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

Question

Responses

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

m∠D=169°

Answer 1

To solve for the angles of quadrilateral ABCD inscribed in circle E, we need to use the properties of cyclic quadrilaterals. The primary property we will utilize is that opposite angles of a cyclic quadrilateral sum to 180°.

Given:

• \( m\angle A = 54° \)
• \( m\angle B = 115° \)
• \( m\angle C = 86° \)

We can calculate \( m\angle D \) using the property of cyclic quadrilaterals:

1. Finding \( m\angle D \):

   • Since \( m\angle A + m\angle C = 54° + 86° = 140° \)
   • Therefore, \( m\angle D = 180° - m\angle A + m\angle C = 180° - 140° = 40° \)

2. Checking Other Statements:

   • Statement 1: \( m\angle AD = 115° \)

     • This is incorrect since we don't have a direct relationship for that angle.

   • Statement 2: \( m\angle ADC = 84.5° \)

     • This is also incorrect as calculated, \( m\angle A + m\angle C = 140° \), \( m\angle B + m\angle D = 115° + 40° = 155° \).

   • Statement 3: \( m\angle B = 95.5° \)

     • This is incorrect, as given \( m\angle B = 115° \).

   • Statement 4: \( m\angle D = 169° \)

     • Clearly incorrect.

From the above calculations, none of the given responses are valid based on our findings that \( m\angle D = 40° \) and existing measures.

Therefore, the correct consequences based on given angles and the properties of cyclic quadrilaterals lead to the conclusion that none of the statements are true.