Answer:
Please see detailed solution below.
Step-by-step explanation:
28.) Given the following facts:
GH bisects < FGI
m < FGH = (3x - 3)°
m <HGI = (4x - 14)°
Since < FGH and < HGI are adjacent angles that have the same vertex, point G:
We can use the Angle Addition Postulate which states that if point G lies in the interior of < FGI then m < FGH + m < HGI = m < FGI.
Essentially, the Angle Addition Postulate implies that the sum of the parts equal the whole.
To find the measure of < FGI:
Establish the following equality statement:
m < FGH = m < HGI
Substitute the given values into the equality statement:
(3x - 3)° = (4x - 14)°
Combine like terms:
3x - 3 = 4x - 14
Subtract 3x from both sides:
3x - 3x - 3 = 4x - 3x - 14
- 3 = x - 14
Add 4 to both sides:
14 - 3 = x - 14 + 14
11 = x
Now that we have the value for x, substitute 11 into the sum of the given values for < FGH and < HGI to solve for m < FGI:
m < FGH + m < HGI = m < FGI
(3x - 3)° + (4x - 14)°= m < FGI
[3(11) - 3]° + [4(11) - 14]° = m < FGI
(33 - 3)° + (44 -14)° = m < FGI
30° + 30° = m < FGI
60° = m < FGI
Therefore:
a.) x = 11; m < FGH = 30°
b.) m < HGI = 30°
c.) m < FGI = 60°
29.) BD Bisects <ABC.
Given that m < ABD = 5x, and m < DBC = 3x + 10:
Applying the Angle Addition Postulate:
m < ABD + m < DBC = m < ABC
Using the same method of solving for the value of x earlier:
m < ABD = m < DBC
5x = 3x + 10
Subtract 3x from both sides:
5x - 3x = 3x - 3x + 10
2x = 10
Divide both sides by 2 to solve for x:
2x/2 = 10/2
x = 5
Substitute the value of x to find the value of m < ABC:
m < ABD + m < DBC = m < ABC
5x + 3x + 10 = m < ABC
5(5) + [3(5) + 10] = m < ABC
25 + (15 + 10) = m < ABC
25 + 25 = m < ABC
x = 5
m < ABC = 50
m < ABD = 25
m < DBC = 25
30) BD Bisects <ABC.
Given that m < ABC = 4x - 12, and m < ABD = 24
Doing the two other given problems, we can assume that m < ABD is equal to m < DBC, and their sum is equal to m < ABC.
Since we're already given the value for m < ABD, then it also means that m < DBC = 24.
Applying the Angle Addition Postulate:
m < ABD + m < DBC = m < ABC
24 + 24 = 4x - 12
48 = 4x - 12
Add 12 to both sides:
48 + 12 = 4x - 12 + 12
60 = 4x
Divide both sides by 4:
60/4 = 4x/4
15 = x
Substitute x into m < ABC to find its measure:
m < ABC = 4x - 12
m < ABC = 4(15) - 12
m < ABC = 60 - 12
m < ABC = 48
x = 15
This proves my statements earlier how the Angle Addition Postulate applies all of the given problems for this post.
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