Question
draw the following diagram: quadrilateral ABCD inscribed in circle E. The mAB = 54 degrees, mBC = 115 degrees, and mCD = 86 degrees. select the true statement
m angleB = 95.5 degrees
m angleD = 169 degrees
mAD = 115 degrees
mADC = 84.5 degrees
m angleB = 95.5 degrees
m angleD = 169 degrees
mAD = 115 degrees
mADC = 84.5 degrees
Answers
GPT-4o mini
To solve the problem, we can use the properties of a cyclic quadrilateral (a quadrilateral inscribed in a circle). One key property is that the opposite angles in a cyclic quadrilateral are supplementary, meaning that their measures add up to 180 degrees.
Given:
- m∠AB = 54°
- m∠BC = 115°
- m∠CD = 86°
We need to find:
- m∠B (which is the same as m∠ABC)
- m∠D (which is the same as m∠ADC)
- mAD (the length of side AD, but we are not explicitly given angles that provide enough information to calculate this without side lengths)
- m∠ADC
### Step 1: Calculate m∠B
Using the property that opposite angles in a cyclic quadrilateral are supplementary:
1. **Finding m∠B**:
\[
m\angle B + m\angle D = 180°
\]
Since we need to compute m∠B, we can find m∠D later.
### Step 2: Calculate m∠D
Now we’ll find m∠D using:
1. **Calculate m∠D**:
\[
m\angle AB + m\angle CD + m\angle BC + m\angle AD = 360°
\]
Thus using angles AB, BC, and CD known:
1.
\[
m\angle AD = 360° - (m\angle AB + m\angle BC + m\angle CD)
\]
\[
m\angle AD = 360° - (54° + 115° + 86°)
\]
2. Calculate:
\[
m\angle AD = 360° - 255° = 105°
\]
### Step 3: Now Calculate m∠B:
Given that m∠B + m∠D = 180°:
\[
m\angle D = 180° - m\angle AB = 180° - 54° = 126°
\]
Since,
\[
m\angle D = 105°
\]
and since we're looking for m∠B using
\[
m\angle ABC + 105° = 180°
\]
Thus,
\[
m\angle ABC = 180° - 105° = 75°
\]
### Analysis of Choices:
1. \( m \angle B = 75° \) **(not 95.5°)**
2. \( m \angle D = 105° \) **(not 169°)**
3. \( m AD = 105° \) **(this can’t be calculated directly without lengths)**
4. \( m \angle ADC = 105° \) **(not 84.5°)**
Based on cyclical quadrilateral properties and the information given above, none of the statements are true based on m figures calculated, m angle B would be correct as 75 but none match provided choices.
**Final True Statement**: None of the original choices are true based on the calculations we've done.
Given:
- m∠AB = 54°
- m∠BC = 115°
- m∠CD = 86°
We need to find:
- m∠B (which is the same as m∠ABC)
- m∠D (which is the same as m∠ADC)
- mAD (the length of side AD, but we are not explicitly given angles that provide enough information to calculate this without side lengths)
- m∠ADC
### Step 1: Calculate m∠B
Using the property that opposite angles in a cyclic quadrilateral are supplementary:
1. **Finding m∠B**:
\[
m\angle B + m\angle D = 180°
\]
Since we need to compute m∠B, we can find m∠D later.
### Step 2: Calculate m∠D
Now we’ll find m∠D using:
1. **Calculate m∠D**:
\[
m\angle AB + m\angle CD + m\angle BC + m\angle AD = 360°
\]
Thus using angles AB, BC, and CD known:
1.
\[
m\angle AD = 360° - (m\angle AB + m\angle BC + m\angle CD)
\]
\[
m\angle AD = 360° - (54° + 115° + 86°)
\]
2. Calculate:
\[
m\angle AD = 360° - 255° = 105°
\]
### Step 3: Now Calculate m∠B:
Given that m∠B + m∠D = 180°:
\[
m\angle D = 180° - m\angle AB = 180° - 54° = 126°
\]
Since,
\[
m\angle D = 105°
\]
and since we're looking for m∠B using
\[
m\angle ABC + 105° = 180°
\]
Thus,
\[
m\angle ABC = 180° - 105° = 75°
\]
### Analysis of Choices:
1. \( m \angle B = 75° \) **(not 95.5°)**
2. \( m \angle D = 105° \) **(not 169°)**
3. \( m AD = 105° \) **(this can’t be calculated directly without lengths)**
4. \( m \angle ADC = 105° \) **(not 84.5°)**
Based on cyclical quadrilateral properties and the information given above, none of the statements are true based on m figures calculated, m angle B would be correct as 75 but none match provided choices.
**Final True Statement**: None of the original choices are true based on the calculations we've done.