Question

1. if (mangle3 = 74^{circ}), find each measure. note: (lparallel m) a. (mangle1=) e. (mangle6=) b. (mangle2=) f. (mangle7=) c. (mangle4=) g. (mangle8=) d. (mangle5=) 17. if (mangle2 = 125^{circ}), (mangle12 = 37^{circ}) and (mangle18 = 102^{circ}), find the measure of each missing angle. note: (aparallel b) a. (mangle1=) f. (mangle7=) k. (mangle13=) b. (mangle3=) g. (mangle8=) l. (mangle14=) c. (mangle4=) h. (mangle9=) m. (mangle15=) d. (mangle5=) i. (mangle10=) n. (mangle16=) e. (mangle6=) j. (mangle11=) o. (mangle17=) for questions 18 - 20, if (lparallel m), find the values of (x) and (y). 18. 19. 20. © gina wilson (all things algebra®, llc), 2014 - 2019

Explanation:

Step1: Identify angle - relationships for question 16

Given \(l\parallel m\) and \(m\angle3 = 74^{\circ}\).
- \(\angle1\) and \(\angle3\) are vertical angles. Vertical angles are equal. So \(m\angle1=m\angle3 = 74^{\circ}\).
- \(\angle2\) and \(\angle3\) are supplementary (linear - pair). So \(m\angle2=180 - m\angle3=180 - 74=106^{\circ}\).
- \(\angle4\) and \(\angle3\) are vertical angles. So \(m\angle4=m\angle3 = 74^{\circ}\).
- \(\angle5\) and \(\angle3\) are corresponding angles. Since \(l\parallel m\), \(m\angle5=m\angle3 = 74^{\circ}\).
- \(\angle6\) and \(\angle2\) are vertical angles. So \(m\angle6=m\angle2 = 106^{\circ}\).
- \(\angle7\) and \(\angle3\) are alternate - interior angles. Since \(l\parallel m\), \(m\angle7=m\angle3 = 74^{\circ}\).
- \(\angle8\) and \(\angle2\) are alternate - exterior angles. Since \(l\parallel m\), \(m\angle8=m\angle2 = 106^{\circ}\).

Step2: Identify angle - relationships for question 17

Given \(a\parallel b\), \(m\angle2 = 125^{\circ}\), \(m\angle12 = 37^{\circ}\), \(m\angle18 = 102^{\circ}\).
- \(\angle1\) and \(\angle2\) are supplementary (linear - pair). So \(m\angle1=180 - m\angle2=180 - 125 = 55^{\circ}\).
- \(\angle3\) and \(\angle2\) are vertical angles. So \(m\angle3=m\angle2 = 125^{\circ}\).
- \(\angle4\) and \(\angle3\) are supplementary (linear - pair). So \(m\angle4=180 - m\angle3=180 - 125 = 55^{\circ}\).
- \(\angle5\) and \(\angle12\) are corresponding angles. Since \(a\parallel b\), \(m\angle5=m\angle12 = 37^{\circ}\).
- \(\angle6\) and \(\angle5\) are vertical angles. So \(m\angle6=m\angle5 = 37^{\circ}\).
- \(\angle7\) and \(\angle12\) are alternate - interior angles. Since \(a\parallel b\), \(m\angle7=m\angle12 = 37^{\circ}\).
- \(\angle8\) and \(\angle12\) are vertical angles. So \(m\angle8=m\angle12 = 37^{\circ}\).
- \(\angle9\) and \(\angle18\) are corresponding angles. Since \(a\parallel b\), \(m\angle9=m\angle18 = 102^{\circ}\).
- \(\angle10\) and \(\angle9\) are vertical angles. So \(m\angle10=m\angle9 = 102^{\circ}\).
- \(\angle11\) and \(\angle12\) are supplementary (linear - pair). So \(m\angle11=180 - m\angle12=180 - 37 = 143^{\circ}\).
- \(\angle13\) and \(\angle18\) are vertical angles. So \(m\angle13=m\angle18 = 102^{\circ}\).
- \(\angle14\) and \(\angle13\) are supplementary (linear - pair). So \(m\angle14=180 - m\angle13=180 - 102 = 78^{\circ}\).
- \(\angle15\) and \(\angle14\) are vertical angles. So \(m\angle15=m\angle14 = 78^{\circ}\).
- \(\angle16\) and \(\angle15\) are supplementary (linear - pair). So \(m\angle16=180 - m\angle15=180 - 78 = 102^{\circ}\).
- \(\angle17\) and \(\angle16\) are vertical angles. So \(m\angle17=m\angle16 = 102^{\circ}\).

Step3: Solve for \(x\) and \(y\) in question 18

Since \(l\parallel m\), \((3x - 16)^{\circ}\) and \((6x + 7)^{\circ}\) are supplementary (same - side interior angles).
So \((3x-16)+(6x + 7)=180\).
Combining like terms: \(3x+6x-16 + 7=180\), \(9x-9 = 180\), \(9x=189\), \(x = 21\).
Also, \((11y-32)^{\circ}\) and \((6x + 7)^{\circ}\) are corresponding angles. Substitute \(x = 21\) into \((6x + 7)\): \(6\times21+7=126 + 7=133\).
Then \(11y-32 = 133\), \(11y=165\), \(y = 15\).

Step4: Solve for \(x\) and \(y\) in question 19

Since \(l\parallel m\), \((8x - 14)^{\circ}\) and \((5x + 34)^{\circ}\) are alternate - exterior angles.
So \(8x-14=5x + 34\).
Subtract \(5x\) from both sides: \(8x-5x-14=5x-5x + 34\), \(3x-14 = 34\).
Add 14 to both sides: \(3x-14 + 14=34 + 14\), \(3x=48\), \(x = 16\).
Also, \((5y + 16)^{\circ}\) and \((5x + 34)^{\circ}\) are corresponding angles. Substitute \(x = 16\) into \((5x + 34)\): \(5\times16+34=80 + 34=114\).
Then \(5y+16 = 114\), \(5y=98\), \(y=\frac{98}{5}=19.6\).

Step5: Solve for \(x\) and \(y\) in question 20

Since \(l\parallel m\), \((2x + 13)^{\circ}\) and \(47^{\circ}\) are alternate - interior angles.
So \(2x+13 = 47\), \(2x=47 - 13=34\), \(x = 17\).
Also, \((5y-23)^{\circ}\) and \((3x)^{\circ}\) are corresponding angles. Substitute \(x = 17\) into \((3x)\): \(3\times17 = 51\).
Then \(5y-23 = 51\), \(5y=74\), \(y=\frac{74}{5}=14.8\).

Answer:

16.
a. \(m\angle1 = 74^{\circ}\)
b. \(m\angle2 = 106^{\circ}\)
c. \(m\angle4 = 74^{\circ}\)
d. \(m\angle5 = 74^{\circ}\)
e. \(m\angle6 = 106^{\circ}\)
f. \(m\angle7 = 74^{\circ}\)
g. \(m\angle8 = 106^{\circ}\)
17.
a. \(m\angle1 = 55^{\circ}\)
b. \(m\angle3 = 125^{\circ}\)
c. \(m\angle4 = 55^{\circ}\)
d. \(m\angle5 = 37^{\circ}\)
e. \(m\angle6 = 37^{\circ}\)
f. \(m\angle7 = 37^{\circ}\)
g. \(m\angle8 = 37^{\circ}\)
h. \(m\angle9 = 102^{\circ}\)
i. \(m\angle10 = 102^{\circ}\)
j. \(m\angle11 = 143^{\circ}\)
k. \(m\angle13 = 102^{\circ}\)
l. \(m\angle14 = 78^{\circ}\)
m. \(m\angle15 = 78^{\circ}\)
n. \(m\angle16 = 102^{\circ}\)
o. \(m\angle17 = 102^{\circ}\)

1. \(x = 21\), \(y = 15\)
2. \(x = 16\), \(y = 19.6\)
3. \(x = 17\), \(y = 14.8\)