Question

For ΔABC, ∠A = 3x - 8, ∠B = 5x - 6, and ∠C = 4x + 2. If ΔABC undergoes a dilation by a scale factor of 1/2 to create ΔA'B'C' with ∠A' = 2x + 8, ∠B' = 90 - x, and ∠C' = 5x - 14, which confirms that ΔABC∼ΔA'B'C by the AA criterion?

A) ∠A = ∠A' = 37° and ∠B = ∠B' = 69°
B) ∠A = ∠A' = 22° and ∠C = ∠C' = 42°
C) ∠B = ∠B' = 37° and ∠C = ∠C' = 33°
D) ∠B = ∠B' = 74° and ∠C = ∠C' = 66°

Answer 1

D. ∠B = ∠B' = 74° and ∠C = ∠C' = 66°

Explanation:

Given,

`\angle A=3x-8\\\angle B=5x-6\\\angle C=4x+2`

We know,sum of all 3 angles of a triangle is
`180`°

Doing so ,

`\angle A+\angle B+\angle C=180\\3x-8+5x-6+4x+2=180`

Simplifying all like terms,

`12x-12=180\\12x-12+12=180+12\\12x=192`

Dividing both side by 12

`x=(192)/(12)`

`x=16`

Now we plug this in each angles and find their exact values.

`\angle A=3x-8=(3*16)-8=48-8=40`°

`\angle B=5x-6=(5*16)-6=80-6 =74`°

`\angle C=4x+2 = (4*16)+2=64+2=66`°

Now substituting the
`x=16` in angles of dilated triangle.

`\angle A'=(2*16)+8=32+8= 40`°

`\angle B'=90-16=74`°

`\angle C'=(5*16)-14=80-14=66`°

We see that all the corresponding angles are equal and as far as the options are concerned only option (D) matches.