Question

1. < z =? given \\( \triangle g h j \cong \triangle x y z \\), find each missing measure. type a response

Explanation:

Step1: Analyze the properties of congruent triangles

Since \(\triangle GHJ\cong\triangle XYZ\), corresponding angles and sides are equal. In \(\triangle GHJ\), \(GH = HJ\) (marked as equal), so \(\triangle GHJ\) is isosceles. Then in \(\triangle XYZ\), \(ZX = ZY\) (corresponding to \(GH = HJ\) in \(\triangle GHJ\)).

Step2: Use the angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle XYZ\), \(\angle Y = 38^{\circ}\). Let \(\angle Z=\angle X\) (because \(ZX = ZY\)). Then \(\angle Z+\angle X+\angle Y=180^{\circ}\). Substituting \(\angle X = \angle Z\), we get \(2\angle Z+38^{\circ}=180^{\circ}\).

Step3: Solve for \(\angle Z\)

First, subtract \(38^{\circ}\) from both sides of the equation \(2\angle Z+38^{\circ}=180^{\circ}\): \(2\angle Z=180^{\circ}- 38^{\circ}=142^{\circ}\). Then divide both sides by 2: \(\angle Z=\frac{142^{\circ}}{2}=71^{\circ}\).

Answer:

\(71^{\circ}\)