Answer:
[tex]m\angle 1=41^{\circ}[/tex]
[tex]m\angle 2=103^{\circ}[/tex]
Step-by-step explanation:
Measure of Angle 1
To find the measure of angle 1, we can use the Intersecting Secants Theorem.
According to the Intersecting Secants Theorem, if two secant segments are drawn to the circle from one exterior point, the angle formed by the two lines is half the positive difference between the measures of the intercepted arcs.
In this case, angle 1 is the angle formed by the two secant segment (BC and DC), and the intercepted arcs are BD and FH.
Therefore:
[tex]m\angle 1=\dfrac{1}{2}\left|\overset{\frown}{BD}-\overset{\frown}{FH}\right|[/tex]
Given that arc BD = 118° and arc FH = 36°, then:
[tex]m\angle 1=\dfrac{1}{2}\left|118^{\circ}-36^{\circ}\right|[/tex]
[tex]m\angle 1=\dfrac{1}{2}\left(82^{\circ}\right)[/tex]
[tex]m\angle 1=41^{\circ}[/tex]
So, the measure of angle 1 is 41°.
[tex]\hrulefill[/tex]
Measure of Angle 2
To find the measure of angle 2, we can use the Angles of Intersecting Chords Theorem.
According to the Angles of Intersecting Chords Theorem, if two chords intersect inside a circle, then the measure of the angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
In this case, chords BF and DH intersect at point K inside circle A, and the intercepted arcs are HB and DF, so:
[tex]m\angle 2=\dfrac{1}{2}\left(\overset{\frown}{HB}+\overset{\frown}{DF}\right)[/tex]
To find the measure of arc DF, we can subtract the measures of the other arcs from 360°:
[tex]\overset{\frown}{DF}=360^{\circ}-\overset{\frown}{FH}-\overset{\frown}{HB}-\overset{\frown}{BD}[/tex]
[tex]\overset{\frown}{DF}=360^{\circ}-36^{\circ}-124^{\circ}-118^{\circ}[/tex]
[tex]\overset{\frown}{DF}=82^{\circ}[/tex]
Therefore:
[tex]m\angle 2=\dfrac{1}{2}\left(124^{\circ}+82^{\circ}\right)[/tex]
[tex]m\angle 2=\dfrac{1}{2}\left(206^{\circ}\right)[/tex]
[tex]m\angle 2=103^{\circ}[/tex]
So, the measure of angle 2 is 103°.
