Question

use the information given in the diagram to prove that ( mangle jgi=\frac{1}{2}(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 ( mangle jhi=\frac{1}{2}b ) and ( mangle gjh=\frac{1}{2}a ) by the (square) angle
( jhi ) is an exterior angle of triangle (square). because the measure of an exterior angle is equal to the sum of the measures of the remote interior angles, ( mangle jhi=)( mangle jgi + mangle gjh ). by the (square), (\frac{1}{2}b)( =mangle jgi+\frac{1}{2}a ). using the subtraction property, ( mangle jgi )( =\frac{1}{2}b-\frac{1}{2}a ). therefore, ( mangle jgi=\frac{1}{2}(b - a) ) by the distributive property.

Explanation:

Step1: Identify the theorem for inscribed angles

Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. So, for angles \( \angle JHI\) and \( \angle GJH\) which are inscribed angles, \(m\angle JHI=\frac{1}{2}b\) (intercepting arc \(JI\)) and \(m\angle GJH = \frac{1}{2}a\) (intercepting arc \(FH\)).

Step2: Identify the triangle

\(\angle JHI\) is an exterior angle of triangle \( \triangle JGI\). By the Exterior Angle Theorem (the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles), \(m\angle JHI=m\angle JGI + m\angle GJH\).

Step3: Substitute the angle measures

Substitute \(m\angle JHI=\frac{1}{2}b\) and \(m\angle GJH=\frac{1}{2}a\) into the equation \(m\angle JHI=m\angle JGI + m\angle GJH\). We get \(\frac{1}{2}b=m\angle JGI+\frac{1}{2}a\).

Step4: Solve for \(m\angle JGI\)

Using the subtraction property of equality, subtract \(\frac{1}{2}a\) from both sides of the equation \(\frac{1}{2}b=m\angle JGI+\frac{1}{2}a\). So, \(m\angle JGI=\frac{1}{2}b-\frac{1}{2}a\).

Step5: Factor out \(\frac{1}{2}\)

By the distributive property \(m\angle JGI=\frac{1}{2}(b - a)\)

Answer:

The blanks are filled as follows:

1. Inscribed Angle Theorem
2. \( \triangle JGI\)
3. Exterior Angle Theorem