Question

parallel lines & transversals puzzle
directions: assume that lines that appear to be parallel are parallel. find the value of x and the measure of angle y for each diagram. then, cut the squares apart and arrange them so the edges match. paste the squares on the template in the correct order.
a 131° (5x - 11)° (3x + 17)° find x find m∠y
b 118° (24x - 2)° (25x - 8)° find x find m∠y
c 77° (7x + 11)° (16x - 38)° find x find m∠y
d 138° (7x + 26)° (10x - 25)° find x find m∠y
e 26° (9x - 5)° (4x + 29)° find x find m∠y
f 83° (6x + 2)° (8x - 18)° find x find m∠y
g 74° (9x + 10)° (15x - 2)° find x find m∠y
h 21° (4x - 8)° 54° (5x - 19)° find x find m∠y
i 145° 41° (7x)° (11x - 5)° find x find m∠y
j 59° (16x - 23)° 62° (10x + 1)° find x find m∠y
k 29° (4x)° 55° (7x - 2)° find x find m∠y
l 152° (14x - 34)° (9x + 1)° find x find m∠y

Explanation:

Step1: Identify angle - relationship for A

Since the angles \((5x - 11)^{\circ}\) and \((3x + 17)^{\circ}\) are vertical angles, they are equal. So we set up the equation \(5x-11 = 3x + 17\).
\[ \begin{align*} 5x-3x&=17 + 11\\ 2x&=28\\ x&=14 \end{align*} \]
The angle \(y\) and \((3x + 17)^{\circ}\) are supplementary. Substitute \(x = 14\) into \((3x + 17)\), we get \(3\times14+17=42 + 17=59^{\circ}\). Then \(y=180 - 59=121^{\circ}\).

Step2: Identify angle - relationship for B

The angles \((24x - 2)^{\circ}\) and \((25x - 8)^{\circ}\) are supplementary. So \((24x - 2)+(25x - 8)=180\).
\[ \begin{align*} 24x+25x&=180 + 2+8\\ 49x&=190\\ x&=\frac{190}{49}\approx3.88 \end{align*} \]
The angle \(y=(24x - 2)\), substituting \(x=\frac{190}{49}\), \(y = 24\times\frac{190}{49}-2=\frac{4560}{49}-2=\frac{4560 - 98}{49}=\frac{4462}{49}\approx91.06^{\circ}\)

Step3: Identify angle - relationship for C

The angles \((7x + 11)^{\circ}\) and \((16x - 38)^{\circ}\) are supplementary. So \((7x + 11)+(16x - 38)=180\).
\[ \begin{align*} 7x+16x&=180-11 + 38\\ 23x&=207\\ x&=9 \end{align*} \]
The angle \(y=(7x + 11)\), substituting \(x = 9\), \(y=7\times9+11=63 + 11=74^{\circ}\)

Step4: Identify angle - relationship for D

The angles \((7x + 26)^{\circ}\) and \((10x - 25)^{\circ}\) are supplementary. So \((7x + 26)+(10x - 25)=180\).
\[ \begin{align*} 7x+10x&=180-26 + 25\\ 17x&=179\\ x&=\frac{179}{17}\approx10.53 \end{align*} \]
The angle \(y=(7x + 26)\), substituting \(x=\frac{179}{17}\), \(y=7\times\frac{179}{17}+26=\frac{1253}{17}+26=\frac{1253+442}{17}=\frac{1695}{17}\approx99.71^{\circ}\)

Step5: Identify angle - relationship for E

The angles \((4x + 29)^{\circ}\) and \((9x - 5)^{\circ}\) are supplementary. So \((4x + 29)+(9x - 5)=180\).
\[ \begin{align*} 4x+9x&=180-29 + 5\\ 13x&=156\\ x&=12 \end{align*} \]
The angle \(y=(4x + 29)\), substituting \(x = 12\), \(y=4\times12+29=48 + 29=77^{\circ}\)

Step6: Identify angle - relationship for F

The angles \((6x + 2)^{\circ}\) and \((8x - 18)^{\circ}\) are supplementary. So \((6x + 2)+(8x - 18)=180\).
\[ \begin{align*} 6x+8x&=180-2 + 18\\ 14x&=196\\ x&=14 \end{align*} \]
The angle \(y=(6x + 2)\), substituting \(x = 14\), \(y=6\times14+2=84 + 2=86^{\circ}\)

Step7: Identify angle - relationship for G

The angles \((9x + 10)^{\circ}\) and \((15x - 2)^{\circ}\) are supplementary. So \((9x + 10)+(15x - 2)=180\).
\[ \begin{align*} 9x+15x&=180-10 + 2\\ 24x&=172\\ x&=\frac{43}{6}\approx7.17 \end{align*} \]
The angle \(y=(9x + 10)\), substituting \(x=\frac{43}{6}\), \(y=9\times\frac{43}{6}+10=\frac{129}{2}+10=\frac{129 + 20}{2}=\frac{149}{2}=74.5^{\circ}\)

Step8: Identify angle - relationship for H

The angles \((4x - 8)^{\circ}\), \(54^{\circ}\) and \((5x - 19)^{\circ}\) form a straight - line. So \((4x - 8)+54+(5x - 19)=180\).
\[ \begin{align*} 4x+5x&=180 + 8-54 + 19\\ 9x&=153\\ x&=17 \end{align*} \]
To find \(y\), we first find one of the adjacent angles. Let's take the angle \((4x - 8)\) with \(x = 17\), \((4\times17-8)=68 - 8 = 60^{\circ}\). Then \(y = 180-(60 + 54)=66^{\circ}\)

Step9: Identify angle - relationship for I

The angles \((7x)^{\circ}\) and \((11x - 5)^{\circ}\) and \(41^{\circ}\) are related such that \((7x)+(11x - 5)+41 = 180\) (since they form a straight - line).
\[ \begin{align*} 7x+11x&=180+5 - 41\\ 18x&=144\\ x&=8 \end{align*} \]
The angle \(y=(7x)\), substituting \(x = 8\), \(y=56^{\circ}\)

Step10: Identify angle - relationship for J

The angles \((16x - 23)^{\circ}\) and \((10x + 1)^{\circ}\) and \(62^{\circ}\) are related such that \((16x - 23)+(10x + 1)+62 = 180\) (since they form a straight - line).
\[ \begin{align*} 16x+10x&=180+23-1 - 62\\ 26x&=140\\ x&=\frac{70}{13}\approx5.38 \end{align*} \]
The angle \(y\) can be found by first finding one of the adjacent angles. Let's find the value of \((16x - 23)\) with \(x=\frac{70}{13}\), \(16\times\frac{70}{13}-23=\frac{1120}{13}-23=\frac{1120-299}{13}=\frac{821}{13}\approx63.15^{\circ}\), then \(y = 180-(63.15 + 62)\approx54.85^{\circ}\)

Step11: Identify angle - relationship for K

The angles \((4x)^{\circ}\), \(55^{\circ}\) and \((7x - 2)^{\circ}\) are related such that \((4x)+55+(7x - 2)=180\) (since they form a straight - line).
\[ \begin{align*} 4x+7x&=180-55 + 2\\ 11x&=127\\ x&=\frac{127}{11}\approx11.55 \end{align*} \]
The angle \(y=(4x)\), substituting \(x=\frac{127}{11}\), \(y=\frac{508}{11}\approx46.18^{\circ}\)

Step12: Identify angle - relationship for L

The angles \((14x - 34)^{\circ}\) and \((9x + 1)^{\circ}\) are complementary (since there is a right - angle). So \((14x - 34)+(9x + 1)=90\).
\[ \begin{align*} 14x+9x&=90+34 - 1\\ 23x&=123\\ x&=\frac{123}{23}\approx5.35 \end{align*} \]
The angle \(y=(14x - 34)\), substituting \(x=\frac{123}{23}\), \(y=14\times\frac{123}{23}-34=\frac{1722}{23}-34=\frac{1722-782}{23}=\frac{940}{23}\approx40.87^{\circ}\)

Answer:

For A: \(x = 14\), \(m\angle y=121^{\circ}\)
For B: \(x=\frac{190}{49}\), \(m\angle y=\frac{4462}{49}\)
For C: \(x = 9\), \(m\angle y=74^{\circ}\)
For D: \(x=\frac{179}{17}\), \(m\angle y=\frac{1695}{17}\)
For E: \(x = 12\), \(m\angle y=77^{\circ}\)
For F: \(x = 14\), \(m\angle y=86^{\circ}\)
For G: \(x=\frac{43}{6}\), \(m\angle y=\frac{149}{2}\)
For H: \(x = 17\), \(m\angle y=66^{\circ}\)
For I: \(x = 8\), \(m\angle y=56^{\circ}\)
For J: \(x=\frac{70}{13}\), \(m\angle y\approx54.85^{\circ}\)
For K: \(x=\frac{127}{11}\), \(m\angle y=\frac{508}{11}\)
For L: \(x=\frac{123}{23}\), \(m\angle y=\frac{940}{23}\)

Answer:

For A: \(x = 14\), \(m\angle y=121^{\circ}\)
For B: \(x=\frac{190}{49}\), \(m\angle y=\frac{4462}{49}\)
For C: \(x = 9\), \(m\angle y=74^{\circ}\)
For D: \(x=\frac{179}{17}\), \(m\angle y=\frac{1695}{17}\)
For E: \(x = 12\), \(m\angle y=77^{\circ}\)
For F: \(x = 14\), \(m\angle y=86^{\circ}\)
For G: \(x=\frac{43}{6}\), \(m\angle y=\frac{149}{2}\)
For H: \(x = 17\), \(m\angle y=66^{\circ}\)
For I: \(x = 8\), \(m\angle y=56^{\circ}\)
For J: \(x=\frac{70}{13}\), \(m\angle y\approx54.85^{\circ}\)
For K: \(x=\frac{127}{11}\), \(m\angle y=\frac{508}{11}\)
For L: \(x=\frac{123}{23}\), \(m\angle y=\frac{940}{23}\)