wo angies in an equilaterai triangie are \( (a+b) \) and \( (2 a-10 b) \) degrees nid the values of \( a \) and \( b \). ne sides of an equilateral triangle are \( y-x) \mathrm{cm},(5 x+3) \mathrm{cm} \) and \( (2+2 y) \mathrm{cm} \). nd the length of each side.

Let's tackle both problems step by step. --- ### **Problem 1: Finding \( a \) and \( b \) in an Equilateral Triangle** **Given:** - Two angles in an equilateral triangle are \( (a + b) \) degrees and \( (2a - 10b) \) degrees. **Understanding the Problem:** - In an **equilateral triangle**, all three angles are equal, and each measures **60 degrees**. **Setting Up Equations:** 1. Since both given expressions represent angles in the triangle: \[ a + b = 60 \quad \text{(Equation 1)} \] \[ 2a - 10b = 60 \quad \text{(Equation 2)} \] **Solving the System of Equations:** 1. **From Equation 1:** \[ a = 60 - b \] 2. **Substitute \( a \) into Equation 2:** \[ 2(60 - b) - 10b = 60 \] \[ 120 - 2b - 10b = 60 \] \[ 120 - 12b = 60 \] \[ -12b = -60 \] \[ b = 5 \] 3. **Find \( a \) using Equation 1:** \[ a = 60 - 5 = 55 \] **Solution:** \[ a = 55^\circ \quad \text{and} \quad b = 5^\circ \] --- ### **Problem 2: Determining the Lengths of Sides in an Equilateral Triangle** **Given:** - The sides of an equilateral triangle are: \[ y - x \ \text{cm}, \quad 5x + 3 \ \text{cm}, \quad \text{and} \quad 2 + 2y \ \text{cm} \] **Understanding the Problem:** - In an **equilateral triangle**, all three sides are equal in length. **Setting Up Equations:** 1. Set the first side equal to the second: \[ y - x = 5x + 3 \quad \text{(Equation 3)} \] 2. Set the first side equal to the third: \[ y - x = 2 + 2y \quad \text{(Equation 4)} \] **Solving the System of Equations:** 1. **From Equation 3:** \[ y - x = 5x + 3 \] \[ y = 6x + 3 \quad \text{(Equation 5)} \] 2. **From Equation 4:** \[ y - x = 2 + 2y \] \[ -x - y = 2 \] \[ x + y = -2 \quad \text{(Equation 6)} \] 3. **Substitute Equation 5 into Equation 6:** \[ x + (6x + 3) = -2 \] \[ 7x + 3 = -2 \] \[ 7x = -5 \] \[ x = -\frac{5}{7} \] 4. **Find \( y \) using Equation 5:** \[ y = 6\left(-\frac{5}{7}\right) + 3 = -\frac{30}{7} + \frac{21}{7} = -\frac{9}{7} \] **Calculating the Length of Each Side:** Using \( y - x \): \[ y - x = \left(-\frac{9}{7}\right) - \left(-\frac{5}{7}\right) = -\frac{4}{7} \ \text{cm} \] Since side lengths cannot be negative, we take the absolute value: \[ \text{Length of each side} = \frac{4}{7} \ \text{cm} \] **Solution:** \[ x = -\frac{5}{7}, \quad y = -\frac{9}{7}, \quad \text{and each side is} \ \frac{4}{7} \ \text{cm long} \] --- ### **Final Answers:** 1. **For the angles:** \[ a = 55^\circ \quad \text{and} \quad b = 5^\circ \] 2. **For the sides of the equilateral triangle:** \[ x = -\frac{5}{7}, \quad y = -\frac{9}{7}, \quad \text{and each side measures} \ \frac{4}{7} \ \text{cm} \]