1. **State the problem:** We are given a rectangle MNOL with sides labeled as follows: top side MN = 26, bottom side LO = 26, left side ML = 5a - 7, right side NO = 2a + 8, and angle at L is 24°. We need to find the measure of angle NMO. 2. **Understand the rectangle properties:** In a rectangle, opposite sides are equal and all angles are 90°. Since ML and NO are opposite sides, they must be equal: $$5a - 7 = 2a + 8$$ 3. **Solve for a:** $$5a - 7 = 2a + 8$$ $$5a - \cancel{7} + \cancel{7} = 2a + 8 + 7$$ $$5a = 2a + 15$$ $$5a - 2a = 15$$ $$3a = 15$$ $$a = \frac{15}{3} = 5$$ 4. **Calculate the side lengths:** Left side ML: $$5a - 7 = 5 \times 5 - 7 = 25 - 7 = 18$$ Right side NO: $$2a + 8 = 2 \times 5 + 8 = 10 + 8 = 18$$ So both vertical sides are 18 units. 5. **Analyze the angle NMO:** Since MNOL is a rectangle, angle M is 90°. Point O is diagonally opposite M. The angle NMO is the angle at M formed by points N and O. 6. **Use coordinates to find angle NMO:** Place rectangle MNOL on coordinate plane: - Let M at origin (0,0) - N at (26,0) since MN = 26 - L at (0,18) since ML = 18 - O at (26,18) Vectors: - Vector MN = N - M = (26,0) - (0,0) = (26,0) - Vector MO = O - M = (26,18) - (0,0) = (26,18) 7. **Calculate angle between vectors MN and MO:** Formula for angle $\theta$ between vectors $\vec{u}$ and $\vec{v}$: $$\cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}$$ Calculate dot product: $$\vec{MN} \cdot \vec{MO} = 26 \times 26 + 0 \times 18 = 676$$ Calculate magnitudes: $$|\vec{MN}| = \sqrt{26^2 + 0^2} = 26$$ $$|\vec{MO}| = \sqrt{26^2 + 18^2} = \sqrt{676 + 324} = \sqrt{1000} = 10\sqrt{10}$$ Calculate cosine: $$\cos \theta = \frac{676}{26 \times 10\sqrt{10}} = \frac{676}{260\sqrt{10}} = \frac{676}{260\sqrt{10}}$$ Simplify numerator and denominator: $$\frac{676}{260} = \frac{676 \div 4}{260 \div 4} = \frac{169}{65}$$ So: $$\cos \theta = \frac{169}{65\sqrt{10}}$$ Rationalize denominator: $$\cos \theta = \frac{169}{65\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{169\sqrt{10}}{65 \times 10} = \frac{169\sqrt{10}}{650}$$ Simplify fraction: $$\frac{169}{650} = \frac{169 \div 13}{650 \div 13} = \frac{13}{50}$$ So: $$\cos \theta = \frac{13\sqrt{10}}{50}$$ 8. **Calculate angle $\theta$:** $$\theta = \cos^{-1} \left( \frac{13\sqrt{10}}{50} \right)$$ Numerical approximation: $$\sqrt{10} \approx 3.1623$$ $$\frac{13 \times 3.1623}{50} = \frac{41.11}{50} = 0.8222$$ $$\theta \approx \cos^{-1}(0.8222) \approx 35.1^\circ$$ **Final answer:** $$\boxed{35.1^\circ}$$ This is the measure of angle NMO.