1. **State the problem:** We need to find the measure of angle $\angle BHD$ given two expressions for angles formed by a transversal crossing two parallel lines. 2. **Given:** - $m\angle BHD = (14x + 21)^\circ$ - $m\angle IHG = (87 - 2x)^\circ$ 3. **Important rule:** Since $BD$ and $FG$ are parallel lines and $BHI$ is a transversal, the angles $\angle BHD$ and $\angle IHG$ are alternate interior angles and therefore congruent. 4. **Set up the equation:** $$14x + 21 = 87 - 2x$$ 5. **Solve for $x$:** $$14x + 2x = 87 - 21$$ $$16x = 66$$ $$x = \frac{66}{16} = 4.125$$ 6. **Find $m\angle BHD$ by substituting $x$ back:** $$m\angle BHD = 14(4.125) + 21 = 57.75 + 21 = 78.75^\circ$$ **Final answer:** $$m\angle BHD = 78.75^\circ$$