✒️CIRCLE
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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]
[tex] \qquad \Large \:\: \rm 1) \; m\overset{\frown}{SN} = 164\degree [/tex]
[tex] \qquad \Large \:\: \rm 2) \; m\overset{\frown}{MT} = 113\degree [/tex]
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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]
Number 1:
The measure of an angle and its vertical is half the sum of the measures of its intercepted arcs.
- [tex] m\angle SPN = \frac12 (m\overset{\frown}{MT} + m\overset{\frown}{SN}) [/tex]
Substitute the given to find the measure of arc SN.
- [tex] 118\degree = \frac12 (72\degree + m\overset{\frown}{SN}) [/tex]
- [tex] (2)118\degree = \frac12 (72\degree + m\overset{\frown}{SN})(2) [/tex]
- [tex] 236\degree = 72\degree + m\overset{\frown}{SN} [/tex]
- [tex] 236\degree - 72\degree = m\overset{\frown}{SN} [/tex]
- [tex] 164\degree = m\overset{\frown}{SN} [/tex]
Therefore, the measure of arc SN is 164°
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Number 2:
The measure of an angle and its vertical is half the sum of the measures of its intercepted arcs.
- [tex] m\angle MPT = \frac12 (m\overset{\frown}{SN} + m\overset{\frown}{MT}) [/tex]
Substitute the given to find the measure of arc MT.
- [tex] 135\degree = \frac12 (157\degree + m\overset{\frown}{MT}) [/tex]
- [tex] (2)135\degree = \frac12 (157\degree + m\overset{\frown}{MT})(2) [/tex]
- [tex] 270\degree = 157\degree + m\overset{\frown}{MT} [/tex]
- [tex] 270\degree - 157\degree = m\overset{\frown}{MT} [/tex]
- [tex] 113\degree = m\overset{\frown}{MT} [/tex]
Therefore, the measure of arc MT is 113°
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