Mathematics, 23.02.2021 14:50, Govind4954

Angle ABD is 52°. State the size of angle DEB, giving a reason for your answer.

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answered: lulustar13

angle DEB = 52 degrees

The short answer is that you use the tangent-chord theorem. A more lengthy answer is given below.

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Explanation:

Refer to the diagram below.

I'm going to move point E so that segment EB becomes a diameter of the circle (it goes through the center point F). Points A,B,C,D will stay in place. The red points and red segments show the changes.

In this slightly redrawn diagram, triangle EDB is a right triangle where angle EDB is 90 degrees. This is due to Thales Theorem, which is a special case of the inscribed angle theorem.

Also, the new segment EB is perpendicular to segment AC because the diameter touching the point of tangency is perpendicular to the tangent line. So angle EBA is 90 degrees.

Since points A, D and B didn't move, this means angle ADB is still 52 degrees. This makes the new angle EBD equal to 90-52 = 38 degrees.

Despite the fact that angle E has moved, it has not effected the size of angle DEB. This angle is the same as its original because we're still subtending the same minor arc DB, and that arc hasn't changed size (since segment DB hasn't changed). So that's why it's valid to use the same x for the old angle DEB in black and the new angle DEB in red. They're the same angle.

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Focus on triangle EBD. We have the following interior angles

B = 38D = 90E = x

For any triangle, the three interior angles must add to 180

B+D+E = 180

38+90+x = 180

128+x = 180

x = 180-128

x = 52

Angle DEB is 52 degrees

It's not a coincidence that we end up with the same angle as shown in the original diagram. The tangent-chord theorem says that the angle between a chord and tangent (as shown in the diagram) is exactly equal to the inscribed angle that subtends the arc formed by the chord. Arc DB is formed by chord DB.

In short, the tangent-chord theorem says that inscribed angle DEB and angle ABD are the same measure (both are 52 degrees).

This demonstration I've shown is one way to prove the tangent-chord theorem. Though a more rigorous proof would involve another variable.