Question

if ( ig = 20 ), ( gh = 19 ), and ( jk = 39 ), find the length of ( overline{lj} ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Find the third angle of each triangle

In \(\triangle IGH\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), the third angle \(\angle H=180^{\circ}-(60^{\circ}+ 53^{\circ}) = 67^{\circ}\).
In \(\triangle LJK\), using the angle - sum property of a triangle (\(A + B + C = 180^{\circ}\)), the third angle \(\angle J=180^{\circ}-(60^{\circ}+67^{\circ})=53^{\circ}\).

Step2: Prove the similarity of triangles

Since \(\angle I=\angle L = 60^{\circ}\), \(\angle G=\angle J = 53^{\circ}\), and \(\angle H=\angle K = 67^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle IGH\sim\triangle LJK\).

Step3: Set up the proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{IG}{LJ}=\frac{GH}{JK}\).
We know that \(IG = 20\), \(GH = 19\), and \(JK = 39\). Substituting these values into the proportion \(\frac{20}{LJ}=\frac{19}{39}\).

Step4: Solve for \(LJ\)

Cross - multiply: \(19\times LJ=20\times39\). Then \(LJ=\frac{20\times39}{19}\).
\(LJ=\frac{780}{19}\approx41.1\).

Answer:

\(41.1\)