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Question

Accepted Answer

### Problem 7 (Rectangle, Diagonals Bisect Each Other)
# Explanation:
## Step1: Recall rectangle diagonal property
In a rectangle, diagonals are equal and bisect each other. So, the segments of the diagonals are equal. Thus, $a - 7=8$ and $10 = 2b - 6$.

## Step2: Solve for $a$
From $a - 7=8$, add 7 to both sides: $a=8 + 7=15$.

## Step3: Solve for $b$
From $10 = 2b - 6$, add 6 to both sides: $10+6 = 2b$, so $16 = 2b$. Divide by 2: $b = 8$.

# Answer:
$a = 15$, $b = 8$

### Problem 8 (Parallelogram, Opposite Sides Equal)
# Explanation:
## Step1: Recall parallelogram side property
In a parallelogram, opposite sides are equal. So, $a + 15=3b - 17$ and $3a + 11=b - 11$.

## Step2: Solve the system of equations
From the second equation $3a + 11=b - 11$, we can express $b=3a + 22$. Substitute $b = 3a + 22$ into the first equation:
$a + 15=3(3a + 22)-17$
$a + 15=9a + 66 - 17$
$a + 15=9a + 49$
Subtract $a$ from both sides: $15 = 8a + 49$
Subtract 49: $15-49 = 8a$
$-34 = 8a$ (Wait, this seems wrong. Maybe I mixed the sides. Let's re - check. Maybe $a + 15=3a + 11$ and $3b - 17=b - 11$. Ah, that's the mistake! In a parallelogram, opposite sides are equal. So, adjacent sides? No, opposite sides. So, $a + 15$ and $3a + 11$ are opposite? Wait, the parallelogram has sides $a + 15$, $3b - 17$, $3a + 11$, $b - 11$. So correct opposite sides: $a + 15=3a + 11$ and $3b - 17=b - 11$.

## Step3: Solve for $a$
From $a + 15=3a + 11$, subtract $a$: $15 = 2a + 11$, subtract 11: $4 = 2a$, so $a = 2$.

## Step4: Solve for $b$
From $3b - 17=b - 11$, subtract $b$: $2b - 17=-11$, add 17: $2b=6$, so $b = 3$.

# Answer:
$a = 2$, $b = 3$

### Problem 9 (Parallelogram, Alternate Interior Angles)
# Explanation:
## Step1: Recall parallelogram angle and diagonal property
In a parallelogram, a diagonal divides it into two congruent triangles. Also, alternate interior angles are equal. So, $x = 50$ (since the diagonal creates alternate interior angles which are equal in the parallelogram).

# Answer:
$x = 50$

### Problem 10 (Parallelogram, Opposite Angles Equal and Consecutive Angles Supplementary)
# Explanation:
## Step1: Recall parallelogram angle properties
In a parallelogram, opposite angles are equal, so $2x + 11=x - 5$ (Wait, no. Wait, consecutive angles are supplementary. Wait, the angles are $(2x + 11)^{\circ}$, $(2y + 3)^{\circ}$, $(x - 5)^{\circ}$. Wait, in a parallelogram, opposite angles are equal. So $(2x + 11)=(x - 5)$? No, that can't be. Wait, maybe $(2x + 11)$ and $(x - 5)$ are consecutive angles? Wait, no, let's re - examine. In a parallelogram, opposite angles are equal, so if one angle is $(2x + 11)^{\circ}$ and another is $(x - 5)^{\circ}$, maybe they are opposite? No, that would give $2x+11=x - 5$, $x=-16$, which is impossible. Wait, maybe consecutive angles: $(2x + 11)+(x - 5)=180$. Let's try that.

## Step2: Solve for $x$
$2x + 11+x - 5 = 180$
$3x+6 = 180$
Subtract 6: $3x = 174$
Divide by 3: $x = 58$

For $y$, since opposite angles are equal, but we need more info? Wait, maybe the other pair: $(2y + 3)$ and... Wait, maybe the problem has a typo, but assuming we only solve for $x$ first.

# Answer:
$x = 58$ (assuming consecutive angles are supplementary)

### Problem 11 (Rhombus or Kite, Diagonals Bisect Each Other and Sides Equal)
# Explanation:
## Step1: Recall diagonal and side properties
In a rhombus (or a kite - like figure with diagonals bisecting), the segments of the diagonals are equal, so $3y - 5 = 4y - 5$ (Wait, no, the vertical segments: $3y - 5$ and $4y - 5$? Wait, the horizontal segments: $x + 9=2x + 7$ and $3y - 5 = 4y - 5$.

## Step2: Solve for $x$
From $x + 9=2x + 7$, subtract $x$ and 7: $9 - 7=2x - x$, so $x = 2$.

## Step3: Solve for $y$
From $3y - 5 = 4y - 5$, subtract $3y$ and add 5: $0=y$.

# Answer:
$x = 2$, $y = 0$

### Problem 12 (Parallelogram, Diagonals Bisect Each Other)
# Explanation:
## Step1: Recall parallelogram diagonal property
In a parallelogram, diagonals bisect each other. So, $4w - 7=2w + 3$ and $3b + 1=2b + 5$.

## Step2: Solve for $w$
From $4w - 7=2w + 3$, subtract $2w$ and add 7: $4w-2w=3 + 7$, $2w = 10$, so $w = 5$.

## Step3: Solve for $b$
From $3b + 1=2b + 5$, subtract $2b$ and 1: $3b-2b=5 - 1$, so $b = 4$.

# Answer:
$w = 5$, $b = 4$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7 (Rectangle, Diagonals Bisect Each Other)

Step1: Recall rectangle diagonal property

Step2: Solve for \(a\)

Step3: Solve for \(b\)

Step1: Recall parallelogram side property

Step2: Solve the system of equations

Step3: Solve for \(a\)

Step4: Solve for \(b\)

Step1: Recall parallelogram angle and diagonal property

Answer:

Problem 8 (Parallelogram, Opposite Sides Equal)