i will be greatefull,if you help me in answering the below problem:
1.)Prove that the bisectors of two adjacent supplementary angles include a right angle.
Let the supplementary angles be
A and B . You can call them also A and 180-A, since they are supplementary.
Draw a straight line and let a second line join it at point O, to divide the straight line into a zero degree line and a 180 degree line. The space above the line is divided into angles A and 180 - A.
The bisector of A is A/2 degrees from the zero degree line.
The bisector of B is A + (180-A)/2 from the zero degree line.
The angle between the two bisectors is
A + (180-A)/2 -A/2 = 90
Prove that the bisector of two adjacent supplementary angle include a right angle
To prove that the bisectors of two adjacent supplementary angles include a right angle, we will follow these steps:
Step 1: Understand the problem
Let's start by understanding the problem statement. We are given two adjacent supplementary angles, which means that they add up to 180 degrees. We need to prove that the bisectors of these angles intersect at a right angle.
Step 2: Draw a diagram
Visualizing the problem can often aid in finding a solution. Draw two adjacent supplementary angles sharing a common vertex, and label them angle A and angle B.
B
/
/
/
/ A
Step 3: Identify the bisectors
Next, identify the bisectors of angles A and B. These are the lines that divide each angle into two equal parts, forming two smaller angles. Label these bisectors as line m and line n.
B
/
/
n
/ A
m
Step 4: Prove that the bisectors intersect
To prove that the bisectors intersect, consider the point where line m intersects line n. Let's call this point P.
Step 5: Apply angle bisector theorem
According to the angle bisector theorem, the line that divides an angle into two equal parts also divides the opposite side of the angle into two proportional segments. In other words, we can say that the ratio of the lengths of the segments formed by the intersection point on the opposite side is equal to the ratio of the lengths of the two adjacent sides of the angle.
In our case, angle A is divided into two equal parts by line m, and angle B is divided into two equal parts by line n. This means that angle APB is half of angle A, and angle APB is also half of angle B.
Step 6: Apply the sum of angles property
Since angle A and angle B are supplementary (they add up to 180 degrees), we can write the equation:
Angle A + Angle B = 180 degrees
Substituting the values in terms of angle APB:
2 * angle APB + 2 * angle APB = 180 degrees
4 * angle APB = 180 degrees
Dividing both sides by 4:
angle APB = 45 degrees
Step 7: Prove that angle APB is a right angle
Since angle APB is 45 degrees, we can conclude that it is a right angle. A right angle measures exactly 90 degrees, and 45 degrees is half of 90 degrees.
Therefore, the bisectors of two adjacent supplementary angles do intersect at a right angle.
Note: In this explanation, we followed a deductive reasoning approach using a combination of geometry principles and theorems. The steps presented here should be followed to prove the given statement accurately, showing the logical progression of the proof.