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Draw the following diagram: quadrilateral ABCD inscribed in circle E. The mAB=54°, mBC=115°, and mCD=86°. Select the true statement.(1 point)

Responses

m∠B=95.5°
m angle upper B equals 95.5 degrees

mAD=115°
m Modifying above upper A upper D with arc equals 115 degrees

mADC=84.5°
m Modifying above upper A upper D upper C with arc equals 84.5 degrees

m∠D=169°

1 answer

To solve the problem, we need to analyze the given angles of the quadrilateral ABCD inscribed in circle E.

  1. Understanding the angles:

    • In a cyclic quadrilateral, the opposite angles are supplementary, meaning they add up to 180°.
    • The given angles are as follows:
      • \(m \angle A + m \angle C = 180°\) (opposite angles)
      • \(m \angle B + m \angle D = 180°\) (opposite angles)

  2. Given Angles:

    • \(m \angle A = 54°\)
    • \(m \angle B = 115°\)
    • \(m \angle C = x\) (unknown angle)
    • \(m \angle D = y\) (unknown angle)

  3. Calculating Opposite Angles: Using the supplementary property of opposite angles, we can find angles C and D.

    a. Finding angle C: \[ m \angle A + m \angle C = 180° \] \[ 54° + x = 180° \implies x = 180° - 54° = 126° \] Therefore, \(m \angle C = 126°\).

    b. Finding angle D: \[ m \angle B + m \angle D = 180° \] \[ 115° + y = 180° \implies y = 180° - 115° = 65° \] Therefore, \(m \angle D = 65°\).

Now we have:

  • \(m \angle A = 54°\)
  • \(m \angle B = 115°\)
  • \(m \angle C = 126°\)
  • \(m \angle D = 65°\)
  1. Evaluating options:
  • Option 1: \(m \angle B = 95.5°\) (False), as \(m \angle B = 115°\).
  • Option 2: \(m AD = 115°\) (False); without further information, we cannot equate the angle AD directly to the measurements given.
  • Option 3: \(m \angle ADC = 84.5°\) (False), as we calculated angles A and D.
  • Option 4: \(m \angle D = 169°\) (False), since \(m \angle D = 65°\).

None of the provided options appear to match the calculations. To ensure we follow logical course:

  • Verifying angles sum: \(54° + 115° + 126° + 65° = 360°\).

In conclusion, the true relationships in cyclic quadrilaterals were confirmed, yet the descriptions provided for responses don’t hold accurate in the context of \(ABC\) being \(D\). Please double-check if assumptions were made against stated angles vs. diagram orientation. Each calculated angle stands independently as dictated under circle properties.