13) ( mangle vuk = 2x + 57 ), ( mangle vut = 101^circ ), and ( mangle kut = 44 + 2x ). find ( mangle kut ).
14) find ( mangle def ) if ( mangle def = 175x + 1 ), ( mangle deh = 26^circ ), and ( mangle hef = 1 + 149x ).
15) ( mangle cbk = x + 39 ), ( mangle cba = 163^circ ), and ( mangle kba = x + 146 ). find ( mangle kba ).
16) ( mangle fgh = 84^circ ), ( mangle lgh = x + 69 ), and ( mangle fgl = x + 37 ). find ( mangle lgh ).
17) ( mangle def = 171^circ ), ( mangle cef = 17x + 14 ), and ( mangle dec = x + 13 ). find ( mangle cef ).
18) ( mangle cde = 15 + 20x ), ( mangle cdz = 10x + 15 ), and ( mangle zde = 80^circ ). find ( mangle cde ).
19) find ( mangle srq ) if ( mangle srd = 95^circ ), ( mangle srq = 75x + 3 ), and ( mangle drq = 79x ).
20) ( mangle cda = 9x - 2 ), ( mangle cde = 17x + 14 ), and ( mangle ade = 80^circ ). find ( mangle cda ).

Question

Accepted Answer

Let's solve problem 13 first:

# Explanation:
## Step1: Use angle addition postulate
We know that $ m\angle VUT = m\angle VUK + m\angle KUT $. So we substitute the given expressions:
$ 101 = (2x + 57) + (44 + 2x) $

## Step2: Simplify and solve for x
Combine like terms:
$ 101 = 4x + 101 $
Subtract 101 from both sides:
$ 0 = 4x $
So $ x = 0 $

## Step3: Find $ m\angle KUT $
Substitute $ x = 0 $ into $ m\angle KUT = 44 + 2x $:
$ m\angle KUT = 44 + 2(0) = 44^\circ $

# Answer:
$ 44^\circ $

Now problem 14:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle DEF = m\angle DEH + m\angle HEF $. Substitute the given expressions:
$ 175x + 1 = 26 + (1 + 149x) $

## Step2: Simplify and solve for x
Simplify the right side: $ 26 + 1 + 149x = 27 + 149x $
So the equation is $ 175x + 1 = 27 + 149x $
Subtract $ 149x $ from both sides: $ 26x + 1 = 27 $
Subtract 1 from both sides: $ 26x = 26 $
Divide by 26: $ x = 1 $

## Step3: Find $ m\angle DEF $
Substitute $ x = 1 $ into $ m\angle DEF = 175x + 1 $:
$ m\angle DEF = 175(1) + 1 = 176^\circ $

# Answer:
$ 176^\circ $

Problem 15:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle CBA = m\angle CBK + m\angle KBA $. Substitute the given expressions:
$ 163 = (x + 39) + (x + 146) $

## Step2: Simplify and solve for x
Combine like terms: $ 163 = 2x + 185 $
Subtract 185 from both sides: $ -22 = 2x $
Divide by 2: $ x = -11 $

## Step3: Find $ m\angle KBA $
Substitute $ x = -11 $ into $ m\angle KBA = x + 146 $:
$ m\angle KBA = -11 + 146 = 135^\circ $

# Answer:
$ 135^\circ $

Problem 16:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle FGH = m\angle FGL + m\angle LGH $. Substitute the given expressions:
$ 84 = (x + 37) + (x + 69) $

## Step2: Simplify and solve for x
Combine like terms: $ 84 = 2x + 106 $
Subtract 106 from both sides: $ -22 = 2x $
Divide by 2: $ x = -11 $

## Step3: Find $ m\angle LGH $
Substitute $ x = -11 $ into $ m\angle LGH = x + 69 $:
$ m\angle LGH = -11 + 69 = 58^\circ $

# Answer:
$ 58^\circ $

Problem 17:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle DEF = m\angle DEC + m\angle CEF $. Substitute the given expressions:
$ 171 = (x + 13) + (17x + 14) $

## Step2: Simplify and solve for x
Combine like terms: $ 171 = 18x + 27 $
Subtract 27 from both sides: $ 144 = 18x $
Divide by 18: $ x = 8 $

## Step3: Find $ m\angle CEF $
Substitute $ x = 8 $ into $ m\angle CEF = 17x + 14 $:
$ m\angle CEF = 17(8) + 14 = 136 + 14 = 150^\circ $

# Answer:
$ 150^\circ $

Problem 18:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle CDE = m\angle CDZ + m\angle ZDE $. Substitute the given expressions:
$ 15 + 20x = (10x + 15) + 80 $

## Step2: Simplify and solve for x
Simplify the right side: $ 10x + 95 $
So the equation is $ 15 + 20x = 10x + 95 $
Subtract $ 10x $ from both sides: $ 15 + 10x = 95 $
Subtract 15 from both sides: $ 10x = 80 $
Divide by 10: $ x = 8 $

## Step3: Find $ m\angle CDE $
Substitute $ x = 8 $ into $ m\angle CDE = 15 + 20x $:
$ m\angle CDE = 15 + 20(8) = 15 + 160 = 175^\circ $

# Answer:
$ 175^\circ $

Problem 19 (assuming the problem is: Find $ m\angle SRQ $ if $ m\angle SRD = 95^\circ $, $ m\angle SRQ = 75x + 3 $, and $ m\angle DRQ = 79x $):

# Explanation:
## Step1: Use angle addition postulate
$ m\angle SRD = m\angle SRQ + m\angle DRQ $. Substitute the given expressions:
$ 95 = (75x + 3) + 79x $

## Step2: Simplify and solve for x
Combine like terms: $ 95 = 154x + 3 $
Subtract 3 from both sides: $ 92 = 154x $
Wait, this seems off. Maybe the angle is $ m\angle SRD = m\angle SRQ - m\angle DRQ $ (if $ \angle DRQ $ is inside $ \angle SRQ $)? Let's re-examine the diagram. If $ RQ $ is a ray inside $ \angle SRD $, then $ m\angle SRQ = m\angle SRD + m\angle DRQ $? Wait, maybe the correct postulate is $ m\angle SRQ = m\angle SRD + m\angle DRQ $ or $ m\angle SRD = m\angle SRQ - m\angle DRQ $. Let's assume the correct relation is $ m\angle SRQ = m\angle SRD + m\angle DRQ $ (if $ D $ is between $ S $ and $ Q $? No, the diagram shows $ Q $, $ R $, $ S $ with $ W $ down. Maybe $ m\angle SRD = m\angle SRQ + m\angle DRQ $ is wrong. Let's check the numbers. If $ 75x + 3 = 95 + 79x $, then $ -4x = 92 $, $ x = -23 $, which is odd. Maybe the correct equation is $ m\angle SRQ + m\angle DRQ = m\angle SRD $? No, that would be $ 75x + 3 + 79x = 95 $, $ 154x = 92 $, $ x = 92/154 = 46/77 $, which is messy. Maybe there's a typo, but assuming the angle addition is $ m\angle SRQ = m\angle SRD + m\angle DRQ $ (maybe the diagram has $ \angle SRD $ as a part of $ \angle SRQ $), then:

$ 75x + 3 = 95 + 79x $

$ -4x = 92 $

$ x = -23 $

Then $ m\angle SRQ = 75(-23) + 3 = -1725 + 3 = -1722^\circ $, which is impossible. So maybe the correct relation is $ m\angle SRD = m\angle SRQ - m\angle DRQ $, so:

$ 95 = (75x + 3) - 79x $

$ 95 = -4x + 3 $

$ -4x = 92 $

$ x = -23 $, still wrong. Maybe the problem is $ m\angle SRD = 95^\circ $, $ m\angle SRQ = 75x + 3 $, $ m\angle DRQ = 7x $ (typo in 79x). Let's assume $ 7x $:

$ 75x + 3 = 95 + 7x $

$ 68x = 92 $

$ x = 92/68 = 23/17 $, still messy. Alternatively, maybe $ m\angle SRD = m\angle SRQ + m\angle DRQ $ with $ m\angle SRD = 180 - 95 = 85^\circ $ (if it's a linear pair). Wait, the diagram shows $ Q $, $ R $, $ S $ with $ W $ down, so maybe $ \angle SRW $ is 90 degrees? Let's re-express. If $ \angle SRW = 90^\circ $, and $ \angle SRD = 95^\circ $, then $ \angle DRW = 5^\circ $, but not helpful. Maybe the original problem has a typo, but since we need to solve, let's proceed with the given numbers. Alternatively, maybe the angle is $ m\angle SRQ = m\angle SRD - m\angle DRQ $, so:

$ 75x + 3 = 95 - 79x $

$ 154x = 92 $

$ x = 92/154 = 46/77 $

Then $ m\angle SRQ = 75*(46/77) + 3 = (3450/77) + 3 = (3450 + 231)/77 = 3681/77 ≈ 47.8^\circ $, which is not nice. Maybe the problem is different, but since we can't see the diagram clearly, we'll skip for now.

Problem 20:

# Explanation:
## Step1: Use angle addition postulate
$ m\angle CDE = m\angle CDA + m\angle ADE $. Substitute the given expressions:
$ 17x + 14 = (9x - 2) + 80 $

## Step2: Simplify and solve for x
Simplify the right side: $ 9x + 78 $
So the equation is $ 17x + 14 = 9x + 78 $
Subtract $ 9x $ from both sides: $ 8x + 14 = 78 $
Subtract 14 from both sides: $ 8x = 64 $
Divide by 8: $ x = 8 $

## Step3: Find $ m\angle CDA $
Substitute $ x = 8 $ into $ m\angle CDA = 9x - 2 $:
$ m\angle CDA = 9(8) - 2 = 72 - 2 = 70^\circ $

# Answer:
$ 70^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Use angle addition postulate

Step2: Simplify and solve for x

Step3: Find \( m\angle KUT \)

Step1: Use angle addition postulate

Step2: Simplify and solve for x

Step3: Find \( m\angle DEF \)

Step1: Use angle addition postulate

Step2: Simplify and solve for x

Step3: Find \( m\angle KBA \)

Answer: