true or false? use the diagram at the right.
13. ∠5 and ∠4 are supplementary angles.
14. ∠6 and ∠5 are adjacent angles.
15. ∠1 and ∠2 are a linear pair.
16. find ( mangle bad ) (diagram with ( 31^circ ) and ( 24^circ ) at ( a ))
17. find ( x ). (diagram with angles ( (2x - 18)^circ ) and ( (4x)^circ ) on a straight line)
18. if ( mangle fhi = 142 ), what are ( mangle fhg ) and ( mangle ghi )? (diagram with angles ( (3x + 6)^circ ) and ( (9x - 8)^circ ))
in the diagram at the right, ( mangle hki = 48 ). find each of the following.
19. ( mangle hkj )
20. ( mangle ikj )
21. ( mangle fkj )
22. ( mangle fkg )
23. ( mangle fkh )
24. ( mangle gki ) (diagram with right angle, ( 48^circ ))

Question

Accepted Answer

### Question 16: Find $ m\angle BAD $
# Explanation:
## Step 1: Identify the angles
$ \angle BAC = 31^\circ $ and $ \angle CAD = 24^\circ $. To find $ \angle BAD $, we add these two angles.
$ m\angle BAD = m\angle BAC + m\angle CAD $
## Step 2: Calculate the sum
$ m\angle BAD = 31^\circ + 24^\circ = 55^\circ $
# Answer:
$ 55^\circ $

### Question 17: Find $ x $
# Explanation:
## Step 1: Identify the relationship
The angles $ (2x - 18)^\circ $ and $ (4x)^\circ $ are supplementary (they form a linear pair), so their sum is $ 180^\circ $.
$ (2x - 18) + 4x = 180 $
## Step 2: Solve for $ x $
Combine like terms: $ 6x - 18 = 180 $
Add 18 to both sides: $ 6x = 198 $
Divide by 6: $ x = \frac{198}{6} = 33 $
# Answer:
$ 33 $

### Question 18: If $ m\angle FHI = 142^\circ $, find $ m\angle FHG $ and $ m\angle GHI $
# Explanation:
## Step 1: Identify the relationship
$ \angle FHG = (3x + 6)^\circ $, $ \angle GHI = (9x - 8)^\circ $, and $ \angle FHI = \angle FHG + \angle GHI = 142^\circ $. Also, $ \angle GHI $ is a right angle? Wait, no, from the diagram, $ \angle GHI $ and $ \angle FHG $ add up to $ 142^\circ $, and also, maybe $ \angle FHG + \angle GHI = 142^\circ $, and also, $ \angle GHI $ is adjacent to a right angle? Wait, no, let's re - express.
Wait, actually, looking at the diagram, $ \angle FHG=(3x + 6)^\circ $, $ \angle GHI=(9x - 8)^\circ $, and $ \angle FHI=\angle FHG+\angle GHI = 142^\circ $
So, $ (3x + 6)+(9x - 8)=142 $
## Step 2: Solve for $ x $
Combine like terms: $ 12x - 2 = 142 $
Add 2 to both sides: $ 12x=144 $
Divide by 12: $ x = 12 $
## Step 3: Find $ m\angle FHG $
Substitute $ x = 12 $ into $ 3x + 6 $: $ 3\times12+6=36 + 6=42^\circ $
## Step 4: Find $ m\angle GHI $
Substitute $ x = 12 $ into $ 9x - 8 $: $ 9\times12-8=108 - 8 = 100^\circ $? Wait, no, that can't be. Wait, maybe $ \angle GHI $ is a right angle? Wait, no, the diagram shows that $ \angle GHI $ is adjacent to a right angle? Wait, I think I made a mistake. Wait, actually, $ \angle FHI = 142^\circ $, and $ \angle GHI $ and $ \angle FHG $ are such that $ \angle FHG+\angle GHI=\angle FHI $, and also, $ \angle GHI $ is a right angle? No, the diagram has $ G $ as a vertical line, $ H $ as the vertex, $ I $ is horizontal. Wait, maybe $ \angle FHG+(9x - 8)=90^\circ $? No, the problem says $ m\angle FHI = 142^\circ $. Let's start over.
Wait, the sum of $ \angle FHG=(3x + 6)^\circ $ and $ \angle GHI=(9x - 8)^\circ $ is $ 142^\circ $
So $ 3x+6 + 9x - 8=142 $
$ 12x-2 = 142 $
$ 12x=144 $
$ x = 12 $
Then $ m\angle FHG=3\times12 + 6=42^\circ $
$ m\angle GHI=9\times12-8 = 100^\circ $? But that doesn't make sense. Wait, maybe $ \angle GHI $ is a right angle? No, the problem must have $ \angle FHG+\angle GHI = 142^\circ $, and maybe $ \angle GHI $ is a right angle? No, the given is $ m\angle FHI = 142^\circ $. So according to the calculation:
$ m\angle FHG = 42^\circ $
$ m\angle GHI=142 - 42=100^\circ $
# Answer:
$ m\angle FHG = 42^\circ $, $ m\angle GHI = 100^\circ $

### Question 19: $ m\angle HKJ $
# Explanation:
## Step 1: Identify the relationship
From the diagram, $ \angle HKJ $ is a right angle (since $ \angle GKH = 90^\circ $ and $ JKG $ is a straight line), so $ m\angle HKJ = 90^\circ $
# Answer:
$ 90^\circ $

### Question 20: $ m\angle IKJ $
# Explanation:
## Step 1: Identify the angles
$ m\angle HKI = 48^\circ $ and $ m\angle HKJ = 90^\circ $. $ \angle IKJ=\angle HKJ+\angle HKI $
## Step 2: Calculate the sum
$ m\angle IKJ=90^\circ + 48^\circ=138^\circ $
# Answer:
$ 138^\circ $

### Question 21: $ m\angle FKJ $
# Explanation:
## Step 1: Identify the relationship
$ \angle FKJ $ and $ \angle IKJ $ are supplementary (they form a linear pair), so $ m\angle FKJ = 180^\circ - m\angle IKJ $
## Step 2: Calculate the value
$ m\angle FKJ=180^\circ - 138^\circ = 42^\circ $
# Answer:
$ 42^\circ $

### Question 22: $ m\angle FKG $
# Explanation:
## Step 1: Identify the relationship
$ \angle FKG $ and $ \angle HKI $ are equal (vertical angles or alternate angles). Since $ m\angle HKI = 48^\circ $, and also, $ \angle FKG $ can be found as $ 90^\circ - \angle FKH $? Wait, no, from the diagram, $ \angle FKG $ and $ \angle HKI $ are vertical angles? Wait, $ \angle FKG $ is adjacent to $ \angle GKH = 90^\circ $. Wait, $ m\angle FKG=90^\circ - m\angle FKH $? No, let's see, $ \angle FKG $ and $ \angle HKI $ are vertical angles? Wait, $ \angle HKI = 48^\circ $, and $ \angle FKG $ is equal to $ 90^\circ - \angle FKH $? No, actually, $ \angle FKG $ is a right angle? No, the diagram shows $ \angle GKH = 90^\circ $, and $ \angle FKG $ is such that $ \angle FKG + \angle FKH=90^\circ $, but $ \angle FKH $ and $ \angle HKI $ are related. Wait, $ m\angle FKG = 90^\circ - 48^\circ=42^\circ $? No, wait, $ \angle FKG $ is equal to $ \angle HKJ - \angle FKH $? No, let's use the fact that $ \angle IKJ = 138^\circ $, and $ \angle FKJ = 42^\circ $, and $ \angle FKG $ is a right angle? No, I think I made a mistake. Wait, $ \angle FKG $ is equal to $ 90^\circ - \angle HKI $? No, $ m\angle HKI = 48^\circ $, and $ \angle FKG $ is $ 90^\circ - 48^\circ = 42^\circ $? No, wait, the correct way: since $ \angle GKH = 90^\circ $ and $ \angle HKI = 48^\circ $, then $ \angle FKG=\angle HKI = 48^\circ $? No, that's not right. Wait, $ \angle FKG $ and $ \angle HKI $ are vertical angles? Wait, the lines $ FI $ and $ JG $ intersect at $ K $, and $ KH \perp JG $. So $ \angle FKG $ and $ \angle HKI $ are complementary to the same angle? Wait, $ \angle FKG + \angle FKH=90^\circ $, and $ \angle HKI+\angle FKH = 90^\circ $, so $ \angle FKG=\angle HKI = 48^\circ $? No, $ \angle HKI = 48^\circ $, $ \angle FKH = 90^\circ - 48^\circ = 42^\circ $, and $ \angle FKG = 90^\circ - \angle FKH=48^\circ $. Wait, I'm confused. Wait, the correct answer: since $ \angle GKH = 90^\circ $ and $ \angle HKI = 48^\circ $, then $ \angle FKG = 90^\circ - 48^\circ=42^\circ $? No, let's look at the diagram again. The angle $ \angle HKI = 48^\circ $, $ KH $ is perpendicular to $ JG $, so $ \angle GKH = 90^\circ $. Then $ \angle FKG $ is equal to $ \angle HKI = 48^\circ $? No, that's not. Wait, $ \angle FKG $ and $ \angle IKJ $ are supplementary? No, $ \angle IKJ = 138^\circ $, $ \angle FKJ = 42^\circ $, and $ \angle FKG = 90^\circ - 42^\circ = 48^\circ $. Ah, yes. So $ m\angle FKG = 48^\circ $
# Answer:
$ 48^\circ $

### Question 23: $ m\angle FKH $
# Explanation:
## Step 1: Identify the relationship
$ \angle FKH $ and $ \angle HKI $ are complementary (since $ KH \perp JG $, $ \angle GKH = 90^\circ $), so $ m\angle FKH=90^\circ - m\angle HKI $
## Step 2: Calculate the value
$ m\angle FKH = 90^\circ - 48^\circ=42^\circ $
# Answer:
$ 42^\circ $

### Question 24: $ m\angle GKI $
# Explanation:
## Step 1: Identify the angles
$ \angle GKI=\angle GKH+\angle HKI $
## Step 2: Calculate the sum
$ m\angle GKI = 90^\circ+48^\circ = 138^\circ $
# Answer:
$ 138^\circ $

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QUESTION IMAGE

Question

Explanation:

Question 16: Find \( m\angle BAD \)

Step 1: Identify the angles

Step 2: Calculate the sum

Step 1: Identify the relationship

Step 2: Solve for \( x \)

Step 1: Identify the relationship

Step 2: Solve for \( x \)

Step 3: Find \( m\angle FHG \)

Step 4: Find \( m\angle GHI \)

Answer:

Question 17: Find \( x \)