Question
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 mADC=84.5° m Modifying above upper A upper D upper C with arc equals 84.5 degrees mAD=115° m Modifying above upper A upper D with arc equals 115 degrees m∠D=169°
Answers
GPT-4o mini
To solve this problem, we need to recall some properties of cyclic quadrilaterals (quadrilaterals inscribed in circles). The main property we will use is that the opposite angles of a cyclic quadrilateral are supplementary, meaning that the sum of the angles will equal 180°.
Let’s denote the angles of quadrilateral ABCD as follows:
- m∠A = a
- m∠B = b
- m∠C = c
- m∠D = d
From the data we have:
- mAB = 54° (this is the arc opposite angle C, so the angle C can be calculated as c= 54°)
- mBC = 115° (this is the arc opposite angle A, so the angle A can be calculated as a= 115°)
- mCD = 86° (this is the arc opposite angle B, so the angle B can be calculated as b= 86°)
- mDA = angle D (yet to find)
### Step 1: Calculate the angles.
Using the properties of a cyclic quadrilateral:
- Angle C + Angle A = 180°
- Angle B + Angle D = 180°
Substituting the values we have:
1. To find m∠A (which equals 115°), calculate m∠C and check the supplementary relationship:
- m∠C = 54°
So:
\[
54° + 115° = 169° \quad \text{(this confirms angles add to supplementary)}
\]
2. To find m∠B (which equals 86°), calculate m∠D:
- m∠B = 86°
Then:
\[
m∠D = 180° - m∠B = 180° - 86° = 94°
\]
### Step 2: Verify the statements.
Now we can check the provided options:
1. m∠B = 95.5° (False)
2. mADC = 84.5° (the external angle, not true either; interior angles do not equal this)
3. mAD = 115° (the length of the arc measures are not equal)
4. m∠D = 169° (False; m∠D = 94°)
So looking through the listed angles, all options provided are false or not possible based on cyclical properties of angle measures summing to 180°
Based on the work, we can firmly conclude that none of the statements provided post-calculation are correct within the scoped quadrilateral ABCD. If you have further details concerning the statements or corrections on angles, feel free to ask!
Let’s denote the angles of quadrilateral ABCD as follows:
- m∠A = a
- m∠B = b
- m∠C = c
- m∠D = d
From the data we have:
- mAB = 54° (this is the arc opposite angle C, so the angle C can be calculated as c= 54°)
- mBC = 115° (this is the arc opposite angle A, so the angle A can be calculated as a= 115°)
- mCD = 86° (this is the arc opposite angle B, so the angle B can be calculated as b= 86°)
- mDA = angle D (yet to find)
### Step 1: Calculate the angles.
Using the properties of a cyclic quadrilateral:
- Angle C + Angle A = 180°
- Angle B + Angle D = 180°
Substituting the values we have:
1. To find m∠A (which equals 115°), calculate m∠C and check the supplementary relationship:
- m∠C = 54°
So:
\[
54° + 115° = 169° \quad \text{(this confirms angles add to supplementary)}
\]
2. To find m∠B (which equals 86°), calculate m∠D:
- m∠B = 86°
Then:
\[
m∠D = 180° - m∠B = 180° - 86° = 94°
\]
### Step 2: Verify the statements.
Now we can check the provided options:
1. m∠B = 95.5° (False)
2. mADC = 84.5° (the external angle, not true either; interior angles do not equal this)
3. mAD = 115° (the length of the arc measures are not equal)
4. m∠D = 169° (False; m∠D = 94°)
So looking through the listed angles, all options provided are false or not possible based on cyclical properties of angle measures summing to 180°
Based on the work, we can firmly conclude that none of the statements provided post-calculation are correct within the scoped quadrilateral ABCD. If you have further details concerning the statements or corrections on angles, feel free to ask!