Question

1. how can you decide if $\triangle def \cong \triangle def$?

Explanation:

Step1: Identify vertex coordinates

$\triangle DEF$: $D(3,5)$, $E(5,1)$, $F(2,1)$
$\triangle D'E'F'$: $D'(-3,-3)$, $E'(0,2)$, $F'(-4,1)$

Step2: Calculate side lengths (distance formula)

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For $\triangle DEF$:
$DE=\sqrt{(5-3)^2+(1-5)^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}$
$EF=\sqrt{(2-5)^2+(1-1)^2}=\sqrt{9+0}=3$
$FD=\sqrt{(3-2)^2+(5-1)^2}=\sqrt{1+16}=\sqrt{17}$
For $\triangle D'E'F'$:
$D'E'=\sqrt{(0-(-3))^2+(2-(-3))^2}=\sqrt{9+25}=\sqrt{34}$
$E'F'=\sqrt{(-4-0)^2+(1-2)^2}=\sqrt{16+1}=\sqrt{17}$
$F'D'=\sqrt{(-3-(-4))^2+(-3-1)^2}=\sqrt{1+16}=\sqrt{17}$

Step3: Compare corresponding sides

No matching set of 3 equal side lengths between the two triangles.

Step4: Verify congruence (SSS rule)

Since all pairs of corresponding sides are not equal, the triangles do not satisfy the Side-Side-Side congruence criterion.

Answer:

We can use the Side-Side-Side (SSS) congruence rule: calculate the lengths of all sides of both triangles using the distance formula, then check if all corresponding sides are equal. For $\triangle DEF$ and $\triangle D'E'F'$, the side lengths do not match, so $\triangle DEF
ot\cong \triangle D'E'F'$.