Question

unknown measures of congruent figures
he 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 congruent triangle properties

Since \(\triangle GHJ\cong\triangle STU\), corresponding angles and sides are equal.
\(\angle J\) corresponds to \(\angle U\). In \(\triangle STU\), \(\angle U\) is equal to \(\angle T\) (isosceles triangle as \(ST = SU\)). In \(\triangle GHJ\), \(\angle H = 67^{\circ}\). The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle GHJ\), \(\angle G=\angle J\). So \(\angle G+\angle J + 67^{\circ}=180^{\circ}\), \(2\angle J=180 - 67=113^{\circ}\), \(\angle J = 56.5^{\circ}\).

Step2: Find \(\angle T\)

\(\angle T\) corresponds to \(\angle G\). From above, \(\angle G = 56.5^{\circ}\), so \(\angle T=56.5^{\circ}\)

Step3: Find \(\angle G\)

As calculated in Step1, \(\angle G = 56.5^{\circ}\)

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:

\(m\angle J = 56.5^{\circ}\), \(m\angle T=56.5^{\circ}\), \(m\angle G = 56.5^{\circ}\), \(ST = 5\mathrm{cm}\), \(HJ = 3.7\mathrm{cm}\)