Question

determine if the two triangles are congruent when ( x = - 4 ). if so, write the congruence statement along with which congruence theorem proves that they are congruent.
(1 point)
( \triangle qrscong\triangle tuv ) by the sas congruence theorem
( \triangle qrscong\triangle tuv ) by the hl congruence theorem
the triangles are not congruent.
( \triangle qrscong\triangle tuv ) by the sss congruence theorem

Explanation:

Step1: Calculate the side lengths

When \(x = - 4\):

• For \(\triangle QRS\):
• \(QS=-x=-(-4) = 4\)
• \(RS=-2x + 1=-2\times(-4)+1=8 + 1=9\)
• \(QR = 6\)
• For \(\triangle TUV\):
• \(TV = 4\)
• \(UV=9\)
• \(TU=x + 10=-4+10 = 6\)

Step2: Apply SSS Congruence Theorem

Since \(QR=TU = 6\), \(QS=TV = 4\), and \(RS=UV = 9\), by the SSS (Side - Side - Side) Congruence Theorem, which states that if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent, we can conclude that \(\triangle QRS\cong\triangle TUV\)

Answer:

$\triangle QRS\cong\triangle TUV$ by the SSS Congruence Theorem