Question

in the diagram, ( \triangle ghj cong \triangle stu ).
what are the missing measures in the diagram?
( mangle j=square^{circ} )
( mangle t=square^{circ} )
( mangle g=square^{circ} )
( st=square mathrm{cm} )
( hj=square mathrm{cm} )

Explanation:

Step1: Use the property of congruent triangles

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

Step2: Find \(m\angle J\)

In \(\triangle STU\), two sides \(ST = SU\) (marked as equal), so it is an isosceles triangle. In \(\triangle GHJ\), two sides \(GH = GJ\) (marked as equal). \(\angle H = 67^{\circ}\), and in an isosceles triangle \(\triangle GHJ\), \(\angle J=\angle H = 67^{\circ}\) (base angles of isosceles triangle).

Step3: Find \(m\angle T\)

In \(\triangle STU\), \(\angle T=\angle U\). In \(\triangle GHJ\), sum of angles in a triangle is \(180^{\circ}\). \(\angle G+\angle H+\angle J = 180^{\circ}\). Since \(\angle H=\angle J = 67^{\circ}\), \(\angle G=180-(67 + 67)=46^{\circ}\). \(\angle T=\angle G = 46^{\circ}\) (corresponding angles of congruent triangles).

Step4: Find \(m\angle G\)

\(\angle G = 46^{\circ}\) (calculated above).

Step5: Find \(ST\)

\(ST = GH\). Given \(GH = 5\mathrm{cm}\), so \(ST = 5\mathrm{cm}\) (corresponding sides of congruent triangles).

Step6: Find \(HJ\)

\(HJ=TU\). Given \(TU = 3.7\mathrm{cm}\), so \(HJ = 3.7\mathrm{cm}\) (corresponding sides of congruent triangles).

Answer:

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