Lines M and T are parallel with a Transversal line P. [Labeling for the angles on line P intersecting M are, top right corner is angle 1, the top left is angle 2, the bottom left is angle 3, and the bottom left is angle 4. This pattern {top left, top right, bottom left, bottom right} is the same for line P intersecting through line T, however the angle numbers are angles 5 to 8.] (exterior angles are 1, 2, 7, and 8. interior angles are 3, 4, 5, and 6)
I know that M<1=5x-16 and M<8=2x+32. I must find all 8 angles in DEGREE form. However i do not know how to do that.
You know ∠5 + ∠8 = 180
∠5 = 180 - (2x+32) = 148-2x
you also know that ∠1 = ∠5
5x - 16 = 148 - 2x
7x = 164
x = 164/7° --- I would would have expected the answer to come out "nicer".
So angle 1 = 5x-16 = 5(164/7) - 16 = 708/7
check my arithmetic, then carefully calculate the other angles.
To find the measures of the angles, you can start by setting up equations using the given angles M<1 and M<8.
1. Use the fact that angles 1 and 5 are corresponding angles formed by the transversal P crossing lines M and T. Due to the parallel lines, corresponding angles are congruent. Set up the equation:
M<1 = M<5
5x-16 = M<5
2. Use the fact that angles 1 and 7 are exterior angles formed by the transversal P crossing lines M and T. Due to the parallel lines, exterior angles are congruent. Set up the equation:
M<1 = M<7
5x-16 = M<7
3. Use the fact that angles 7 and 8 are corresponding angles formed by the transversal P crossing lines M and T. Due to the parallel lines, corresponding angles are congruent. Set up the equation:
M<7 = M<8
M<7 = 2x+32
Now you have three equations, which you can solve to find the values of x and subsequently the measures of the angles.
4. Solve equation 2 for M<7:
5x-16 = M<7
5. Substitute the value of M<7 from equation 4 into equation 3:
5x-16 = 2x+32
6. Solve the equation to find the value of x:
3x = 48
x = 16
Now that you have found the value of x, you can substitute it back into the equations to find the measures of the angles.
7. Substitute x=16 into equation 1 to find M<5:
5(16)-16 = 64
M<5 = 64 degrees
8. Substitute x=16 into equation 2 to find M<7:
5(16)-16 = 64
M<7 = 64 degrees
9. Substitute x=16 into equation 3 to find M<8:
2(16)+32 = 64
M<8 = 64 degrees
Now you have found the measures of angles 1, 5, 7, and 8. To find the measures of the remaining angles, you can use the properties of angles formed by parallel lines and a transversal.
10. Use the fact that angles 1 and 3 are alternate interior angles formed by the transversal P crossing lines M and T. Due to the parallel lines, alternate interior angles are congruent. Set up the equation:
M<1 = M<3
M<3 = 5x-16 = 64 degrees
11. Use the fact that angles 2 and 4 are alternate interior angles formed by the transversal P crossing lines M and T. Due to the parallel lines, alternate interior angles are congruent. Set up the equation:
M<2 = M<4
M<4 = 5x-16 = 64 degrees
12. Use the fact that angles 3 and 5 are corresponding angles formed by the transversal P crossing lines M and T. Due to the parallel lines, corresponding angles are congruent. Set up the equation:
M<3 = M<5
M<3 = 5x-16 = 64 degrees
13. Use the fact that angles 4 and 6 are corresponding angles formed by the transversal P crossing lines M and T. Due to the parallel lines, corresponding angles are congruent. Set up the equation:
M<4 = M<6
M<6 = 5x-16 = 64 degrees
Now you have found the measures of all 8 angles as follows:
M<1 = 64 degrees
M<2 = 64 degrees
M<3 = 64 degrees
M<4 = 64 degrees
M<5 = 64 degrees
M<6 = 64 degrees
M<7 = 64 degrees
M<8 = 64 degrees