DF has a length of 36 units. The triangles can be proven similar by angle-angle similarity
Explanation:
First we need to find the missing angles in each of the triangles to ascertain they have same angles:
From triangle ABC:
∠A + ∠B + ∠C = 180°
68 + 44 + ∠C = 180
112 + ∠C = 180
∠C = 180 -112 = 68°
from triangle DEF:
∠D + ∠E + ∠F = 180°
68 + ∠E + 44 = 180
∠E = 180 -(112) = 68°
The three angles in one triangle corresponds to the three angles of the second triangle
AB = 12, AC = 9
DE = 27, DF = ?
For two triangles to be similar, the ratio of the corresponding sides will be equal
∠A = ∠D, ∠B = ∠F, ∠C = ∠E
DE correponds to AC,
BC corresponds to EF
AB corresponds to DF
The ratio:
[tex]\begin{gathered} \frac{AC}{DE}\text{ = }\frac{BC}{EF}\text{ = }\frac{AB}{DF} \\ \frac{AC}{DE}=\text{ }\frac{AB}{DF}\text{ (we use this due to the values we were given)} \\ \frac{9}{27}\text{ = }\frac{12}{DF} \end{gathered}[/tex][tex]\begin{gathered} 9(DF)\text{ = 27(12)} \\ DF\text{ = }\frac{27(12)}{9} \\ DF\text{ =3}6 \end{gathered}[/tex]
Since two angles in one triangle corresponds to two angles in the second triangle, it is similar by angle-angle similarity
DF has a length of 36 units. The triangles can be proven similar by angle-angle similarity
