Answer:
1)
A. m arc QRS = 100°
B. m arc QRT = 255°
C. m arc UTS = 180°
D. m arc UTR = 205°
2)
1] The major arc CA = 255°
2] The minor arc AB = 90°
Step-by-step explanation:
In a circle:
- The measure of an central angle is equal to the measure of its subtended arc
- The measure of a circle is 360°
- Any chord divides a circle into two arcs, a minor arc its measure is < 180° and a major arc its measure is > 180°, the sum of the measures of the minor and major arcs is 360°
- If the measures of the minor and major arcs are equal, then each arc represents a semi-circle
1)
In circle O
A.
∵ m∠QOR = 75°
∵ ∠QOR subtended by arc QR
- By using the 1st note above
∴ m of arc QR = m∠QOR
∴ m of arc QR = 75°
∵ m∠ROS = 25°
∵ ∠ROS subtended by arc RS
∴ m of arc RS = m∠ROS
∴ m of arc RS = 25°
The measure of arc QRS is the sum of the measures of arcs QR and RS
∴ m arc QRS = 75° + 25° = 100°
B.
∵ m∠SOT = 155°
∵ ∠SOT subtended by arc ST
∴ m of arc ST = m∠SOT
∴ m of arc ST = 155°
The measure of arc QRT is the sum of the measures of arcs QR, RS and ST
∴ m arc QRT = 75° + 25° + 155 °= 255°
C.
∵ m∠UOT = 25°
∵ ∠UOT subtended by arc UT
∴ m of arc UT = m∠UOT
∴ m of arc RUT = 25°
The measure of arc UTS is the sum of the measures of arcs UT and TS
∴ m arc UTS = 25° + 155° = 180°
D.
The measure of arc UTR is the sum of the measures of arcs UT, TS and SR
∴ m arc UTR = 25° + 155° + 25 = 205°
2)
1]
∵ The sum of the measures of the minor and major arcs is 360°
∵ m of minor arc AC = 105°
- Subtract 105 from 360° to find the measure of the major arc CA
∴ m of major arc CA = 360° - 105°
∴ m of major arc CA = 255°
2]
∵ m of major arc AB = 270°
- Subtract 270° from 360° to find the measure of the minor arc AB
∴ m of minor arc AB = 360° - 270°
∴ m of minor arc AB = 90°
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