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Question

Find the values of x, y, and z. The diagram is not to scale.
Triangle A B C is drawn with point D on side B C. A segment is drawn from A to D. Angle A C D measures 63 degrees. Angle C A D measures 36 degrees. Angle D A B measures 13 degrees. Angle C D A measures x degrees. Angle C B A measures y degrees. Angle A D B measures z degrees.
The smaller triangle formed on the left has a top vertex angle of 36 degrees, bottom left vertex angle of 63 degrees, and a bottom right vertex angle of x degrees.
The smaller triangle formed on the right has a top vertex angle of 13 degrees, bottom left vertex angle of z degrees, and a bottom right vertex angle of y degrees.
(3 points)

Answer 1

To find the values of x, y, and z, we can use the fact that the sum of the angles in a triangle is always 180 degrees.

In triangle ACD, we know that angle ACD is 63 degrees, angle CAD is 36 degrees, and angle CDA is x degrees. Using the fact that the sum of the angles in a triangle is 180 degrees, we can write the equation:

63 + 36 + x = 180

Simplifying this equation, we get:

99 + x = 180

Subtracting 99 from both sides, we find:

x = 81

So, x = 81.

In triangle ABC, we know that angle CAB is 36 degrees, angle CBA is y degrees, and angle ABC is z degrees. Using the fact that the sum of the angles in a triangle is 180 degrees, we can write the equation:

36 + y + z = 180

Simplifying this equation, we get:

36 + y + z = 180

Subtracting 36 from both sides, we find:

y + z = 144

So, y + z = 144.

In triangle ADB, we know that angle DAB is 13 degrees, angle ADB is z degrees, and angle ABD is x degrees. Using the fact that the sum of the angles in a triangle is 180 degrees, we can write the equation:

13 + z + x = 180

Simplifying this equation, we get:

13 + z + 81 = 180

Subtracting 94 from both sides, we find:

z + x = 87

So, z + x = 87.

From the equations y + z = 144 and z + x = 87, we can subtract the second equation from the first one to find:

(y + z) - (z + x) = 144 - 87

Simplifying this equation, we get:

y - x = 57

So, y - x = 57.

Now we have two equations: y + z = 144 and y - x = 57. We can solve these equations simultaneously by adding them:

(y + z) + (y - x) = 144 + 57

Simplifying this equation, we get:

2y = 201

Dividing both sides by 2, we find:

y = 100.5

So, y = 100.5.

Substituting the value of y into the equation y - x = 57, we can solve for x:

100.5 - x = 57

Subtracting 100.5 from both sides, we find:

-x = -43.5

Multiplying both sides by -1, we get:

x = 43.5

So, x = 43.5.

Finally, substituting the values of x = 43.5 and y = 100.5 into the equation z + x = 87, we can solve for z:

z + 43.5 = 87

Subtracting 43.5 from both sides, we find:

z = 43.5

So, z = 43.5.

In conclusion, the values of x, y, and z are: x = 43.5, y = 100.5, and z = 43.5.