<math><mstyle displaystyle="true"><mtable class="shapeTable" columnlines="solid" rowlines="solid"><mtr><mtd><mi>Side</mi></mtd><mtd><mi>Angle</mi></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow><mi>b</mi><mo>=</mo></mrow></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mrow><mi>c</mi><mo>=</mo></mrow></mtd><mtd><mn>13</mn></mtd></mtr><mtr><mtd><mrow><mi>a</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr></mtable></mtd><mtd><mtable><mtr><mtd><mrow><mi>A</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd><mrow><mi>B</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd><mrow><mi>C</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr></mtable></mtd></mtr></mtable></mstyle></math>

Assume that angle <math><mstyle displaystyle="true"><mi>C</mi><mo>=</mo><mn>90</mn></mstyle></math> .

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

Solve the equation for <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

Substitute the actual values into the equation.

Raise <math><mstyle displaystyle="true"><mn>13</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>169</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>144</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>12</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical, assuming positive real numbers.

The angle <math><mstyle displaystyle="true"><mi>B</mi></mstyle></math> can be found using the inverse sine function.

Substitute in the values of the opposite side to angle <math><mstyle displaystyle="true"><mi>B</mi></mstyle></math> and hypotenuse <math><mstyle displaystyle="true"><mn>13</mn></mstyle></math> of the triangle.

Evaluate <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>13</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

The sum of all the angles in a triangle is <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> degrees.

Solve the equation for <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>22.61986494</mn></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> to the right side of the equation.

Subtract <math><mstyle displaystyle="true"><mn>112.61986494</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>112.61986494</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

These are the results for all angles and sides for the given triangle.

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