Two triangles are inside a circle as shown in the following diagram. Prove that the triangles are congruent. Determine the measure of LON

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To get the measure of angle LON, let's work on angles LNO, YNL and the given, 80 degrees. We can let LNO = YNL, both are one of the base angles of the congruent triangles. So taking the two angles and the given, 80 degrees. We can formulate and equation, knowing that those three angles will form a straight line having 180 degrees as measure of its angle. The working equation is:

80 + LNO + YNL = 180

Since LNO = YNL, let x = LNO = YNL

so 80 + x + x = 180

Combining similar terms:

2x = 180 - 80 (move 80 to the right side will change its sign)

2x = 100 (divide both sides by 2 to solve for x)

(2x)/2 = 100/2

x = 50

Now that we know x and that the triangle LON is an isosceles triangle, the other base angle, OLN is equal to x.

x = angle OLN = 50

Taking into consideration the property of triangle that the sum of all its angles is 180 degrees.

We already the measure of the base angles which is 50.

So,

180 = 50 + 50 + LON

combine similar terms

180 - 50 - 50 = LON

Therefore, measure of angle LON = 80 degrees.

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