Answer:
(the answers are written in bold)
1. Line HP was divided into two by the midpoint (point E), so adding the steps from H to E and steps from E to P will result to the steps from H to P.
(9x) + (x + 4)
= (10x + 4) steps
2.
(3x + 2) + (5x)
= (8x + 3) steps
3.
m∠HOE = 30
m∠OEH = 90 (since line OE is perpendicular to line HP)
m∠H = ?
m∠HOE + m∠OEH + m∠H = 180 (a triangle's sum of interior angles is equal to 180°)
30 + 90 + m∠H = 180
120 + m∠H = 180
m∠H = 180 - 120 (transfer 120 to the other side to isolate m∠H)
m∠H = 60
4.
m∠HOE = x
m∠POE = 2x - 30
x = ?
m∠HOE = m∠POE (line OE bisects angle POH, therefore, the measure of angle HOE and angle POE is equal)
x = 2x - 30
x + 30 = 2x (transpose)
30 = 2x - x (transfer x to the other side of the equation)
30 = x
5.
m∠OHE = 2(m∠HOE)
m∠OEH = 90 (again, as we can see, this is a right angle)
m∠HOE = ?
m∠OHE + m∠OEH + m∠HOE = 180 (again, a triangle's sum of interior angles is equal to 180°)
2(m∠HOE) + 90 + m∠HOE = 180
3(m∠HOE) + 90 = 180 (add like terms)
3(m∠HOE) = 180 - 90 (transpose)
3(m∠HOE) = 90 (divide both sides by 3 to isolate m∠HOE)
m∠OHE = 2(30)
m∠OHE = 60
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Answers & Comments
Answer:
(the answers are written in bold)
1. Line HP was divided into two by the midpoint (point E), so adding the steps from H to E and steps from E to P will result to the steps from H to P.
(9x) + (x + 4)
= (10x + 4) steps
2.
(3x + 2) + (5x)
= (8x + 3) steps
3.
m∠HOE = 30
m∠OEH = 90 (since line OE is perpendicular to line HP)
m∠H = ?
m∠HOE + m∠OEH + m∠H = 180 (a triangle's sum of interior angles is equal to 180°)
30 + 90 + m∠H = 180
120 + m∠H = 180
m∠H = 180 - 120 (transfer 120 to the other side to isolate m∠H)
m∠H = 60
4.
m∠HOE = x
m∠POE = 2x - 30
x = ?
m∠HOE = m∠POE (line OE bisects angle POH, therefore, the measure of angle HOE and angle POE is equal)
x = 2x - 30
x + 30 = 2x (transpose)
30 = 2x - x (transfer x to the other side of the equation)
30 = x
5.
m∠OHE = 2(m∠HOE)
m∠OEH = 90 (again, as we can see, this is a right angle)
m∠HOE = ?
m∠OHE + m∠OEH + m∠HOE = 180 (again, a triangle's sum of interior angles is equal to 180°)
2(m∠HOE) + 90 + m∠HOE = 180
3(m∠HOE) + 90 = 180 (add like terms)
3(m∠HOE) = 180 - 90 (transpose)
3(m∠HOE) = 90 (divide both sides by 3 to isolate m∠HOE)
m∠HOE = 30
m∠OHE = 2(m∠HOE)
m∠OHE = 2(30)
m∠OHE = 60