1. for the problems below, find the value of the missing variable. show the steps used to find your answer.
4. a. complete the proof that (mangle vux + mangle uxy=180^{circ}).
statement reason
1. (overleftrightarrow{wy}paralleloverleftrightarrow{fv})
2. (angle uxycongangle suv)
3. (mangle suv + mangle vux = 180^{circ})
4. (mangle vux + mangle uxy = 180^{circ})
b. given: (ac = 19) prove: (x=-6)
|statements|reasons|
|----|----|
| |given|
| (ab=x + 18)|
| (bc = 3x+25)|
| (ab + bc=ac)|segment addition postulate|
| (x + 18+3x + 25=19)|
| (4x+43 = 19)|
| |subtraction p.o.e.|
| (x=-6)|

Question

Accepted Answer

### a.
# Explanation:
## Step1: Use vertical - angle property
Vertical angles are equal. So, $8x=2x + 72$.
## Step2: Solve the equation for $x$
Subtract $2x$ from both sides: $8x-2x=2x + 72-2x$, which gives $6x=72$. Then divide both sides by 6: $x = 12$.
# Answer:
$x = 12$

### b.
# Explanation:
## Step1: Note the right - angle
Since $\angle ABC = 90^{\circ}$, then $4x+(7x - 9)=90$.
## Step2: Combine like terms
$4x+7x-9 = 90$, so $11x-9 = 90$.
## Step3: Add 9 to both sides
$11x-9 + 9=90 + 9$, getting $11x=99$.
## Step4: Divide by 11
$x = 9$
# Answer:
$x = 9$

### c.
# Explanation:
## Step1: Use the angle - sum property of a triangle
The sum of angles in a triangle is $180^{\circ}$. So, $80+(8x + 2)+6x=180$.
## Step2: Combine like terms
$80+8x+2 + 6x=180$, which simplifies to $14x+82 = 180$.
## Step3: Subtract 82 from both sides
$14x+82-82=180-82$, giving $14x=98$.
## Step4: Divide by 14
$x = 7$
# Answer:
$x = 7$

### d.
# Explanation:
## Step1: Note the straight - angle
The sum of angles on a straight line is $180^{\circ}$. So, $x+3x+2x=180$.
## Step2: Combine like terms
$6x=180$.
## Step3: Divide by 6
$x = 30$
# Answer:
$x = 30$

### e.
# Explanation:
## Step1: Use the angle - sum property of a straight - line
The sum of angles on a straight line is $180^{\circ}$. So, $x + 40+x+60=180$.
## Step2: Combine like terms
$2x+100 = 180$.
## Step3: Subtract 100 from both sides
$2x+100-100=180-100$, getting $2x=80$.
## Step4: Divide by 2
$x = 40$
# Answer:
$x = 40$

### f.
# Explanation:
## Step1: Note the right - angle and equal angles
Since there is a right - angle ($90^{\circ}$) and three equal angles $y$, then $3y=90$.
## Step2: Divide by 3
$y = 30$
# Answer:
$y = 30$

### 4.a.
# Explanation:
## Step1: Given
$\overleftrightarrow{WY}\parallel\overleftrightarrow{ZV}$ (Given).
## Step2: Corresponding angles are equal
$\angle UXY=\angle SUV$ (Corresponding angles postulate).
## Step3: Linear - pair property
$m\angle SUV+m\angle VUX = 180^{\circ}$ (Linear - pair postulate).
## Step4: Substitution
Since $\angle UXY=\angle SUV$, then $m\angle VUX+m\angle UXY = 180^{\circ}$ (Substitution property).
# Answer:
1. Given; 2. Corresponding angles postulate; 3. Linear - pair postulate; 4. Substitution property

### 4.b.
# Explanation:
## Step1: Given
$AC = 19$ (Given).
## Step2: Given segment lengths
$AB=x + 18$ and $BC=3x + 25$ (Given).
## Step3: Segment addition postulate
$AB + BC=AC$ (Segment addition postulate).
## Step4: Substitute
$x + 18+3x+25=19$ (Substitution).
## Step5: Combine like terms
$4x+43 = 19$ (Combining like terms).
## Step6: Subtract 43 from both sides
$4x+43-43=19-43$ (Subtraction property of equality), getting $4x=-24$.
## Step7: Divide by 4
$x=-6$ (Division property of equality).
# Answer:
Statements: 1. $AC = 19$; 2. $AB=x + 18$; 3. $BC=3x + 25$; 4. $AB + BC=AC$; 5. $x + 18+3x+25=19$; 6. $4x+43 = 19$; 7. $4x=-24$; 8. $x=-6$
Reasons: 1. Given; 2. Given; 3. Given; 4. Segment addition postulate; 5. Substitution; 6. Combining like terms; 7. Subtraction property of equality; 8. Division property of equality

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

a.

Step1: Use vertical - angle property

Step2: Solve the equation for \(x\)

Step1: Note the right - angle

Step2: Combine like terms

Step3: Add 9 to both sides

Step4: Divide by 11

Step1: Use the angle - sum property of a triangle

Step2: Combine like terms

Step3: Subtract 82 from both sides

Step4: Divide by 14

Answer:

b.