Question

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Explanation:

1. First, note the property of an isosceles - triangle:

• In the given triangle, since two sides are marked equal, the angles opposite those sides are equal. So, the angle with measure \((2x + 11)^{\circ}\) and the angle with measure \((3x-11)^{\circ}\) are equal.
• Set up the equation based on the equality of these angles:
• \(2x + 11=3x - 11\).

1. Then, solve the equation for \(x\):

• Subtract \(2x\) from both sides of the equation:
• \(2x+11-2x=3x - 11-2x\).
• \(11=x - 11\).
• Add 11 to both sides of the equation:
• \(11 + 11=x-11 + 11\).
• \(x = 22\).

1. Next, use the angle - sum property of a triangle:

• The sum of the interior angles of a triangle is \(180^{\circ}\).
• Substitute \(x = 22\) into the expressions for the angles:
• The first two angles are \(2x + 11=2\times22+11=44 + 11=55^{\circ}\) and \(3x - 11=3\times22-11=66 - 11=55^{\circ}\).
• Let the third angle be \(2y^{\circ}\).
• Then, \(55+55 + 2y=180\).
• Combine like - terms: \(110+2y=180\).
• Subtract 110 from both sides: \(110+2y-110=180 - 110\).
• \(2y=70\).
• Divide both sides by 2: \(y = 35\).

Step1: Set equal angles equation

\(2x + 11=3x - 11\)

Step2: Solve for \(x\)

\(x=22\)

Step3: Use angle - sum property

\(55+55 + 2y=180\)

Step4: Solve for \(y\)

\(y = 35\)

Answer:

\(x = 22,y = 35\)