Question

scribe how to prove the given triangles are congruent terms included angle and side-angle-side congruence your answer. scribe how to prove the given triangles are congr terms included side and angle-side-angle congru your answer.

Explanation:

Step1: Find the lengths of sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(\triangle ABC\):

• \(AC\): \(A(-6,7)\), \(C(-6,4)\). \(d=\sqrt{(-6+6)^2+(4 - 7)^2}=\sqrt{0 + 9}=3\)
• \(BC\): \(B(-2,4)\), \(C(-6,4)\). \(d=\sqrt{(-2 + 6)^2+(4 - 4)^2}=\sqrt{16+0}=4\)

For \(\triangle XYZ\):

• \(XZ\): \(X(2,-3)\), \(Z(6,-3)\). \(d=\sqrt{(6 - 2)^2+(-3+3)^2}=\sqrt{16 + 0}=4\)
• \(YZ\): \(Y(6,-9)\), \(Z(6,-3)\). \(d=\sqrt{(6 - 6)^2+(-3 + 9)^2}=\sqrt{0+36}=6\)

Step2: Check included angles

\(\angle C\) in \(\triangle ABC\) and \(\angle Z\) in \(\triangle XYZ\) are right angles (\(90^{\circ}\)).

Step3: Apply SAS

In \(\triangle ABC\) and \(\triangle ZYX\) (re - label \(\triangle XYZ\) as \(\triangle ZYX\) for correspondence):

• \(AC = YZ=3\)
• \(\angle C=\angle Z = 90^{\circ}\)
• \(BC=XZ = 4\)

Answer:

By calculating the side lengths using the distance formula (\(AC = 3\), \(BC = 4\) in \(\triangle ABC\); \(YZ=3\), \(XZ = 4\) in \(\triangle XYZ\)) and noting the right - angled included angles (\(\angle C\) and \(\angle Z\)), we can apply the Side - Angle - Side (SAS) congruence criterion to prove that \(\triangle ABC\cong\triangle ZYX\)