How Do I Calculate Angle BAC if Angle CBA is 69 Degrees?

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Homework Help Overview

The discussion revolves around calculating angle BAC in a geometric context involving a quadrilateral inscribed in a circle, with angle CBA given as 69 degrees. Participants are exploring the relationships between the angles and the implications of the inscribed angle theorem.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the implications of the inscribed angle theorem and the relationships between angles in the quadrilateral. There is uncertainty about the marked angles and their interpretations, particularly regarding the angle measures and their relationships.

Discussion Status

The discussion is active, with participants questioning the assumptions made about the angles and the geometry involved. Some guidance is offered regarding the relationships between the angles, but there is no explicit consensus on the correct interpretation or calculation of angle BAC.

Contextual Notes

There is mention of potential confusion regarding the marked angles and the implications of changing the lengths of chords in the geometry. Participants are also considering how the positioning of points A and D might affect the angles.

Natasha1
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Homework Statement
The diagram shows a circle with centre O.
Relevant Equations
A, B, C and D lie on the circumference of the circle. Work out angle BAC
Please see attached drawing which I have written my work (in colour).

Despite spending 1h on it I still can’t understand how to find angle BAC.

776B6095-1913-4498-8E36-8FE05139D586.jpeg
 
Last edited by a moderator:
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I found it! It’s 48 degrees
 
1) For a quadrilateral inscribed in a circle, the sum of opposite angles equals to 180 degrees.
2) Inscribed angle theorem
 
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Natasha1 said:
I found it! It’s 48 degrees
In that case BD would be parallel to BA which it is plainly not. There is a 48 degree angle where you've marked 58. I'm not sure we can solve for BAC since chord DA's length can be shrunk or stretched without materially affecting the rest of the given geometry. Sum of angles BAC and ABD should be 48 by exterior angle thm.
 
Last edited:
neilparker62 said:
I'm not sure we can solve for BAC since chord DA's length can be shrunk or stretched without materially affecting the rest of the given geometry.
Not by moving A because that would change ABC, and if you move D then BAC doesn't change anyway.
neilparker62 said:
In that case BD would be parallel to BA

How do you deduce that? I agree with 48 degrees.
 
Maybe I am misinterpreting the marked angles ? Why has the OP marked 58 degrees when the other 2 angles in the triangle are 63 and 69 degrees. I think the confusion arises because I thought (and it looks like the OP also initially thought) that the measure of angle CBD was 69 degrees. But now I see it's CBA which is 69 degrees. Then the 48 degree answer makes sense.

If a person thought CBD was 69 degrees, he/she would also mark CAD as 69 degrees as appears to be the case.
 
Last edited:
neilparker62 said:
it's CBA which is 69 degrees
Yes.
 

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