Answers for problem 24:
Angle EBG = 16 degrees
Angle BGC = 58 degrees
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Step-by-step explanation:
Let
x = measure of angle EBG
y = measure of angle EGB
z = measure of angle BGC
Angle ABF = 74, angle FBG = 90, and angle EBG = x all combine to form a straight angle which is 180 degrees. Add these angles up, set the sum equal to 180, and solve for x
(angle ABF)+(angle FBG) + (angle EBG) = 180
74+90+x = 180
164+x = 180
164+x-164 = 180-164 ... subtract 164 from both sides
x = 16
Therefore angle EBG is 16 degrees
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Now focus on triangle BEG. We found angle B to be 16 degrees (this was angle EBG). Angle E is given to be 42. Angle G is unknown which I've defined with the variable y. The three angles of any triangle add to 180
B+E+G = 180
16+42+y = 180
58+y = 180
58+y-58 = 180-58 .... subtract 58 from both sides
y = 122
Therefore, angle EGB is 122 degrees
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Use the value of y to find z. The two angles y and z add to 180 (they form a straight angle), so,
y+z = 180
z = 180-y
z = 180-122 .... replace y with 122
z = 58
Therefore, angle BGC is 58 degrees