Question

are the polygons below similar?
options: yes, no

Explanation:

Step1: Check Angle Conditions

For a quadrilateral, the sum of interior angles is \(360^\circ\). In the first quadrilateral (STUV), we know two angles are \(133^\circ\). Let the other two angles (at S and U) be \(x\). So \(2\times133^\circ + 2x=360^\circ\), \(266^\circ+2x = 360^\circ\), \(2x=94^\circ\), \(x = 47^\circ\)? Wait, no, wait the second quadrilateral (BCDE) has angles \(49^\circ\). Wait, maybe I made a mistake. Wait, looking at the markings: the first quadrilateral (STUV) has two angles of \(133^\circ\) and the other two angles (marked with red) – let's recalculate. Wait, the second quadrilateral (BCDE) has angles \(49^\circ\) and the other two angles (marked with double red) – let's check the angle measures. Wait, maybe the first quadrilateral: angles at T and V are \(133^\circ\), so the other two angles (S and U) should be \(\frac{360 - 2\times133}{2}=\frac{360 - 266}{2}=\frac{94}{2}=47^\circ\). But the second quadrilateral has angles \(49^\circ\). Wait, no, maybe the markings: the first quadrilateral has sides all 27 (ST=27, TU=27, UV=27, VS=27? Wait, ST=27, TU=27, UV=27, VS=27? So it's a rhombus? Wait, no, ST=27, TU=27, UV=27, VS=27? So all sides equal? Wait, the second quadrilateral has sides all 18 (BE=18, BC=18, CD=18, DE=18)? So it's a rhombus. Now, check angles: in the first rhombus, angles at T and V are \(133^\circ\), so the adjacent angles (S and U) are \(180 - 133 = 47^\circ\)? But the second rhombus has angles \(49^\circ\). Wait, that's a problem. Wait, maybe the angle markings: the red single mark on S (first quadrilateral) and the \(49^\circ\) on B and D (second quadrilateral) – maybe I misread. Wait, no, looking again: the first quadrilateral (STUV) has angles at T: \(133^\circ\), V: \(133^\circ\), and the other two angles (S and U) are equal (marked with red single arc). The second quadrilateral (BCDE) has angles at B: \(49^\circ\), D: \(49^\circ\), and the other two angles (E and C) are equal (marked with red double arc). Wait, maybe the first quadrilateral's angles at S and U: let's calculate again. Sum of angles in quadrilateral: \(360^\circ\). So \(133 + 133 + \angle S + \angle U = 360\). So \(\angle S + \angle U = 360 - 266 = 94\). Since it's a quadrilateral with all sides equal (a rhombus), opposite angles are equal, so \(\angle S=\angle U = 47^\circ\). The second quadrilateral: all sides equal (a rhombus), angles at B and D are \(49^\circ\), so opposite angles are equal, and adjacent angles are \(180 - 49 = 131^\circ\)? Wait, no, the markings on the second quadrilateral: the double red arcs are at E and C, so those angles should be equal, and the single red arcs (at B and D) are \(49^\circ\). Wait, this is confusing. Wait, maybe the problem is that the angle measures don't match. Wait, but maybe I made a mistake. Wait, the first quadrilateral: sides are all 27, so it's a rhombus with two angles \(133^\circ\) and two angles \(47^\circ\) (since \(133 + 133 + 47 + 47 = 360\)). The second quadrilateral: sides are all 18, so it's a rhombus with two angles \(49^\circ\) and two angles \(131^\circ\) (since \(49 + 49 + 131 + 131 = 360\)). Wait, but the angle measures don't match. Wait, but maybe the markings are different. Wait, the first quadrilateral: the red single arcs are at S and U, and the second quadrilateral's red single arcs are at B and D (49°), and double red at E and C. Wait, maybe the first quadrilateral's angles at S and U are 49°? Wait, no, \(133 + 133 + 49 + 49 = 133*2 + 49*2 = 266 + 98 = 364 eq 360\). Oh, I see my mistake! Wait, the sum of interior angles of a quadrilateral is \(360^\circ\). So for the first quadrilateral (STUV): angles at T and V are \(133^\circ\), so the other two angles (S and U) must satisfy \(133 + 133 + \angle S + \angle U = 360\), so \(\angle S + \angle U = 360 - 266 = 94\). If the quadrilateral is a rhombus (all sides equal), then opposite angles are equal, so \(\angle S = \angle U = 47^\circ\). The second quadrilateral (BCDE): angles at B and D are \(49^\circ\), so \(\angle B + \angle D + \angle E + \angle C = 360\), so \(\angle E + \angle C = 360 - 98 = 262\), so if it's a rhombus (all sides equal), opposite angles are equal, so \(\angle E = \angle C = 131^\circ\). Now, check the angle correspondence: the first quadrilateral has angles \(133^\circ\) and \(47^\circ\), the second has \(49^\circ\) and \(131^\circ\). These angle measures don't match. Wait, but maybe I misread the angle markings. Wait, looking at the diagram again: the first quadrilateral (STUV) – the red single arcs are at S and U, and the second quadrilateral (BCDE) – the red single arcs are at B and D (49°), and the double red arcs are at E and C. Wait, maybe the first quadrilateral's angles at S and U are 49°? Wait, no, \(133*2 + 49*2 = 266 + 98 = 364 eq 360\). That's not possible. Wait, maybe the first quadrilateral is not a rhombus? Wait, ST=27, TU=27, UV=27, VS=27? So all sides equal, so it's a rhombus. The second quadrilateral: BE=18, BC=18, CD=18, DE=18? All sides equal, rhombus. Now, in a rhombus, adjacent angles are supplementary. So in the first rhombus, if angle at T is \(133^\circ\), then adjacent angle (S) is \(180 - 133 = 47^\circ\). In the second rhombus, angle at B is \(49^\circ\), so adjacent angle (E) is \(180 - 49 = 131^\circ\). Since \(47^\circ eq 49^\circ\) and \(133^\circ eq 131^\circ\), the angles are not equal. Wait, but maybe the problem is that I misread the angle values. Wait, the second quadrilateral's angles at B and D are 49°, and the first quadrilateral's angles at S and U – maybe the first quadrilateral's angles at S and U are 49°? Let's check: \(133*2 + 49*2 = 266 + 98 = 364\), which is more than 360. So that's impossible. Therefore, the angles are not equal, so the polygons are not similar? Wait, but wait, maybe the first quadrilateral: let's count the sides. ST=27, TU=27, UV=27, VS=27 – all sides equal. The second: BE=18, BC=18, CD=18, DE=18 – all sides equal. So the ratio of sides is \(27/18 = 3/2\), so the sides are in proportion. Now, check angles: in similar polygons, corresponding angles must be equal. So we need to check if the angle measures are equal. Let's recalculate the angles of the first quadrilateral correctly. Wait, maybe the first quadrilateral: angles at T and V are \(133^\circ\), so the other two angles (S and U) – let's look at the diagram again. Wait, the second quadrilateral has angles 49°, and the first has angles marked with red (maybe 49°? Wait, the red single arc in the first quadrilateral – maybe the angle is 49°? Wait, maybe I made a mistake in the sum. Wait, \(133 + 133 + 49 + 49 = 133*2 + 49*2 = 266 + 98 = 364\), which is wrong. Wait, the sum must be 360. So that can't be. Wait, maybe the first quadrilateral is not a quadrilateral? Wait, no, it's a quadrilateral. Wait, maybe the angles at S and U are 47°, and the second quadrilateral's angles at B and D are 49°, so they are not equal. Therefore, the corresponding angles are not equal, so the polygons are not similar? Wait, but wait, maybe the diagram is different. Wait, maybe the first quadrilateral: angles at T and V are 133°, and the other two angles (S and U) are 49°? Let's check: \(133*2 + 49*2 = 266 + 98 = 364\), which is 4° more than 360. So that's impossible. Therefore, the angles are not equal, so the polygons are not similar? Wait, but maybe I messed up the angle calculation. Wait, another approach: similar polygons have corresponding angles equal and corresponding sides in proportion. The sides are in proportion (27/18 = 3/2). Now check angles: in the first quadrilateral, let's find the angle at S. The quadrilateral has sides ST=27, TU=27, UV=27, VS=27 – so it's a rhombus, so opposite angles are equal. Angles at T and V are 133°, so angles at S and U are (360 - 2*133)/2 = (360 - 266)/2 = 94/2 = 47°. In the second quadrilateral, sides BE=18, BC=18, CD=18, DE=18 – rhombus, angles at B and D are 49°, so angles at E and C are (360 - 2*49)/2 = (360 - 98)/2 = 262/2 = 131°. Since 47° ≠ 49° and 133° ≠ 131°, the corresponding angles are not equal. Therefore, the polygons are not similar.

Answer:

no