A circle is inscribed inside a right triangle as shown. If angle CAB =

How do you know he measure of the two are 90?

Since the circle is inscribed into the triangle, the lines AC and AB are tangent to the circle. A line that is tangent to a circle must have the same angle to the radius as the circle itself. And that angle is 90°.

Radii OD and OE must be perpendicular to the tangents AC and AB respectively.
Thus Angle ODA and Angle OEA = 90 each
Angle DAE + Angle EOD = 180
Angle EOD = 120
Answer A

Answer A. 120.

In Quadrilateral ADOC
One angle given angle A = 60 degree.
two tangents 90+90=180
Hence sum of angles in quad =360 deg
180+60 + angle 0 = 360
angle O = 120

OFFICIAL SOLUTION:

(A) Figure AEOD is a quadrilateral, so the sum of its interior angles must be 360º.

Therefore, angle EOD = 360º – angle DAE – angle ADO – angle AEO.

We know that both angles ADO and AEO are equal to 90º since the angle created by a radius and a tangent line is always 90º.

Now we can solve for angle EOD:
Angle EOD = 360º – 60º – 90º – 90º = 120º.

The correct answer is choice (A).

Since the circle is inscribed inside the triangle, the sides of the triangle form the tangents to the circle. The angle between the tangent and the radius is always 90 degrees.

Also, the line drawn from the external point (from which the tangents are drawn) to the centre of the circle, bisects the angles made at the external point as well as at the centre.
This means, if A is connected to O with a line segment,
Angle DAO = Angle EAO and
Angle DOA = Angle EOA

It is also known that angles ODA and OEA are right angles and hence equal to 90 degrees.
Therefore, triangles ODA and OEA are 30-60-90 right triangles.

Note that angle CAB = angle DAO + angle EAO = 60 degrees.
Since Angle DAO = angle EAO, each of them will be equal to 30 degrees.

Therefore, Angle DOA = Angle EOA = 60 degrees
Angle EOD = Angle EOA + Angle DOA = 60 + 60 = 120 degrees.

The correct answer option is A.

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