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Question

Triangle A B C is shown with an angle bisector and two congruent sides. Side A B is congruent to Side A C. Segment A D is the bisector of angle A. Point D is on side B C. Angle B A D is labeled y degrees. Angle B D A is labeled x degrees. Angle A C D is labeled 47 degrees.

Find the values of x and y.

Answer 1

To find the values of x and y, we can use the angle bisector theorem.

Let's start with angle ABD. Since BD bisects angle A, we can use the angle bisector theorem to determine that angle ABD is congruent to angle ACD. Therefore, y = 47 degrees.

Now, let's look at the triangle ABD. We know that angle ABD is y degrees and angle BDA is x degrees. We also know that angle B is congruent to angle C (since sides AB and AC are congruent). Therefore, we can use the triangle angle sum theorem to write an equation: x + y + (angle B) = 180 degrees.

Since angles B and C are congruent, we can substitute angle B with angle C in our equation: x + y + (angle C) = 180 degrees.

Since angles B and C are congruent, we can also substitute angle C with y degrees in our equation: x + y + y = 180 degrees.

Simplifying the equation, we get: x + 2y = 180 degrees.

Now, we have two equations:
1) y = 47 degrees
2) x + 2y = 180 degrees

Substitute y = 47 degrees into equation 2): x + 2(47) = 180 degrees.

Simplify the equation: x + 94 = 180 degrees.

Next, subtract 94 from both sides of the equation: x = 180 degrees - 94 = 86 degrees.

Therefore, the values of x and y are:
x = 86 degrees
y = 47 degrees.