4. the diagram shows $overline{eh}$ intersecting $overline{cd}$, $overline{af}$, and $overline{ab}$ at point $a$. use the diagram to complete parts (a) and (b).
a. describe the relevant angle relationship.
b. write an equation for the angle relationship and solve for $x$.

Question

Accepted Answer

### Part (a)
# Brief Explanations:
Looking at the diagram, we see that $ \angle FAD = 20^\circ $, $ \angle FAE $ is a right angle ($ 90^\circ $), and the angles around point $ A $ on one side of the straight line $ EH $ (or the angles related to the right angle and the given angles) should sum up appropriately. Also, the angles $ x^\circ $, $ \frac{5}{3}x^\circ $, along with the right angle and the $ 20^\circ $ angle, might be related to supplementary or complementary angles, but more precisely, since $ EH $ is a straight line (so a straight angle is $ 180^\circ $) and there's a right angle ($ 90^\circ $) between $ E $ and $ F $ (or related rays), the relevant angle relationship is that the sum of the right angle ($ 90^\circ $), the $ 20^\circ $ angle, $ x^\circ $, and $ \frac{5}{3}x^\circ $ should equal $ 180^\circ $ (since they are on a straight line, forming a linear pair or supplementary angles with the right angle considered). Alternatively, the angles $ x $ and $ \frac{5}{3}x $ along with the $ 20^\circ $ and the right angle make up a straight angle (180 degrees). So the relevant angle relationship is that the sum of $ 90^\circ $, $ 20^\circ $, $ x^\circ $, and $ \frac{5}{3}x^\circ $ is $ 180^\circ $ (they are supplementary as they lie on a straight line, forming a linear pair with the right angle - related angles).
# Answer:
The relevant angle relationship is that the sum of the right angle ($ 90^\circ $), $ 20^\circ $, $ x^\circ $, and $ \frac{5}{3}x^\circ $ equals $ 180^\circ $ (these angles lie on a straight line, so they are supplementary).

### Part (b)
# Explanation:
## Step 1: Set up the equation
From the angle relationship in part (a), we know that $ 90 + 20 + x + \frac{5}{3}x = 180 $. First, combine the constant terms: $ 90 + 20 = 110 $, so the equation becomes $ 110 + x + \frac{5}{3}x = 180 $.
## Step 2: Combine like terms (the $ x $ terms)
To combine $ x $ and $ \frac{5}{3}x $, we can write $ x $ as $ \frac{3}{3}x $. So $ \frac{3}{3}x + \frac{5}{3}x = \frac{8}{3}x $. Now the equation is $ 110 + \frac{8}{3}x = 180 $.
## Step 3: Solve for $ x $
Subtract 110 from both sides: $ \frac{8}{3}x = 180 - 110 = 70 $. Then multiply both sides by $ \frac{3}{8} $ to isolate $ x $: $ x = 70\times\frac{3}{8} = \frac{210}{8} = \frac{105}{4} = 26.25 $. Wait, no, wait, maybe I made a mistake in the angle relationship. Wait, looking again: the right angle is between $ E $ and $ F $, so the angles on the right side of the right angle (towards $ D $, $ B $, $ C $): the straight line is $ EH $, so from $ E $ to the right is 180 degrees. The right angle is $ 90^\circ $ (between $ E $ and $ F $), then $ \angle FAD = 20^\circ $, then $ \angle DAB = \frac{5}{3}x $, and $ \angle BAC = x $. Wait, maybe the correct angle relationship is that the sum of $ 20^\circ $, $ \frac{5}{3}x^\circ $, $ x^\circ $, and the right angle's complement? No, wait, the right angle is $ 90^\circ $, so the angles from $ F $ to $ B $ to $ C $ to the right side of $ EH $: actually, the straight line is $ EH $, so the total angle on one side (from $ E $ to the right) is $ 180^\circ $. The right angle is $ 90^\circ $ (between $ E $ and $ F $), so the angles from $ F $ to $ H $ (the right side) should sum to $ 90^\circ $? Wait, no, the diagram shows a right angle symbol between $ E $ and $ F $, so $ \angle EAF = 90^\circ $. Then, the angles $ \angle FAD = 20^\circ $, $ \angle DAB = \frac{5}{3}x $, $ \angle BAC = x $, and these three angles plus $ \angle EAF $ should sum to $ 180^\circ $ (since $ EH $ is a straight line). So $ 90 + 20 + \frac{5}{3}x + x = 180 $. Let's recalculate:

## Step 1: Correct equation setup
$ 90 + 20 + \frac{5}{3}x + x = 180 $
Combine constants: $ 90 + 20 = 110 $
Combine $ x $ terms: $ x + \frac{5}{3}x = \frac{3}{3}x + \frac{5}{3}x = \frac{8}{3}x $
So equation: $ 110 + \frac{8}{3}x = 180 $

## Step 2: Subtract 110 from both sides
$ \frac{8}{3}x = 180 - 110 = 70 $

## Step 3: Multiply both sides by $ \frac{3}{8} $
$ x = 70\times\frac{3}{8} = \frac{210}{8} = \frac{105}{4} = 26.25 $? Wait, that seems odd. Wait, maybe the right angle is between $ F $ and the horizontal line? No, the diagram shows a right angle at $ A $ between the horizontal line ( $ E $ to the right) and $ F $. So the horizontal line is $ EH $, with $ E $ on the left and $ H $ on the right? Wait, no, the arrow for $ E $ is left, and the right side is the other direction. Wait, maybe the straight line is $ EH $, so from $ E $ (left) to $ H $ (right) is 180 degrees. The right angle is between $ E $ and $ F $, so $ \angle EAF = 90^\circ $, so the angles from $ F $ to $ H $ (right side) should sum to $ 90^\circ $. Wait, that's the mistake! The right angle is $ 90^\circ $, so $ \angle EAF = 90^\circ $, so the angles $ \angle FAD = 20^\circ $, $ \angle DAB = \frac{5}{3}x $, and $ \angle BAC = x $ should sum to $ 90^\circ $, because $ \angle EAF = 90^\circ $, and $ \angle FAD + \angle DAB + \angle BAC = \angle FAC $ (or related) which is complementary to $ \angle EAF $? No, wait, the diagram: the right angle is between $ E $ (left) and $ F $ (up), then $ D $ is another ray, then $ B $, then $ C $, then back to the horizontal line. Wait, maybe the correct angle relationship is that the sum of $ 20^\circ $, $ \frac{5}{3}x $, and $ x $ is equal to $ 90^\circ $, because the right angle is $ 90^\circ $, so those three angles are complementary to the right angle? No, that doesn't make sense. Wait, let's look at the diagram again: the right angle is at $ A $ between the horizontal line (left to right) and the ray $ AF $. Then, the rays $ AD $, $ AB $, $ AC $ are on the other side of $ AF $ from $ E $. So the angle between $ AF $ and $ AD $ is $ 20^\circ $, between $ AD $ and $ AB $ is $ \frac{5}{3}x $, between $ AB $ and $ AC $ is $ x $, and the angle between $ AC $ and the horizontal line (right side) is... Wait, no, the horizontal line is $ EH $, with $ E $ on the left and $ H $ on the right. The right angle is between $ E $ (left) and $ F $ (up), so the angle from $ F $ to $ H $ (right) is $ 90^\circ $. So the angles $ \angle FAD = 20^\circ $, $ \angle DAB = \frac{5}{3}x $, and $ \angle BAC = x $ are inside this $ 90^\circ $ angle? Wait, that would mean $ 20 + \frac{5}{3}x + x = 90 $. Let's try that.

## Step 1: Correct equation setup (revised)
Since $ \angle EAF = 90^\circ $ (right angle), and the angles $ \angle FAD = 20^\circ $, $ \angle DAB = \frac{5}{3}x $, $ \angle BAC = x $ are adjacent and form a linear pair or sum to $ 90^\circ $ (because $ \angle EAF = 90^\circ $ and the total from $ F $ to $ C $ to the horizontal line is $ 90^\circ $)? Wait, no, the horizontal line is $ EH $, so from $ E $ to $ H $ is $ 180^\circ $. The right angle is $ 90^\circ $ ( $ \angle EAF $ ), so the angle from $ F $ to $ H $ is $ 90^\circ $. Therefore, $ \angle FAD + \angle DAB + \angle BAC = 90^\circ $, because they are within the $ 90^\circ $ angle from $ F $ to $ H $. So:

$ 20 + \frac{5}{3}x + x = 90 $

## Step 2: Combine like terms
$ x + \frac{5}{3}x = \frac{3}{3}x + \frac{5}{3}x = \frac{8}{3}x $

So equation: $ 20 + \frac{8}{3}x = 90 $

## Step 3: Subtract 20 from both sides
$ \frac{8}{3}x = 90 - 20 = 70 $

## Step 4: Multiply both sides by $ \frac{3}{8} $
$ x = 70\times\frac{3}{8} = \frac{210}{8} = \frac{105}{4} = 26.25 $. Wait, but that still gives $ x = 26.25 $, but let's check the original diagram again. Wait, maybe the right angle is between $ F $ and the vertical line? No, the diagram shows a right angle symbol between the horizontal line ( $ E $ to right) and $ F $ (up), so $ \angle EAF = 90^\circ $. Then the angles from $ F $ to $ D $ is $ 20^\circ $, $ D $ to $ B $ is $ \frac{5}{3}x $, $ B $ to $ C $ is $ x $, and $ C $ to the horizontal line (right) is... Wait, no, the horizontal line is $ EH $, so the total angle from $ E $ to $ H $ is $ 180^\circ $. The right angle is $ 90^\circ $ ( $ \angle EAF $ ), so the angle from $ F $ to $ H $ is $ 90^\circ $, so $ \angle FAD + \angle DAB + \angle BAC = 90^\circ $, which is what we used. So the equation is $ 20 + \frac{5}{3}x + x = 90 $, leading to $ x = \frac{105}{4} = 26.25 $. But let's verify:

If $ x = 26.25 $, then $ \frac{5}{3}x = \frac{5}{3}\times26.25 = 43.75 $. Then $ 20 + 43.75 + 26.25 = 20 + 70 = 90 $, which matches the right angle. So that's correct.

Wait, but initially, I thought the sum was 180, but that was a mistake. The right angle is $ 90^\circ $, so the angles from $ F $ to $ C $ (through $ D $, $ B $) sum to $ 90^\circ $, because $ \angle EAF = 90^\circ $, so the remaining angle from $ F $ to $ H $ (the right side) is $ 90^\circ $. So the correct equation is $ 20 + \frac{5}{3}x + x = 90 $, leading to $ x = 26.25 $ (or $ \frac{105}{4} $).

But let's re - examine the problem statement: "EH intersecting CD, AF, and AB at point A". So EH is a straight line, AF is a ray, AB is a ray, CD is a line. The right angle is between E and F, so $ \angle EAF = 90^\circ $. Then, the angles on the straight line EH: $ \angle EAF + \angle FAD + \angle DAB + \angle BAC = 180^\circ $, because EH is a straight line (180 degrees). So $ 90 + 20 + \frac{5}{3}x + x = 180 $. Let's calculate that:

$ 90 + 20 = 110 $, $ \frac{5}{3}x + x = \frac{8}{3}x $, so $ 110 + \frac{8}{3}x = 180 $

Subtract 110: $ \frac{8}{3}x = 70 $

Multiply by $ \frac{3}{8} $: $ x = 70\times\frac{3}{8} = \frac{210}{8} = 26.25 $. So both ways, we get $ x = 26.25 $, but the key is the angle relationship. So the correct equation is $ 90 + 20 + x + \frac{5}{3}x = 180 $ (since EH is a straight line, 180 degrees), and solving it gives $ x = 26.25 $.

# Answer:
## Equation:
$ 90 + 20 + x + \frac{5}{3}x = 180 $ (or simplified as $ 110 + \frac{8}{3}x = 180 $)

## Solution for $ x $:
### Step 1: Combine constant terms
$ 90 + 20 = 110 $, so the equation becomes $ 110 + x + \frac{5}{3}x = 180 $.

### Step 2: Combine $ x $ - terms
$ x+\frac{5}{3}x=\frac{3}{3}x + \frac{5}{3}x=\frac{8}{3}x $, so $ 110+\frac{8}{3}x = 180 $.

### Step 3: Isolate the $ x $ - term
Subtract 110 from both sides: $ \frac{8}{3}x=180 - 110 = 70 $.

### Step 4: Solve for $ x $
Multiply both sides by $ \frac{3}{8} $: $ x = 70\times\frac{3}{8}=\frac{210}{8}=\frac{105}{4} = 26.25 $.

So the value of $ x $ is $ \boldsymbol{26.25} $ (or $ \frac{105}{4} $).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Set up the equation

Step 2: Combine like terms (the \( x \) terms)

Step 3: Solve for \( x \)

Step 1: Correct equation setup

Step 2: Subtract 110 from both sides

Step 3: Multiply both sides by \( \frac{3}{8} \)

Answer:

Part (b)