Short Answer
Note: Your teacher will grade your responses to questions 4–8 to ensure you receive proper credit for your answers.
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.
The exterior angle of an angle in a triangle is equal to the sum of the other two interior angles. In this case, the exterior angle is 105 degrees, and one of the interior angles is 24 degrees. Let's call the third angle of the triangle y degrees.
Therefore, we can write the equation:
105 = 24 + y
To solve for y, we subtract 24 from both sides:
81 = y
So, the third angle of the triangle is 81 degrees.
Since the sum of the angles in a triangle is 180 degrees, we can use this information to find the value of x:
180 = 81 + 90 + x
Subtracting 81 and 90 from both sides, we have:
180 - 81 - 90 = x
x = 9
Therefore, the value of x is 9 degrees.
To find the value of x, we need to use the fact that the angles in a triangle add up to 180 degrees.
First, we know that the exterior angle 105 degrees is equal to the sum of the two angles that are inside the triangle and adjacent to it. Those angles are the right base angle (x degrees) and the third angle of the triangle.
So, we can write an equation:
x + third angle = 105 degrees
Now, let's focus on the second triangle that forms the bottom point of the star. Since the star is symmetrical, the top angle of this triangle is also 24 degrees. And again, the three angles of this triangle add up to 180 degrees.
Therefore, we can also write an equation for this triangle:
24 degrees + right base angle + third angle = 180 degrees
Now, we can solve these two equations simultaneously to find the value of x.
First, let's simplify the second equation:
24 degrees + x + third angle = 180 degrees
Subtracting 24 degrees from both sides, we get:
x + third angle = 180 degrees - 24 degrees
x + third angle = 156 degrees
Now, we have a system of equations:
x + third angle = 105 degrees
x + third angle = 156 degrees
Since both equations are equal to the same sum, x must be equal to 105 degrees and 156 degrees. However, this is not possible as x cannot have two different values.
Therefore, there is no single value of x that satisfies both equations.
To find the value of x, we can use the fact that the exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles. In this case, we have an exterior angle of 105 degrees.
The triangle with the unlabeled angle is a right triangle, which means one of its angles is 90 degrees. We can call this angle "y" for simplicity.
Using the given information, we can set up an equation: x + y + 90 = 105. Rearranging the equation, we get x + y = 15.
Now we need to find the value of y. Since the top angle of the triangle is labeled 24 degrees, we can set up another equation: y + 24 + 90 = 180. Simplifying, we have y + 114 = 180.
Subtracting 114 from both sides, we get y = 66.
Now we can substitute the value of y into the equation x + y = 15: x + 66 = 15.
Subtracting 66 from both sides, we find x = -51.
Therefore, the value of x is -51 degrees.