A star-patterned quilt has a star with the angles shown. What is the value of x? The diagram is not to scale.
A five-pointed star is formed when two congruent right triangles overlap. The top triangle formed to make the top point of the star has two labeled angles. Its top angle is labeled 24 degree-sign, and its right base angle is labeled x degree-sign. The third angle of this triangle is unlabeled, but its exterior angle is labeled 105 degree-sign.
To find the value of x, we can use the fact that the sum of the angles in a triangle is 180 degrees.
Since the top angle of the triangle is labeled as 24 degrees, and the exterior angle of the triangle is labeled as 105 degrees, we can find the third angle of the triangle by subtracting the exterior angle from 180 degrees.
Third angle = 180 degrees - exterior angle
Third angle = 180 degrees - 105 degrees
Third angle = 75 degrees
Now, we can use the fact that the sum of the angles in a triangle is 180 degrees to find the value of x.
Sum of angles in the triangle = top angle + right base angle + third angle
180 degrees = 24 degrees + x degrees + 75 degrees
Now, we can solve for x. Subtracting 24 degrees and 75 degrees from both sides of the equation, we have:
180 degrees - 24 degrees - 75 degrees = x degrees
81 degrees = x
Therefore, the value of x is 81 degrees.
To find the value of x, we can use the properties of angles in a triangle and the exterior angle of a triangle.
1. In a triangle, the sum of all interior angles is 180 degrees.
Let's call the third angle of the triangle formed by the top point of the star as angle A.
Angle A + Angle B + Angle C = 180 degrees.
2. The exterior angle of a triangle is equal to the sum of its two opposite interior angles. So, the exterior angle of angle A is equal to Angle B + Angle C.
Angle D = Angle B + Angle C.
Given information:
Angle D (exterior angle) = 105 degrees
Angle B = 24 degrees
Using the above information, we can set up the equation as follows:
105 degrees = 24 degrees + Angle C
Solving for Angle C:
Angle C = 105 degrees - 24 degrees
Angle C = 81 degrees
So, the value of x (Angle B) is 24 degrees.
Therefore, the value of x is 24 degrees.
To find the value of x, we first need to recognize that the sum of the interior angles of any triangle is always 180 degrees.
In this case, we have a triangle with a top angle of 24 degrees and a right base angle of x degrees. So, let's calculate the unlabeled angle using the fact that the sum of angles in a triangle is 180 degrees:
Unlabeled angle + 24 degrees + x degrees = 180 degrees
To find the value of the unlabeled angle, we can subtract 24 degrees and x degrees from both sides of the equation:
Unlabeled angle = 180 degrees - 24 degrees - x degrees
Now, we know that the exterior angle of the triangle is labeled as 105 degrees. According to the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. In this case, the unlabeled angle is one of the non-adjacent angles.
So, the sum of the unlabeled angle and x degrees should equal the exterior angle, which is 105 degrees:
Unlabeled angle + x degrees = 105 degrees
Finally, we can substitute the expression for the unlabeled angle from the previous equation into this equation to solve for x:
180 degrees - 24 degrees - x degrees + x degrees = 105 degrees
Simplifying the equation gives us:
180 degrees - 24 degrees = 105 degrees + x degrees
156 degrees = 105 degrees + x degrees
Subtracting 105 degrees from both sides gives us:
156 degrees - 105 degrees = x degrees
51 degrees = x
Therefore, the value of x is 51 degrees.