Question

1. if m∠deg = 5x − 4, m∠gef = 7x − 8, m∠deh = 9y + 5, find the values of x and y.

Explanation:

Step1: Recognize the straight angle

∠DEG and ∠GEF form a straight angle, so their measures add up to \(180^\circ\).
\[
(5x - 4) + (7x - 8) = 180
\]

Step2: Solve for x

Combine like terms:
\[
12x - 12 = 180
\]
Add 12 to both sides:
\[
12x = 192
\]
Divide by 12:
\[
x = 16
\]

Step3: Recognize vertical angles

∠DEG and ∠DEH? Wait, no. Wait, ∠DEG and ∠HEF? Wait, no, looking at the diagram, ∠DEG and ∠HEF are vertical? Wait, no, actually, ∠DEG and ∠GEF are a linear pair (sum to 180), and then ∠DEH should be equal to ∠GEF (vertical angles) or ∠DEG? Wait, let's check the diagram again. Points: G---E---H is a straight line? Wait, no, the lines are GD and FH intersecting at E. So ∠DEG and ∠HEF are vertical angles? Wait, no, ∠DEG and ∠GEF are adjacent and form a straight line (since G, E, H? Wait, no, the lines are GD (with G and D) and FH (with F and H) intersecting at E. So ∠DEG and ∠GEF: D---E---G? Wait, no, the arrows: G is left, D is down-left, F is up-right, H is right. So the line GEH? Wait, G to E to H is a straight line? And D to E to F is another straight line. So ∠DEG and ∠GEF are adjacent angles on the straight line DEF? Wait, D---E---F is a straight line, so ∠DEG and ∠GEF are adjacent and form a straight angle (180 degrees). Then, ∠DEH: since G---E---H is a straight line, ∠DEH and ∠DEG? Wait, no, ∠DEH and ∠GEF are vertical angles? Wait, when two lines intersect, vertical angles are equal. So line GD (G-E-D) and line FH (F-E-H) intersect at E. So ∠DEG and ∠HEF are vertical angles, ∠GEF and ∠DEH are vertical angles. So m∠DEH = m∠GEF.

So since we found x = 16, m∠GEF = 7x - 8 = 716 - 8 = 112 - 8 = 104. Wait, no, wait: ∠DEG + ∠GEF = 180, so 5x -4 +7x -8 = 180 => 12x -12=180 => 12x=192 => x=16. Then m∠DEG = 516 -4=80-4=76, m∠GEF=7*16 -8=112-8=104. Then ∠DEH is equal to ∠GEF (vertical angles), so m∠DEH = m∠GEF = 104. So 9y +5 = 104.

Step4: Solve for y

\[
9y + 5 = 104
\]
Subtract 5:
\[
9y = 99
\]
Divide by 9:
\[
y = 11
\]

Answer:

x = 16, y = 11