Question

1. describe how to prove the given triangles are congruent. use the key terms included angle and side - angle - side congruence theorem in your answer.

Explanation:

Step1: Calculate side lengths

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

• \(AC=\sqrt{(-7 + 7)^2+(7 - 3)^2}=4\)
• \(BC=\sqrt{(-2+7)^2+(3 - 3)^2}=5\)

For \(\triangle XYZ\):

• \(XZ=\sqrt{(6 - 2)^2+(-3+3)^2}=4\)
• \(YZ=\sqrt{(6 - 6)^2+(-3 + 7)^2}=4\)

Step2: Check included angles

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

Step3: Apply SAS

Since \(AC = XZ = 4\), \(\angle C=\angle Z = 90^{\circ}\), \(BC = YZ = 5\), by the Side - Angle - Side (SAS) Congruence Theorem, \(\triangle ABC\cong\triangle XYZ\).

Answer:

First, use the distance formula to find the lengths of the sides. For \(\triangle ABC\), \(AC = 4\) and \(BC = 5\). For \(\triangle XYZ\), \(XZ = 4\) and \(YZ = 5\). Then, note that \(\angle C\) and \(\angle Z\) are right angles (\(90^{\circ}\)). Since two sides and the included angle of \(\triangle ABC\) are equal to two sides and the included angle of \(\triangle XYZ\) (\(AC = XZ\), \(\angle C=\angle Z\), \(BC = YZ\)), by the Side - Angle - Side (SAS) Congruence Theorem, \(\triangle ABC\) and \(\triangle XYZ\) are congruent.