Question

Given: ∠AOB is a central angle and ∠ACB is a circumscribed angle.
Prove: △ACO ≅ △BCO

We are given that angle AOB is a central angle of circle O and that angle ACB is a circumscribed angle of circle O. We see that AO ≅ BO because ____

We also know that AC ≅ BC since ___

Using the reflexive property, we see that ____

Therefore, we conclude that △ACO is congruent to △BCO by the _____

Answer 1

Answer: We can fill all the blanks with help of below explanation.

Explanation:

Given: $\angle AOB$ is a central angle and $\angle ACB$ is a circumscribed angle.  (where, AC and CB are two tangent lines from an external point C.)

Prove: $\triangle ACO\cong\triangle BCO$

We are given that angle AOB is a central angle of circle O and that angle ACB is a circumscribed angle of circle O. We see that$AO\cong BO$ because both OA and OB make the radius of the circle O. (Since,  A and B are points on the circumference of circle O So, they are equally far from the center O

We also know that $AC\cong BC$ since both are the tangent line from a common point C So, they must be equal.(By the property of tangent line)

Using the reflexive property, we see that $OC\cong OC$

Therefore, we conclude that $\triangle ACO\cong\triangle BCO$ by the SSS postulate.

Answer 2

The answers to the dashed parts of the questions to prove that △ACO ≅ △BCO are respectively;

- All radii of the same circle are congruent

- Tangents to a circle that intersect are congruent

- Side CO is congruent to side CO

- SSS congruency theorem

The image of the Triangle and circle has been attached.

Now, from the attached image we can see that;

• Point O is the center of the circle and as such AO and BO are both radius of the circle.

Thus, it means that AO ≅ BO because; All radii of the same circle are congruent

• Since AO and BO are equal, it means that since point C is the intersection point of tangents from points A and B, then we can say that;

AC ≅ BC because; tangents to a circle that intersect are congruent.

• Reflexive property shows that the length of a side is congruent to itself. Now, in △ACO and △BCO, we see that they share a common side which is CO. Thus, CO is congruent to side CO.

• We have been able to establish that;

AO ≅ BO

AC ≅ BC

CO ≅ CO

This means all three corresponding sides of both △ACO and △BCO are congruent. Thus, we can conclude that;

△ACO is congruent to △BCO by the "SSS Congruency theorem"

Read more about SSS Congruency theorem at; https://brainly.com/question/2102943