. So all sides are in proportion (scale factor 4), and the angles (marked with the same number of arcs) are equal. Therefore, the polygons are similar.

Step2: Check angle and side conditions

For two polygons to be similar, two conditions must be met: 1) All corresponding angles are congruent, 2) All corresponding sides are proportional (same ratio).

- **Angle Condition**: The angle marks indicate that the corresponding angles (those with the same number of arcs) are equal. So the angle at V (in VUTW) corresponds to the angle at I (in JKIH), angle at U to angle at K, angle at T to angle at H, angle at W to angle at J (or whatever the correct correspondence is, but the marks show congruent angles).
- **Side Condition**: We calculated the ratios of corresponding sides. Taking the sides in order (matching the angle correspondences), the ratios are \( \frac{16}{4} = 4 \), \( \frac{12}{3} = 4 \), \( \frac{16}{4} = 4 \), \( \frac{28}{7} = 4 \). All ratios are equal, so the sides are proportional.

Since both conditions (congruent corresponding angles and proportional corresponding sides) are satisfied, the polygons are similar.

Answer:

Yes, the polygons are similar because their corresponding angles are congruent (marked with equal arcs) and their corresponding sides are proportional (all ratios are 4: \( \frac{16}{4} = 4 \), \( \frac{12}{3} = 4 \), \( \frac{16}{4} = 4 \), \( \frac{28}{7} = 4 \)).