Line MN is perpendicular to line NP. Point R is in the interior of the angle MNP. If angle MNR = 5x+12 and angle PNR =6x+23. Find both angle measures
MNR = 5x+12.
PNR = 6x+23.
MNR+PNR = 90o.
(5x+12) + (6x+23) = 90
11x = 90-35 = 55
X = 5.
MNR = 5*5 + 12 = 37o.
PNR = 6*5 + 23 = 53o.
Wich line is peroendicular to line NP?
To find both angle measures, we need to understand the properties of perpendicular lines and the angles formed by them. When two lines intersect to form a pair of perpendicular lines, they create four angles.
In this case, we have lines MN and NP, with point R in the interior of angle MNP. This means we have two angles, MNR and PNR, that are adjacent (next to) each other. According to the properties of perpendicular lines, these angles are complementary, which means their sum is 90 degrees.
So, to find both angle measures, we need to set up an equation and solve for x.
Given: angle MNR = 5x + 12
angle PNR = 6x + 23
Since the angles are complementary, we can set up the equation:
(5x + 12) + (6x + 23) = 90
Simplifying the equation, we have:
11x + 35 = 90
Next, we isolate the variable x by subtracting 35 from both sides:
11x = 90 - 35
11x = 55
Finally, we solve for x by dividing both sides by 11:
x = 55 / 11
x = 5
Now that we have the value of x, we can substitute it back into the given equations to find the angle measures:
angle MNR = 5x + 12
= 5(5) + 12
= 25 + 12
= 37 degrees
angle PNR = 6x + 23
= 6(5) + 23
= 30 + 23
= 53 degrees
Therefore, angle MNR measures 37 degrees, and angle PNR measures 53 degrees.