Question

problem 1: segment addition postulate
on line segment ab, point c is between a and b.
if ac = 2x + 5
cb = 3x - 2
ab = 31
find the value of x and the length of ac. show all work.
work space:
problem 2: angle addition postulate
in the diagram, ∠lmo and ∠nmo are adjacent angles that form ∠lmn.
m∠lmo = 3x + 3
m∠nmo = 7x - 11
m∠lmn = 22°
solve for m∠lmo and m∠nmo. explain each step.
work space (steps & explanation):

Explanation:

Problem 1

Step1: Apply segment - addition postulate

Since $AC + CB=AB$, we substitute the given expressions: $(2x + 5)+(3x-2)=31$.

Step2: Simplify the left - hand side

Combine like terms: $2x+3x + 5-2=31$, which gives $5x + 3=31$.

Step3: Isolate the variable term

Subtract 3 from both sides: $5x=31 - 3$, so $5x=28$.

Step4: Solve for x

Divide both sides by 5: $x=\frac{28}{5}=5.6$.

Step5: Find the length of AC

Substitute $x = 5.6$ into the expression for $AC$: $AC=2x+5=2\times5.6 + 5=11.2+5=16.2$.

Step1: Apply angle - addition postulate

Since $m\angle{LMO}+m\angle{NMO}=m\angle{LMN}$, we substitute the given expressions: $(3x + 3)+(7x-11)=22$.

Step2: Simplify the left - hand side

Combine like terms: $3x+7x+3 - 11=22$, which gives $10x-8=22$.

Step3: Isolate the variable term

Add 8 to both sides: $10x=22 + 8$, so $10x=30$.

Step4: Solve for x

Divide both sides by 10: $x = 3$.

Step5: Find $m\angle{LMO}$

Substitute $x = 3$ into the expression for $m\angle{LMO}$: $m\angle{LMO}=3x+3=3\times3+3=9 + 3=12^{\circ}$.

Step6: Find $m\angle{NMO}$

Substitute $x = 3$ into the expression for $m\angle{NMO}$: $m\angle{NMO}=7x-11=7\times3-11=21-11 = 10^{\circ}$.

Answer:

$x = 5.6$, $AC=16.2$

Problem 2