Angelo has a triangle for an art project labeled igh. he reduced the size of the triangle by a factor of 5 to fit a smaller frame and labeled that similar triangle dfe. triangle g h i. side g h is 16 inches, h i is 15 inches, g i is 10 inches. angle g is 65 degrees, h is 48 degrees, i is 67 degrees. triangle d e f. side d e is 3 inches, e f is 3.2 inches, d f is 2 inches. angle d is 67 degrees, e is 48 degrees, f is 65 degrees. select the correct similarity statement about these triangles. triangle e f d is congruent to triangle h i g triangle d e f is similar to triangle g i h triangle f e d = triangle g i h triangle d f e is similar to triangle i g h
The Correct Answer and Explanation is :
To determine the correct similarity statement about triangles ( \triangle igh ) and ( \triangle def ), we start by analyzing their properties based on the information provided.
- Given Angles:
- For triangle ( \triangle igh ):
- Angle ( g = 65^\circ )
- Angle ( h = 48^\circ )
- Angle ( i = 67^\circ )
- For triangle ( \triangle def ):
- Angle ( d = 67^\circ )
- Angle ( e = 48^\circ )
- Angle ( f = 65^\circ )
- Angle Correspondence:
From the angle measures, we can observe the following correspondence between the angles of the triangles:
- Angle ( g ) corresponds to angle ( f ) (both ( 65^\circ ))
- Angle ( h ) corresponds to angle ( e ) (both ( 48^\circ ))
- Angle ( i ) corresponds to angle ( d ) (both ( 67^\circ ))
Since all corresponding angles are equal, this confirms that triangles ( \triangle igh ) and ( \triangle def ) are similar by the Angle-Angle (AA) similarity criterion.
- Sides and Ratios:
The sides of triangle ( \triangle igh ) are ( g h = 16 ) inches, ( h i = 15 ) inches, and ( g i = 10 ) inches. The sides of triangle ( \triangle def ) are ( d e = 3 ) inches, ( e f = 3.2 ) inches, and ( d f = 2 ) inches. To check if the triangles are indeed similar, we can compute the ratios of corresponding sides:
- Ratio ( \frac{g h}{d e} = \frac{16}{3} )
- Ratio ( \frac{h i}{e f} = \frac{15}{3.2} = 4.6875 )
- Ratio ( \frac{g i}{d f} = \frac{10}{2} = 5 )
The ratios are not consistent, but the angles confirm similarity.
- Correct Statement:
The correct similarity statement is ( \triangle def \sim \triangle igh ). Therefore, we can conclude:
The correct answer is: triangle ( def ) is similar to triangle ( g i h ).
This reflects that the triangles share the same shape but not necessarily the same size, consistent with the properties of similar triangles.