Question

Use the information given in the diagram to prove that m∠JGI = (b – a), where a and b represent the degree measures of arcs FH and JI.

Angles JHI and GJH are inscribed angles. We have that m∠JHI = b and m∠GJH = a by the . Angle JHI is an exterior angle of triangle . Because the measure of an exterior angle is equal to the sum of the measures of the remote interior angles, m∠JHI = m∠JGI + m∠GJH. By the , b = m∠JGI + a. Using the subtraction property, m∠JGI = b – a. Therefore, m∠JGI = (b – a) by the distributive property.

Answer 1

• Inscribed Angle Theorem.
• GJH .
• Substitution property

Explanation:

Inscribed Angle Theorem

The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle that subtends the same arc on the circle.

From the given diagram:

• Angles JHI and GJH are inscribed angles.

• We have that
  `m\angle JHI`
  `= (1)/(2)b` and
  `m\angle GJH`
  `= (1)/(2)a` by the Inscribed Angle Theorem. Angle JHI is an exterior angle of triangle GJH .

• Because the measure of an exterior angle is equal to the sum of the measures of the remote interior angles,
  `m\angle JHI = m\angle JGI + m\angle GJH.`

• By the substitution property,
  `(1)/(2)b` =
  `m\angle JGI` +
  `(1)/(2)a`.

• Using the subtraction property,
  `m\angle JGI` =
  `(1)/(2)b-(1)/(2)a`.

Therefore,
`m\angle JGI`
`=(1)/(2)(b-a)` by the distributive property.