EC bisects BED, mLABB = 11x-12 and m LCED = 4x+ 1. Find m LAEC .
To find the measure of angle LAEC, we first need to find the measure of angle BED.
Since EC bisects BED, that means angle BEC is congruent to angle DEC.
We can write this as:
m BEC = m DEC.
We are also given that angle BEC is 11x - 12 and angle DEC is 4x + 1.
So we have:
11x - 12 = 4x + 1.
Subtracting 4x from both sides:
7x - 12 = 1.
Adding 12 to both sides:
7x = 13.
Dividing both sides by 7:
x = 13/7.
Now we can substitute this value of x back into one of the angle measures to find the measure of angle BEC.
Using angle BEC = 11x - 12:
m BEC = 11(13/7) - 12,
m BEC = 143/7 - 84/7,
m BEC = 59/7.
Since angle LAEC is supplementary to angle BEC, the measure of angle LAEC is:
180 - m BEC = 180 - 59/7 = (1260 - 59)/7 = 1201/7.
To find m(LAEC), we can start by using the fact that EC bisects BED. This means that angles BED and CED are congruent.
Since EC bisects angle BED, we can set the measures of m(LABB) and m(LCED) equal to each other, and solve for x:
m(LABB) = m(1) + m(2)
11x - 12 = 4x + 1
To solve for x, we can start by subtracting 4x from both sides of the equation:
11x - 4x - 12 = 4x - 4x + 1
7x - 12 = 1
Next, we can add 12 to both sides:
7x - 12 + 12 = 1 + 12
7x = 13
Finally, we can divide both sides by 7 to solve for x:
7x/7 = 13/7
x = 13/7
Now that we have the value of x, we can substitute it back into the equation m(LAEC) = 11x - 12 to find the measure of m(LAEC):
m(LAEC) = 11(13/7) - 12
m(LAEC) = 143/7 - 84/7
m(LAEC) = 59/7
Therefore, m(LAEC) = 59/7.
To find the measure of angle LAEC, we need to use the angle bisector theorem. According to the angle bisector theorem, when an angle bisector intersects a side of a triangle, it divides that side into segments that are proportional to the lengths of the other two sides of the triangle.
In this case, EC bisects angle BED, which means it divides the side BD into segments that are proportional to the lengths of side BE and side ED. Let's assign variables to these segments to help us solve the problem.
Let's assume the length of BE is represented by 'a' and the length of ED is represented by 'b'. Thus, the length of BD can be represented as (a + b).
According to the angle bisector theorem, we have the following proportion:
BD / DC = BE / EC
Substituting the values from the given information, we have:
(a + b) / DC = a / EC
Simplifying this equation, we get:
(a + b) * EC = a * DC
Now, let's consider the lengths of the segments given in the problem. The measure of angle LABB is given as 11x - 12, and the measure of angle LCED is given as 4x + 1.
From the given information, we can determine that angle LAC is equal to angle LEC since EC is the angle bisector. Therefore, mLABB + mLEC = 180 degrees.
So, 11x - 12 + 4x + 1 = 180
Simplifying the equation, we get:
15x - 11 = 180
Adding 11 to both sides, we have:
15x = 191
Dividing both sides by 15, we get:
x = 191/15
Now that we have the value of x, we can substitute it back into the equation for mLABB to find its measure.
mLABB = 11(191/15) - 12
Simplifying this expression with the given value of x, we find:
mLABB = 141.067
Similarly, we can substitute the value of x into the equation for mLCED to find its measure.
mLCED = 4(191/15) + 1
Simplifying this expression with the given value of x, we find:
mLCED = 52.067
Since angle LAC is equal to angle LEC, we can find mLAEC by subtracting the sum of the measures of angles LABB and LCED from 180 degrees.
mLAEC = 180 - (mLABB + mLCED)
Substituting the values we found earlier, we get:
mLAEC = 180 - (141.067 + 52.067)
Simplifying this expression, we find:
mLAEC = 180 - 193.134
Therefore, the measure of angle LAEC is -13.134 degrees.