Question
Opposite angles of an inscribed quadrilateral can be proven to be(1 point)
Responses
supplementary.
supplementary.
congruent.
congruent.
complementary.
complementary.
acute.Question
Quadrilateral AEIU
is inscribed in circle O
. Given m∠A=27°
and m∠E=2(m∠A)
, find m∠I
and m∠U
.(1 point)
Responses
m∠I=126°
and m∠U=153°
m angle upper I equals 126 degrees and m angle upper U equals 153 degrees
m∠I=153°
and m∠U=153°
m angle upper I equals 153 degrees and m angle upper U equals 153 degrees
m∠I=126°
and m∠U=126°
m angle upper I equals 126 degrees and m angle upper U equals 126 degrees
m∠I=153°
and m∠U=126°
Identify the true statement about inscribed angles and their intercepted arc.(1 point)
Responses
The intercepted arc is half the measure of the inscribed angle.
The intercepted arc is half the measure of the inscribed angle.
The inscribed angle is twice the measure of its intercepted arc.
The inscribed angle is twice the measure of its intercepted arc.
The intercepted arc is twice the measure of the inscribed angle.
The intercepted arc is twice the measure of the inscribed angle.
The intercepted arc is equal in measure to its inscribed angle.
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
mAD=115°
m Modifying above upper A upper D with arc equals 115 degrees
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
m∠D=169°
Question
Quadrilateral PQRS
is inscribed in circle O
. Opposite angles P
and R
measure (x+16)°
and (6x−4)°
, respectively. Determine the measure of the smaller angle.(1 point)
Responses
24°
24°
56°
56°
20°
20°
40°
Responses
supplementary.
supplementary.
congruent.
congruent.
complementary.
complementary.
acute.Question
Quadrilateral AEIU
is inscribed in circle O
. Given m∠A=27°
and m∠E=2(m∠A)
, find m∠I
and m∠U
.(1 point)
Responses
m∠I=126°
and m∠U=153°
m angle upper I equals 126 degrees and m angle upper U equals 153 degrees
m∠I=153°
and m∠U=153°
m angle upper I equals 153 degrees and m angle upper U equals 153 degrees
m∠I=126°
and m∠U=126°
m angle upper I equals 126 degrees and m angle upper U equals 126 degrees
m∠I=153°
and m∠U=126°
Identify the true statement about inscribed angles and their intercepted arc.(1 point)
Responses
The intercepted arc is half the measure of the inscribed angle.
The intercepted arc is half the measure of the inscribed angle.
The inscribed angle is twice the measure of its intercepted arc.
The inscribed angle is twice the measure of its intercepted arc.
The intercepted arc is twice the measure of the inscribed angle.
The intercepted arc is twice the measure of the inscribed angle.
The intercepted arc is equal in measure to its inscribed angle.
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
mAD=115°
m Modifying above upper A upper D with arc equals 115 degrees
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
m∠D=169°
Question
Quadrilateral PQRS
is inscribed in circle O
. Opposite angles P
and R
measure (x+16)°
and (6x−4)°
, respectively. Determine the measure of the smaller angle.(1 point)
Responses
24°
24°
56°
56°
20°
20°
40°
Answers
GPT-4o mini
Let's break down the questions one by one:
1. **Opposite angles of an inscribed quadrilateral can be proven to be:**
- The correct response is **supplementary**. Opposite angles in an inscribed quadrilateral always add up to 180°.
2. **Given quadrilateral AEIU is inscribed in circle O, with m∠A=27° and m∠E=2(m∠A), find m∠I and m∠U:**
- First, calculate m∠E:
\[
m∠E = 2(m∠A) = 2(27°) = 54°
\]
- By the property of inscribed quadrilaterals, m∠A + m∠C = 180° and m∠E + m∠I = 180°.
\[
m∠I = 180° - m∠E = 180° - 54° = 126°
\]
\[
m∠U = 180° - m∠A = 180° - 27° = 153°
\]
- The correct response is **m∠I = 126° and m∠U = 153°**.
3. **Identify the true statement about inscribed angles and their intercepted arc:**
- The correct statement is **The intercepted arc is twice the measure of the inscribed angle.**
4. **For the inscribed quadrilateral ABCD with given angles:**
- We have:
\[
m∠A = 54°, \quad m∠B = 115°, \quad m∠C = 86°
\]
- Therefore:
\[
m∠D = 180° - m∠B = 180° - 115° = 65°
\]
- Check if mADC (the angle measure based on arcs) results in the true statement. Based on the angles and the properties of quadrilaterals inscribed in circles:
- So the relevant statements need to be checked. The correct statement is **m∠D = 169°** does not fit, meanwhile mAD can be compared with others; as m∠D seems to feedback with angle measures from circumferences and not odd.
- However, if pondering with other intricacies, this response might not yield exact agreeable ones without visual assessments; usually those on mAD would circling back to arcs reflect increasingly vary results need proper checks adaptable angles constructs.
5. **For quadrilateral PQRS inscribed in circle O with angles P and R measuring given expressions:**
- Set up the equation:
\[
(x + 16) + (6x - 4) = 180
\]
\[
7x + 12 = 180
\]
\[
7x = 168 \implies x = 24
\]
- Find the measures:
\[
m∠P = x + 16 = 24 + 16 = 40°
\]
\[
m∠R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140°
\]
- The measure of the smaller angle is **40°**.
So, the final answers are:
1. Supplementary.
2. m∠I = 126° and m∠U = 153°.
3. The intercepted arc is twice the measure of the inscribed angle.
4. m∠D=169° is the affirmed reflection.
5. 40°.
1. **Opposite angles of an inscribed quadrilateral can be proven to be:**
- The correct response is **supplementary**. Opposite angles in an inscribed quadrilateral always add up to 180°.
2. **Given quadrilateral AEIU is inscribed in circle O, with m∠A=27° and m∠E=2(m∠A), find m∠I and m∠U:**
- First, calculate m∠E:
\[
m∠E = 2(m∠A) = 2(27°) = 54°
\]
- By the property of inscribed quadrilaterals, m∠A + m∠C = 180° and m∠E + m∠I = 180°.
\[
m∠I = 180° - m∠E = 180° - 54° = 126°
\]
\[
m∠U = 180° - m∠A = 180° - 27° = 153°
\]
- The correct response is **m∠I = 126° and m∠U = 153°**.
3. **Identify the true statement about inscribed angles and their intercepted arc:**
- The correct statement is **The intercepted arc is twice the measure of the inscribed angle.**
4. **For the inscribed quadrilateral ABCD with given angles:**
- We have:
\[
m∠A = 54°, \quad m∠B = 115°, \quad m∠C = 86°
\]
- Therefore:
\[
m∠D = 180° - m∠B = 180° - 115° = 65°
\]
- Check if mADC (the angle measure based on arcs) results in the true statement. Based on the angles and the properties of quadrilaterals inscribed in circles:
- So the relevant statements need to be checked. The correct statement is **m∠D = 169°** does not fit, meanwhile mAD can be compared with others; as m∠D seems to feedback with angle measures from circumferences and not odd.
- However, if pondering with other intricacies, this response might not yield exact agreeable ones without visual assessments; usually those on mAD would circling back to arcs reflect increasingly vary results need proper checks adaptable angles constructs.
5. **For quadrilateral PQRS inscribed in circle O with angles P and R measuring given expressions:**
- Set up the equation:
\[
(x + 16) + (6x - 4) = 180
\]
\[
7x + 12 = 180
\]
\[
7x = 168 \implies x = 24
\]
- Find the measures:
\[
m∠P = x + 16 = 24 + 16 = 40°
\]
\[
m∠R = 6x - 4 = 6(24) - 4 = 144 - 4 = 140°
\]
- The measure of the smaller angle is **40°**.
So, the final answers are:
1. Supplementary.
2. m∠I = 126° and m∠U = 153°.
3. The intercepted arc is twice the measure of the inscribed angle.
4. m∠D=169° is the affirmed reflection.
5. 40°.