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

. 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, ( mangle jhi=)( mangle jgi+mangle gjh ). by the

, ( \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:

• For the first blank: The formula \(m\angle JHI=\frac{1}{2}b\) and \(m\angle GJH = \frac{1}{2}a\) is based on the inscribed - angle theorem. The inscribed - angle theorem states that an inscribed angle in a circle is half the measure of its intercepted arc.
• For the second blank: Angle \(JHI\) is an exterior angle of triangle \(GJH\). In a triangle, an exterior angle is equal to the sum of the two non - adjacent (remote) interior angles.
• For the third blank: After substituting \(m\angle JHI=\frac{1}{2}b\) and \(m\angle GJH=\frac{1}{2}a\) into the equation \(m\angle JHI=m\angle JGI + m\angle GJH\), we use substitution property of equality.

Answer:

1. Inscribed - angle theorem
2. \(GJH\)
3. Substitution property of equality