In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary, meaning that their measures add up to 180 degrees.
Given the measures of angles P and R as:
- Angle P = (x + 16)°
- Angle R = (6x - 4)°
We can set up the equation:
\[ (x + 16) + (6x - 4) = 180 \]
Simplifying this, we have:
\[ x + 16 + 6x - 4 = 180 \] \[ 7x + 12 = 180 \] \[ 7x = 180 - 12 \] \[ 7x = 168 \] \[ x = 24 \]
Now we can find the measures of angles P and R by substituting \(x\) back into their expressions:
-
Angle P: \[ P = x + 16 = 24 + 16 = 40° \]
-
Angle R: \[ R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140° \]
Now we found Angle P as 40° and Angle R as 140°.
Next, we can find the remaining angles Q and S, which must also be supplementary to angles P and R, respectively:
- Angle Q = 180° - Angle P = 180° - 40° = 140°
- Angle S = 180° - Angle R = 180° - 140° = 40°
Thus, we have:
- Angle P = 40°
- Angle Q = 140°
- Angle R = 140°
- Angle S = 40°
The smaller angles among these are 40° and 40°.
Therefore, the measure of the smaller angle is 40°.
1 answer