Question

1. if ( mangle deg = (5x - 4)^circ ), ( mangle gef = (7x - 8)^circ ), ( mangle deh = (9y + 5)^circ ), find the values of ( x ) and ( y ). (diagram: intersecting lines at e with rays d, g, f, h)

Explanation:

Step1: Find x (∠DEG and ∠GEF are complementary? Wait, no, from the diagram, ∠DEG and ∠GEF form a linear pair? Wait, no, looking at the lines: D-E-F? Wait, no, the lines are intersecting at E. Wait, ∠DEG and ∠GEF: wait, maybe ∠DEG and ∠GEF are complementary? Wait, no, maybe ∠DEG and ∠GEF are adjacent and form a right angle? Wait, no, the problem says "find the values of x and y". Let's re-examine: m∠DEG = (5x - 4)°, m∠GEF = (7x - 8)°, m∠DEH = (9y + 5)°. Wait, maybe ∠DEG and ∠GEF are complementary (sum to 90°) because they look like adjacent angles forming a right angle? Wait, or maybe ∠DEG and ∠GEF are supplementary? Wait, no, let's think again. Wait, the diagram: D---E---F? No, the lines are two intersecting lines: one is D-E-H? Wait, no, the arrows: D and H are on one line? Wait, no, the diagram has four rays: D, G, F, H with E as the intersection. Wait, maybe ∠DEG and ∠GEF are adjacent and form a right angle (90°), so their sum is 90°. So:

(5x - 4) + (7x - 8) = 90

Step2: Solve for x:

Combine like terms: 5x + 7x - 4 - 8 = 90 → 12x - 12 = 90

Add 12 to both sides: 12x = 102

Divide by 12: x = 102 / 12 = 8.5? Wait, that seems odd. Wait, maybe they are supplementary (sum to 180°)? Let's check:

(5x - 4) + (7x - 8) = 180 → 12x - 12 = 180 → 12x = 192 → x = 16. Then, ∠DEG = 516 -4 = 76°, ∠GEF = 716 -8 = 104°, sum 180°, which makes sense as a linear pair. Then, ∠DEH: since ∠DEH and ∠GEF are vertical angles? Wait, no, ∠DEH and ∠GEF: if D-E-F is a straight line, and G-E-H is another straight line, then ∠DEH and ∠GEF are vertical angles? Wait, no, ∠DEG and ∠HEF are vertical, ∠GEF and ∠DEH are vertical. So m∠DEH = m∠GEF. So 9y + 5 = 7x - 8.

Wait, let's correct the first step. If D-E-F is a straight line (180°), then ∠DEG + ∠GEF = 180° (linear pair). So:

5x - 4 + 7x - 8 = 180

12x - 12 = 180

12x = 192

x = 16

Then, ∠GEF = 7*16 -8 = 104°, so ∠DEH (vertical angle to ∠GEF) is also 104°, so:

9y + 5 = 104

9y = 99

y = 11

Yes, that makes sense. So first, find x by setting ∠DEG + ∠GEF = 180° (linear pair), then find y by setting ∠DEH = ∠GEF (vertical angles).

Step1: Solve for x (linear pair)

∠DEG and ∠GEF are a linear pair, so their sum is 180°.
$$(5x - 4) + (7x - 8) = 180$$
Combine like terms:
$$12x - 12 = 180$$
Add 12 to both sides:
$$12x = 192$$
Divide by 12:
$$x = \frac{192}{12} = 16$$

Step2: Solve for y (vertical angles)

∠DEH and ∠GEF are vertical angles, so they are equal.
$$9y + 5 = 7x - 8$$
Substitute x = 16:
$$9y + 5 = 7(16) - 8$$
Calculate right side:
$$7*16 = 112; 112 - 8 = 104$$
So:
$$9y + 5 = 104$$
Subtract 5:
$$9y = 99$$
Divide by 9:
$$y = \frac{99}{9} = 11$$

Answer:

x = 16, y = 11