Question

△abc ~ △def. find each missing measurement.
a
6
72°
12
c
9
b
d
12
76°
e
f
a. de =
b. ef =
c. m∠d =
d. m∠e =

Explanation:

Step1: Find the scale - factor

Since $\triangle ABC\sim\triangle DEF$, and $AC = 6$, $DF=12$, the scale - factor $k=\frac{DF}{AC}=\frac{12}{6} = 2$.

Step2: Find $DE$

$AB = 12$, and since the scale - factor $k = 2$, $DE=AB\times k=12\times2 = 24$.

Step3: Find $EF$

$BC = 9$, and using the scale - factor $k = 2$, $EF=BC\times k=9\times2 = 18$.

Step4: Find $m\angle D$

Corresponding angles of similar triangles are equal. So $m\angle D=m\angle A = 72^{\circ}$.

Step5: Find $m\angle E$

The sum of angles in a triangle is $180^{\circ}$. In $\triangle ABC$, $m\angle B=180^{\circ}-(72^{\circ}+(180^{\circ}-72^{\circ}-76^{\circ}))=32^{\circ}$. Since corresponding angles of similar triangles are equal, $m\angle E=m\angle B = 32^{\circ}$.

Answer:

a. $DE = 24$
b. $EF = 18$
c. $m\angle D = 72^{\circ}$
d. $m\angle E = 32^{\circ}$