Answer:
[tex]x=-16\text{ or } x=7[/tex]
Step-by-step explanation:
Since ΔABC is mapped onto ΔDEF, we can write that:
[tex]\Delta ABC\cong \Delta DE F[/tex]
By CPCTC:
[tex]\angle A\cong \angle D[/tex]
And since ΔABC is isosceles with Vertex C:
[tex]\angle A \cong \angle B[/tex]
We are given that:
[tex]m\angle D=34[/tex]
Hence:
[tex]m\angle A=34=m\angle B[/tex]
We are also given that:
[tex]m\angle C=x^2+9x[/tex]
The interior angles of a triangle must sum to 180°. Thus:
[tex]m\angle A+m\angle B+m\angle C=180[/tex]
Substitute:
[tex](34)+(34)+(x^2+9x)=180[/tex]
Simplify:
[tex]68+x^2+9x=180[/tex]
Isolate the equation:
[tex]x^2+9x-112=0[/tex]
Factor:
[tex](x+16)(x-7)=0[/tex]
Zero Product Property:
[tex]x+16=0\text{ or } x-7=0[/tex]
Solve for each case:
[tex]x=-16\text{ or } x=7[/tex]
Testing the solutions, we can see that both yields C = 112°.
Hence, our solutions are:
[tex]x=-16\text{ or } x=7[/tex]
