➛ ∠COD is a central angle, so its subtended arc [tex]\sf{ \overset\frown{CD}}[/tex] = COD

so, [tex]\sf{ \overset\frown{CD}}[/tex] = z = 55°

[tex]{}[/tex]

➛ AD is a diameter, so ∠AOD = 180°

[tex]{}[/tex]

➛ ∠AOD = ∠AOB+ ∠BOC+ ∠COD

180° = x + y + 55°

x + y = 125°

[tex]{}[/tex]

➛ [tex]\sf{ \overset\frown{AB}}[/tex] subtends central angle ∠AOB, so ∠AOB = [tex]\sf{ \overset\frown{AB}}[/tex] = 25°

so, x = 25°

[tex]{}[/tex]

➛ y = 125° - x

y = 125° - 25°

so, y = 100°

[tex]{}[/tex]

✏️ ANSWER:

• x = 25°
• y = 100°
• z = 55°

✒️CIRCLE

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[tex] \large\underline{\mathbb{PROBLEM}:} [/tex]

• Segment AD is a diameter. Find the value of x, y and z.

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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]

[tex] \qquad \Large \:\: \rm x = 25\degree [/tex]

[tex] \qquad \Large \:\: \rm y = 100\degree [/tex]

[tex] \qquad \Large \:\: \rm z = 55\degree [/tex]

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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]

» The measure of the central angle is as same as the measure of its intercepted arc. By the given figure, we can say that.

• [tex] m\angle AOB = m\overset{\frown}{AB} [/tex]

• [tex] m\angle COD = m\overset{\frown}{CD} [/tex]

» Find the measure of angle x which is also the measure of angle AOB.

• [tex] m\angle AOB = m\overset{\frown}{AB} [/tex]

• [tex] x = 25\degree[/tex]

» Find the measure of arc z which is also the measure of arc CD.

• [tex] m\overset{\frown}{CD} = m\angle COD [/tex]

• [tex] z = 55\degree [/tex]

» Therefore the measure of angle x is 25° and the measure of arc z is 55°

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» Since segment AD is a diameter and O is the center, then angles AOB, BOC, and COD are supplementary.

• [tex] \small m\angle AOB + m\angle BOC + m\angle COD = 180\degree [/tex]

» Find the measure of angle y which is also the measure of angle BOC. Substitute the given.

• [tex] 25\degree + y + 55\degree = 180\degree [/tex]

• [tex] y + 80\degree = 180\degree [/tex]

• [tex] y = 180\degree - 80\degree [/tex]

• [tex] y = 100\degree [/tex]

» Therefore the measure of angle y is 100°

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