Question

solve for x
find m∠jkm
find the measure of the exterior angle
find m∠1

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. For the first triangle with angles 55°, \(x\), and \(x + 24\), we have the equation \(55+x+(x + 24)=180\).

Step2: Simplify the equation

Combine like - terms: \(55+2x + 24=180\), which simplifies to \(2x+79 = 180\).

Step3: Solve for \(x\)

Subtract 79 from both sides: \(2x=180 - 79=101\). Then divide by 2: \(x=\frac{101}{2}=50.5\).

For the second triangle, use the exterior - angle property. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, for the triangle with interior angles \(x\) and 70° and exterior angle \((2x - 5)\), we have the equation \(x + 70=2x-5\).

Step4: Solve for \(x\) in the second triangle

Subtract \(x\) from both sides: \(70=x - 5\). Then add 5 to both sides: \(x = 75\).

For the third triangle, using the exterior - angle property, if the exterior angle is \((6x-16)\) and the non - adjacent interior angles are \((x + 8)\) and \(4x\), we have the equation \((x + 8)+4x=6x-16\).

Step5: Simplify the equation

Combine like - terms: \(5x+8=6x - 16\).

Step6: Solve for \(x\)

Subtract \(5x\) from both sides: \(8=x - 16\). Then add 16 to both sides: \(x = 24\).

For the fourth triangle, using the exterior - angle property, if the exterior angle is \((5x - 10)\) and the non - adjacent interior angles are 45° and \(3x\), we have the equation \(45+3x=5x-10\).

Step7: Simplify the equation

Subtract \(3x\) from both sides: \(45 = 2x-10\).

Step8: Solve for \(x\)

Add 10 to both sides: \(55=2x\). Then divide by 2: \(x=\frac{55}{2}=27.5\).

Answer:

For the first triangle: \(x = 50.5\); For the second triangle: \(x = 75\); For the third triangle: \(x = 24\); For the fourth triangle: \(x = 27.5\)