Question

determine if triangle fgh and triangle ijk are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
triangle fgh: h to g is 16, h to f is 14, angle at h is 82 degrees.
triangle ijk: k to j is 64, k to i is 56, angle at k is 82 degrees, angle at i is 53 degrees.
answer attempt 1 out of 2
the triangles are similar because

Explanation:

Step1: Check Angles for Triangle FGH

In triangle \( FGH \), we know one angle is \( 82^\circ \) (at \( H \)). The sum of angles in a triangle is \( 180^\circ \). Let's find the third angle. Wait, first, let's check the sides and angles of triangle \( IJK \). In triangle \( IJK \), angles are \( 82^\circ \) (at \( K \)), \( 53^\circ \) (at \( I \)), so the third angle at \( J \) is \( 180 - 82 - 53 = 45^\circ \)? Wait, no, wait. Wait, in triangle \( FGH \), sides \( FH = 14 \), \( HG = 16 \), angle at \( H \) is \( 82^\circ \). In triangle \( IJK \), sides \( KI = 56 \), \( KJ = 64 \), angle at \( K \) is \( 82^\circ \). Let's check the ratios of sides. \( \frac{KI}{FH} = \frac{56}{14} = 4 \), \( \frac{KJ}{HG} = \frac{64}{16} = 4 \). So the sides around the \( 82^\circ \) angle are in proportion (ratio 4), and the included angle is equal (\( 82^\circ = 82^\circ \)). So by SAS (Side - Angle - Side) similarity criterion, the triangles are similar. Also, let's check angles. In triangle \( FGH \), angle at \( H \) is \( 82^\circ \), sides \( FH = 14 \), \( HG = 16 \). In triangle \( IJK \), angle at \( K \) is \( 82^\circ \), sides \( KI = 56 \), \( KJ = 64 \). The ratio of \( KI/FH = 56/14 = 4 \), \( KJ/HG = 64/16 = 4 \). So SAS similarity: if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar. Also, let's check the third angle. In triangle \( FGH \), angle at \( F \): sum of angles is \( 180 \). Let's calculate angle at \( F \): \( 180 - 82 - \) angle at \( G \). Wait, no, in triangle \( IJK \), angle at \( I \) is \( 53^\circ \), angle at \( K \) is \( 82^\circ \), so angle at \( J \) is \( 180 - 82 - 53 = 45^\circ \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, no, let's recalculate. Wait, in triangle \( FGH \), sides \( FH = 14 \), \( HG = 16 \), angle at \( H \) is \( 82^\circ \). In triangle \( IJK \), sides \( KI = 56 \), \( KJ = 64 \), angle at \( K \) is \( 82^\circ \). The ratio of \( KI/FH = 56/14 = 4 \), \( KJ/HG = 64/16 = 4 \). So the sides adjacent to the \( 82^\circ \) angle are in proportion, and the included angle is equal. So by SAS similarity, the triangles are similar. Also, let's check the angles. In triangle \( FGH \), let's find angle at \( F \). Wait, maybe the angle at \( I \) is \( 53^\circ \), and in triangle \( FGH \), angle at \( F \): let's see, the sides \( FH = 14 \), \( HG = 16 \), angle at \( H = 82^\circ \). In triangle \( IJK \), sides \( KI = 56 \), \( KJ = 64 \), angle at \( K = 82^\circ \), angle at \( I = 53^\circ \). Let's check angle at \( F \): using the Law of Sines in triangle \( FGH \): \( \frac{\sin F}{HG} = \frac{\sin H}{FG} \). Wait, maybe easier: since the sides around the included angle are proportional (SAS) and the included angle is equal, so SAS similarity. So the triangles are similar by SAS similarity (or also, we can check angles: in triangle \( FGH \), angle at \( H = 82^\circ \), sides \( FH = 14 \), \( HG = 16 \); in triangle \( IJK \), angle at \( K = 82^\circ \), sides \( KI = 56 \), \( KJ = 64 \), ratio \( 56/14 = 4 \), \( 64/16 = 4 \), so SAS similarity). Also, let's check the angles. In triangle \( IJK \), angle at \( I = 53^\circ \), angle at \( K = 82^\circ \), so angle at \( J = 180 - 82 - 53 = 45^\circ \)? Wait, no, that can't be. Wait, maybe I miscalculated. Wait, \( 82 + 53 = 135 \), \( 180 - 135 = 45 \). In triangle \( FGH \), angle at \( H = 82^\circ \), sides \( FH = 14 \), \( HG = 16 \). Let's find angle at \( F \): using Law of Sines: \( \frac{\sin F}{HG} = \frac{\sin H}{FG} \). But \( FG \) is unknown. Wait, but the sides \( KI = 56 \), \( KJ = 64 \), \( FH = 14 \), \( HG = 16 \). \( 56/14 = 4 \), \( 64/16 = 4 \), so the sides are proportional, included angle equal, so SAS similarity. So the triangles are similar by SAS (Side - Angle - Side) similarity criterion. Also, we can check the angles. In triangle \( FGH \), angle at \( F \): let's calculate. Wait, maybe the angle at \( I \) is \( 53^\circ \), and in triangle \( FGH \), angle at \( F \) should be equal to angle at \( I \) if they are similar. Let's check: in triangle \( IJK \), angles are \( 82^\circ \), \( 53^\circ \), and \( 45^\circ \)? No, that can't be. Wait, no, I think I made a mistake. Wait, \( 82 + 53 = 135 \), \( 180 - 135 = 45 \). But in triangle \( FGH \), angle at \( H = 82^\circ \), sides \( FH = 14 \), \( HG = 16 \). Let's find angle at \( F \): using Law of Cosines on triangle \( FGH \): \( FG^2 = FH^2 + HG^2 - 2 \times FH \times HG \times \cos(82^\circ) \). \( FG^2 = 14^2 + 16^2 - 2 \times 14 \times 16 \times \cos(82^\circ) \). \( 14^2 = 196 \), \( 16^2 = 256 \), sum is \( 452 \). \( 2 \times 14 \times 16 = 448 \), \( \cos(82^\circ) \approx 0.1392 \), so \( 448 \times 0.1392 \approx 62.36 \). So \( FG^2 \approx 452 - 62.36 = 389.64 \), \( FG \approx 19.74 \). In triangle \( IJK \), \( KI = 56 \), \( KJ = 64 \), angle at \( K = 82^\circ \), so \( IJ^2 = KI^2 + KJ^2 - 2 \times KI \times KJ \times \cos(82^\circ) \). \( 56^2 = 3136 \), \( 64^2 = 4096 \), sum is \( 7232 \). \( 2 \times 56 \times 64 = 7168 \), \( 7168 \times 0.1392 \approx 997.78 \). So \( IJ^2 \approx 7232 - 997.78 = 6234.22 \), \( IJ \approx 78.96 \). Now, check the ratio of \( IJ/FG \approx 78.96 / 19.74 \approx 4 \), which matches the ratio of the other sides (4). So the sides are in proportion (14:56 = 1:4, 16:64 = 1:4, 19.74:78.96 = 1:4), and the included angle is equal (82°), so by SAS similarity, the triangles are similar. Also, the angles: in triangle \( IJK \), angle at \( I = 53^\circ \), let's check angle at \( F \) using Law of Sines. In triangle \( FGH \), \( \frac{\sin F}{HG} = \frac{\sin H}{FG} \). \( \sin F = \frac{HG \times \sin H}{FG} = \frac{16 \times \sin(82^\circ)}{19.74} \). \( \sin(82^\circ) \approx 0.9903 \), so \( 16 \times 0.9903 \approx 15.8448 \), \( 15.8448 / 19.74 \approx 0.8027 \), so \( F \approx \arcsin(0.8027) \approx 53.3^\circ \), which is approximately \( 53^\circ \), matching angle at \( I \). So the angles are equal (AA similarity also holds, since two angles would be equal: 82° and ~53°). So the triangles are similar by SAS (or AA) similarity.

Step2: Conclusion

Since two sides of triangle \( IJK \) are proportional to two sides of triangle \( FGH \) (ratio 4) and the included angle (82°) is equal, by the SAS (Side - Angle - Side) similarity criterion, the triangles are similar. Also, the angles can be checked to be equal (AA similarity), confirming similarity.

Answer:

The triangles are similar because two sides of one triangle are proportional to two sides of the other triangle (ratio \( \frac{56}{14} = \frac{64}{16} = 4 \)) and the included angle (\( 82^\circ \)) is equal, so by the SAS (Side - Angle - Side) similarity criterion (or also by AA similarity as the corresponding angles are equal).