MHB Proving angles are equal in a triangle in a circle.

Click For Summary
To prove that angles CAB and CBA are equal in a triangle inscribed in a circle, the discussion highlights the use of the inscribed angle theorem and the alternate segment theorem. By connecting the angles to the measures of arcs AC and CB, it is established that these arcs are equal, leading to the conclusion that angles CAB and CBA are congruent. The alternate segment theorem is referenced to show that the angle between a tangent and a chord equals the angle in the alternate segment, reinforcing the isosceles nature of triangle CBA. Thus, the proof demonstrates the equality of the angles based on the properties of circles and triangles. The discussion effectively illustrates the geometric relationships involved in the proof.
markosheehan
Messages
133
Reaction score
0
View attachment 6094

hi can someone help me work this out. i think it has something to do with the exterior angle of a triangle is equal to the sum of the interior angles but i can't work it.
 

Attachments

  • WIN_20161016_19_27_26_Pro.jpg
    WIN_20161016_19_27_26_Pro.jpg
    58 KB · Views: 142
Mathematics news on Phys.org
Prove that $\angle CAB=\angle CBA$ by connecting these angles to measures of arcs using the inscribed angle theorem and its corollary about tangent lines.
 
how do you know the arcs AC and CB are the same
 
Last edited:
markosheehan said:
how do you know the arcs AC and CB are the same

Are you familiar with the Alternate Segment theorem, which states that

The angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.

View attachment 6099

So now do you see a similarity between the angle marked in blue in the above diagram & the angle marked in blue in your diagram.

$\therefore$ It can be said that $\angle CAT = \angle CBA$ using alternate segment theorem

Now what can be said about the triangle CBA using it's angles?
 

Attachments

  • 05c51d46693988123b38a8bb6e6e4ac08213509d.gif
    05c51d46693988123b38a8bb6e6e4ac08213509d.gif
    4 KB · Views: 121
It's isosceles
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

High School In a triangle, is the angle between the points of contact of the inscribed circle with the sides 120 degrees?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Geometry problem of interest with a 3-4-5 triangle
  • · Replies 59 ·
2
Replies
59
Views
24K
High School A Question Relating Sum of Angles and Breaking of the Parallel Postulate
  • · Replies 6 ·
Replies
6
Views
2K
MHB Difference of angles in a triangle
  • · Replies 4 ·
Replies
4
Views
2K
MHB Acute angle of right triangles
  • · Replies 4 ·
Replies
4
Views
1K
MHB Can you prove that Triangle $ABC$ is Isosceles?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad What is the name of this geometry theorem?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Help with area formula using direction cosines
  • · Replies 3 ·
Replies
3
Views
2K
MHB Proving Equality of Angles in an Acute Triangle
  • · Replies 4 ·
Replies
4
Views
1K