Question
The folding chair has different settings that change the angles formed by its parts. Suppose mangle2 is 31 and mangle3 is 72. Find mangle1. The diagram is not to scale.
The image of a quadrilateral and its extended sides and diagonals is shown to resemble a folding chair.
· The left side of the quadrilateral is extended up to form the back of the folding chair.
· The top side of the quadrilateral is extended to the right to form the arm rest of the folding chair.
· The right side of the quadrilateral is extended down to form the front leg of the folding chair.
· The diagonal between the upper right vertex of the quadrilateral and the lower left vertex is extended towards the left to form the back leg of the folding chair.
· Angle 1 is formed by the extended left side and the top side of the quadrilateral so that angle 1 is outside of the quadrilateral.
· Two angles are created by the diagonal.
· Angle 2 is formed by the diagonal and the top side of the quadrilateral.
· Angle 3 is formed by the diagonal and the left side of the quadrilateral.
(1 point)
Responses
123
113
93
103
We can solve this problem using the properties of angles in a quadrilateral.
Since angle 1 is formed by the extended left side and the top side of the quadrilateral, it is an exterior angle. The sum of the interior and exterior angles at a vertex is always 180 degrees. Therefore, angle 1 = 180 - mangle2 = 180 - 31 = 149 degrees.
Similarly, angle 3 is formed by the diagonal and the left side of the quadrilateral. Again, it is an exterior angle. Therefore, angle 3 = 180 - mangle3 = 180 - 72 = 108 degrees.
Therefore, the measure of angle 1 is 149 degrees.
To find mangle1, we need to use the properties of the angles formed by the extended sides and diagonals of the quadrilateral.
Since angle 2 is formed by the diagonal and the top side of the quadrilateral, and we know that mangle2 is 31, we can directly say that mangle2 = mangle1.
Similarly, since angle 3 is formed by the diagonal and the left side of the quadrilateral, and we know that mangle3 is 72, we can directly say that mangle3 = mangle1.
Therefore, mangle1 = mangle2 = mangle3.
So, mangle1 = 31 degrees.
To find angle 1, we need to use the angle sum property of a quadrilateral. In a quadrilateral, the sum of all angles is equal to 360 degrees.
In this case, we are given that angle 2 is 31 degrees and angle 3 is 72 degrees. We need to find angle 1.
To do this, we'll first find the sum of angles 2 and 3:
Angle 2 + Angle 3 = 31 + 72 = 103 degrees
Now, we'll use the angle sum property to find angle 1:
Angle 1 + Angle 2 + Angle 3 + Angle 4 = 360 degrees
Since angle 4 is not given, we can assume it is 0 degrees, as it is not included in the given information.
So, we have:
Angle 1 + 31 + 72 + 0 = 360 degrees
Simplifying the equation:
Angle 1 + 103 = 360
Subtracting 103 from both sides:
Angle 1 = 360 - 103
Calculating:
Angle 1 = 257 degrees
Therefore, angle 1 is 257 degrees.