Question

angle relationships: color - by - number
date:
directions: use the properties of angle relationships to find each angle measure. show all of your work.

1. m∠dbc =
2. m∠dec =
3. m∠abd =
4. m∠abd =
5. m∠abd =
6. m∠aeb =
7. m∠aed =
8. m∠dbc =
9. m∠abd =
10. m∠abd =
11. m∠dbc =
12. m∠dec =

color:
blue: 107°
pink: 94°
light purple: 32°
light green: 62°
purple: 98°
dark green: 74°
gray: 29°
black: 86°
red: 72°
yellow: 87°
light brown: 54°
dark brown: 117°

Explanation:

Step1: Identify angle - pair relationships

For example, if two angles are vertical angles, they are equal. If they form a linear - pair, they are supplementary (sum to 180°).

Step2: Set up equations

Problem 1

Since ∠DBC and the two given angles form a straight - line (linear - pair), we have the equation: \((3x + 67)+(4x + 50)=180\).
Combining like terms: \(7x+117 = 180\).
Subtracting 117 from both sides: \(7x=180 - 117=63\).
Dividing both sides by 7: \(x = 9\).
Then \(m\angle DBC=3x + 67=3\times9+67=27 + 67 = 94\).

Problem 2

Since ∠DEC and the two given angles are vertical angles, we set up the equation: \(9x-18=3x + 42\).
Subtracting \(3x\) from both sides: \(6x-18 = 42\).
Adding 18 to both sides: \(6x=60\).
Dividing by 6: \(x = 10\).
Then \(m\angle DEC=9x-18=9\times10 - 18=72\).

Problem 3

Since the two given angles are vertical angles, we have the equation: \(7x-17=3x + 18\).
Subtracting \(3x\) from both sides: \(4x-17 = 18\).
Adding 17 to both sides: \(4x=35\).
Dividing by 4: \(x=\frac{35}{4}\).
Then \(m\angle ABD=7x-17=7\times\frac{35}{4}-17=\frac{245}{4}-17=\frac{245 - 68}{4}=\frac{177}{4}=44.25\).

Problem 4

Since ∠ABD and the two given angles form a linear - pair, we have the equation: \((x - 17)+(5x-39)=180\).
Combining like terms: \(6x-56 = 180\).
Adding 56 to both sides: \(6x=236\).
Dividing by 6: \(x=\frac{118}{3}\).
Then \(m\angle ABD=x - 17=\frac{118}{3}-17=\frac{118 - 51}{3}=\frac{67}{3}\approx22.33\).

Problem 5

Since ∠ABD and the given angle are vertical angles, we have the equation: \(7x + 40=2x+41\).
Subtracting \(2x\) from both sides: \(5x+40 = 41\).
Subtracting 40 from both sides: \(5x = 1\).
Dividing by 5: \(x=\frac{1}{5}\).
Then \(m\angle ABD=7x + 40=7\times\frac{1}{5}+40=\frac{7}{5}+40=\frac{7 + 200}{5}=\frac{207}{5}=41.4\).

Problem 6

Since ∠AEB and the two given angles are vertical angles, we set up the equation: \(5x+14=2x + 50\).
Subtracting \(2x\) from both sides: \(3x+14 = 50\).
Subtracting 14 from both sides: \(3x=36\).
Dividing by 3: \(x = 12\).
Then \(m\angle AEB=5x+14=5\times12+14=60 + 14 = 74\).

Problem 7

Since ∠AED and the two given angles are vertical angles, we set up the equation: \(9x-23=4x + 42\).
Subtracting \(4x\) from both sides: \(5x-23 = 42\).
Adding 23 to both sides: \(5x=65\).
Dividing by 5: \(x = 13\).
Then \(m\angle AED=9x-23=9\times13-23=117 - 23 = 94\).

Problem 8

Since the two given angles are vertical angles, we have the equation: \(5x-17=8x-26\).
Subtracting \(5x\) from both sides: \(-17 = 3x-26\).
Adding 26 to both sides: \(9 = 3x\).
Dividing by 3: \(x = 3\).
Then \(m\angle DBC=8x-26=8\times3-26=-2\) (This is incorrect, there may be a mis - labeling in the problem. Assuming a linear - pair relationship with a right - angle, if we consider \((5x - 17)+m\angle DBC+(8x - 26)=180\) and assume the right - angle is part of the relationship, we need more information. Let's assume the vertical - angle relationship is correct for now).

Problem 9

Since the two given angles are vertical angles, we have the equation: \(6x-9=2x + 61\).
Subtracting \(2x\) from both sides: \(4x-9 = 61\).
Adding 9 to both sides: \(4x=70\).
Dividing by 4: \(x=\frac{35}{2}\).
Then \(m\angle ABD=6x-9=6\times\frac{35}{2}-9=105 - 9 = 96\).

Problem 10

Since ∠ABD and ∠DBC are complementary, we have the equation: \((10x + 12)+(9x + 40)=180\).
Combining like terms: \(19x+52 = 180\).
Subtracting 52 from both sides: \(19x=128\).
Dividing by 19: \(x=\frac{128}{19}\).
Then \(m\angle ABD=10x + 12=10\times\frac{128}{19}+12=\frac{1280+228}{19}=\frac{1508}{19}\approx79.37\).

Problem 11

Since ∠ABD and ∠DBC form a linear - pair, we have the equation: \((14x + 26)+(7x+70)=180\).
Combining like terms: \(21x+96 = 180\).
Subtracting 96 from both sides: \(21x=84\).
Dividing by 21: \(x = 4\).
Then \(m\angle DBC=7x + 70=7\times4+70=28 + 70 = 98\).

Problem 12

Since ∠AEB and ∠DEC are vertical angles, we have the equation: \(18x+17=6x + 77\).
Subtracting \(6x\) from both sides: \(12x+17 = 77\).
Subtracting 17 from both sides: \(12x=60\).
Dividing by 12: \(x = 5\).
Then \(m\angle DEC=6x + 77=6\times5+77=30 + 77 = 107\).

Answer:

1. \(m\angle DBC = 94\)
2. \(m\angle DEC=72\)
3. \(m\angle ABD = 44.25\)
4. \(m\angle ABD\approx22.33\)
5. \(m\angle ABD = 41.4\)
6. \(m\angle AEB = 74\)
7. \(m\angle AED = 94\)
8. (There may be an issue with the problem setup, assuming vertical - angle relationship \(x = 3\))
9. \(m\angle ABD = 96\)
10. \(m\angle ABD\approx79.37\)
11. \(m\angle DBC = 98\)
12. \(m\angle DEC = 107\)