Question

finding unknown measures of congruent figures
in the diagram, \\( \triangle g h j \cong \triangle s t u \\).
what are the missing measures in the diagram?
\\( m \angle j = \\)
\\( m \angle t = \\)
\\( m \angle g = \\)
\\( s t = \mathrm{cm} \\)
\\( h j = \mathrm{cm} \\)

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle GHJ\cong\triangle STU\), corresponding angles and sides are equal.

Step2: Find \(m\angle J\)

\(\angle J\) corresponds to \(\angle U\). In \(\triangle STU\), \(\angle U = 67^{\circ}\) (by the property of congruent triangles), so \(m\angle J=67^{\circ}\)

Step3: Find \(m\angle T\)

In \(\triangle GHJ\), \(\angle H = 67^{\circ}\), \(\angle G\) and \(\angle T\) are corresponding angles. Let's first find \(\angle G\). Using the angle - sum property of a triangle (\(\angle G+\angle H+\angle J = 180^{\circ}\)). Substitute \(\angle H = 67^{\circ}\) and \(\angle J = 67^{\circ}\), we get \(\angle G=180-(67 + 67)=46^{\circ}\). Since \(\angle G\) and \(\angle T\) are corresponding angles (because \(\triangle GHJ\cong\triangle STU\)), \(m\angle T = 46^{\circ}\)

Step4: Find \(ST\)

\(ST\) corresponds to \(GH\). Given \(GH = 5\mathrm{cm}\), so \(ST = 5\mathrm{cm}\)

Step5: Find \(HJ\)

\(HJ\) corresponds to \(TU\). Given \(TU=3.7\mathrm{cm}\), so \(HJ = 3.7\mathrm{cm}\)

Answer:

\(m\angle J = 67^{\circ}\), \(m\angle T=46^{\circ}\), \(ST = 5\mathrm{cm}\), \(HJ = 3.7\mathrm{cm}\)