In \( \triangle G H I, \angle I \cong \angle H, H I=10 \) and \( G H=18 \). Find \( I G \).

In triangle \( \triangle GHI \), since \( \angle I \cong \angle H \), it is an isosceles triangle where the sides opposite to these angles are equal. Thus, \( IG = GH = 18 \) and the remaining side \( HI \) is given as \( 10 \). You can use the properties of isosceles triangles to determine the unknown side. The sum of angles in a triangle is 180 degrees, so: \[ \angle G = 180 - (\angle H + \angle I) = 180 - (x + x) = 180 - 2x \] From the Law of Cosines or using the formula for the area of an isosceles triangle, you can establish relationships between the sides given. However, without needing complex calculations, you already know that in an isosceles triangle, the equal sides are those opposite equal angles. Thus, \( IG \) must simply equal \( GH \). Therefore, \( IG = 18 \). So, the lengths of sides are: - \( IG = 18 \) - \( GH = 18 \) - \( HI = 10 \) This completes our triangle.