In quadrilateral PQRS, if angle PQR measures 7x + 2 and angle PSR measures 5x + 14, what are the measures of angles PQR and PSR?

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Solution to Find the Measures of Angles PQR and PSR

In a quadrilateral, the sum of all interior angles is always 360 degrees. This means we can use this property to find the unknown angles in quadrilateral PQRS.

Given:

  • Angle PQR = 7x + 2
  • Angle PSR = 5x + 14

Find the sum of the angles:

Assuming the other two angles (let’s call them QR and RS) are known or can be expressed in some way:

We can express the equation as:

    (7x + 2) + (5x + 14) + angle QR + angle RS = 360

Combining like terms:

If we suppose that the two remaining angles (QR and RS) equal some constants, we can rearrange the equation after combining.

For instance, if angle QR and angle RS are not specified, let’s temporarily denote:

    angle QR + angle RS = y

Thus, we can simplify it to:

    12x + 16 + y = 360

Solving for y:

(assuming angle QR + angle RS = 0, just for simplicity):

    12x + 16 = 360
    12x = 360 - 16
    12x = 344
    x = 344 / 12
    x = 28.67

Now calculate each angle:

1. Angle PQR:

    Angle PQR = 7x + 2 = 7(28.67) + 2 = 200.69 degrees (approx)

2. Angle PSR:

    Angle PSR = 5x + 14 = 5(28.67) + 14 = 164.35 degrees (approx)

Conclusion:

The measures of the angles in quadrilateral PQRS would then be approximately:

  • Angle PQR: 200.69 degrees
  • Angle PSR: 164.35 degrees

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