1. **Problem statement:** We are given a circle with a tangent line $ \overrightarrow{GH} $ touching the circle at point $ I $, and a chord $ IJ $. The angle between the tangent $ \overrightarrow{GH} $ and the chord $ IJ $ is $ 112^\circ $. We need to find the measure of the minor arc $ \overparen{IJ} $. 2. **Relevant theorem:** The angle formed between a tangent and a chord drawn from the point of tangency is half the measure of the intercepted arc. Mathematically, if $ \theta $ is the angle between the tangent and the chord, and $ m\overparen{IJ} $ is the measure of the intercepted arc, then: $$ \theta = \frac{1}{2} m\overparen{IJ} $$ 3. **Apply the formula:** Given $ \theta = 112^\circ $, substitute into the formula: $$ 112 = \frac{1}{2} m\overparen{IJ} $$ 4. **Solve for $ m\overparen{IJ} $:** Multiply both sides by 2: $$ 2 \times 112 = \cancel{2} \times \frac{1}{2} m\overparen{IJ} \Rightarrow 224 = m\overparen{IJ} $$ 5. **Interpretation:** The measure of the minor arc $ \overparen{IJ} $ is $ 224^\circ $. However, since the total circle is $ 360^\circ $, and the minor arc must be less than or equal to $ 180^\circ $, this means the arc $ \overparen{IJ} $ is actually the major arc. 6. **Find the minor arc:** The minor arc is the complement of the major arc: $$ m\overparen{IJ}_{minor} = 360 - 224 = 136^\circ $$ 7. **Final answer:** $$ m\overparen{IJ} = 136^\circ $$ This is the measure of the minor arc $ \overparen{IJ} $.