Answer :
Answer:
In the figure ∠ABO and ∠BCO have measures equal to 35°.
Step-by-step explanation:
The complete question is
For circle O, m CD=125° and m∠ABC = 55°
In the figure<____, (AOB, ABO, BOA) and <_____ (BCO, OBC,BOC) have measures equal to 35°
The picture in the attached figure
step 1
Find the measure of angle COB
we know that
[tex]m\angle COB=arc\ CD[/tex] ----> by central angle
we have
[tex]arc\ CD=125^o[/tex]
therefore
[tex]m\angle COB=125^o[/tex]
step 2
we know that
AB is a tangent to the circle O at point A
so
ABC and ABO are right triangles
In the right triangle ABC
Find the measure of angle BCA
Remember that
[tex]m\angle BCA+m\angle\ ABC=90^o[/tex] ---> by complementary angles in a right triangle
we have
[tex]m\angle ABC=55^o[/tex]
substitute
[tex]m\angle BCA+55^o=90^o[/tex]
[tex]m\angle BCA=90^o-55^o=35^o\\[/tex]
step 3
In the triangle BCO
Find the measure of angle CBO
we know that
[tex]m\angle CBO+m\angle COB+m\angle BCO=180^o[/tex] ---> the sum of the interior angles in any triangle must be equal to 180 degrees
we have
[tex]m\angle COB=125^o[/tex]
[tex]m\angle BCO=m\angle BCA=35^o[/tex] -----> have measure equal to 35 degrees
substitute
[tex]m\angle CBO+125^o+35^o=180^o[/tex]
[tex]m\angle CBO=180^o-160^o=20^o[/tex]
step 4
Find the measure of angle ABO
In the right triangle ABO
we know that
[tex]m\angle ABC=m\angle CBO+m\angle ABO[/tex] ----> by angle addition postulate
we have
[tex]m\angle ABC=55^o[/tex]
[tex]m\angle CBO=20^o[/tex]
substitute
[tex]55^o=20^o+m\angle ABO[/tex]
[tex]m\angle ABO=55^o-20^o=35^o[/tex] ----> have measure equal to 35 degrees
therefore
In the figure ∠ABO and ∠BCO have measures equal to 35°.
Answer:
In the figure ∠ABO and ∠BCO have measures equal to 35°.
Step-by-step explanation: