How to solve this geometry problem, with proof?

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  • 3 years ago
    Best Answer

    i) By angle sum property from triangle MKN, <MKN = 55°

    As given <MKP = 10°, <PKN = 45°

    ==> KN = PN ------ (1)

    ii) <KNS = 90 - 20 = 70°

    So from triangle KNS, by angle sum property, <KSN = 55°

    ==> Triangle KNS is isosceles. [<SKN = <KSN]

    ==> KN = SN ----- (2)

    iii) Thus from (1) & (2): PN = SN

    So triangle PSN is isosceles.

    ==> <NPS = <NSP = (180 - 20)/2 = 80°

    So applying exterior angle property, from triangle PSM,

    <PSM = 80° - <PMS = 80° - 35° = 45°

    Thus <PSM = 45°

    NOTE: Irrespective of belated reply, wish you may acknowledge or provide a feed back subject to finding this solution satisfactory. Thanks.

  • 3 years ago

    Recall the rule: the sum of 3 angles of whatever the triangle is always 180 °.

    To be clearer, we are going to use the colors instead name of angle.

    In the triangle KNM, you know that this triangle is a right triangle at N.

    red + cyan = 90

    cyan = 90 - red → given: red = 20

    cyan = 70

    In the triangle KNM, you apply the rule.

    yellow + red + cyan + blue + green = 180 → we've just seen that: red + cyan = 90

    yellow + 90 + blue + green = 180 → given: yellow = 35

    35 + 90 + blue + green = 180 → given: green = 10

    35 + 90 + blue + 10 = 180

    135 + blue = 180

    blue = 45

    In the triangle KPM, you apply the rule.

    purple + green + yellow = 180 → given: yellow = 35

    purple + green + 35 = 180 → given: green = 10

    purple + 10 + 35 = 180

    purple + 45 = 180

    purple = 135

    In the triangle KAN, you apply the rule.

    cyan + blue + orange = 180 → we've seen that: cyan = 70

    70 + blue + orange = 180 → we've seen that: blue = 45

    70 + 45 + orange = 180

    115 + orange = 180

    orange = 65

    At the point A, you can deduce that:

    grey = orange → we've just seen that: orange = 65

    grey = 65

    At the point A, you can see that:

    grey + dark_green = 180 ← because it is a flat angle

    dark_green = 180 - grey → we've just seen that: orange = 65

    dark_green = 115

    In the triangle KAS, you apply the rule.

    dark_green + green + brown = 180 → we've just seen that: dark_green = 115

    115 + green + brown = 180 → given: green = 10

    115 + 10 + brown = 180

    125 + brown = 180

    brown = 55

    At the point S, you can deduce that:

    brown + pink = 180 ← because it is a flat angle

    brown + pink = 180 → we've just seen that: brown = 55

    55 + pink = 180

    pink = 125

    At the point P, you can deduce that:

    purple + white = 180 ← because it is a flat angle

    purple + white = 180 → we've seen that: purple = 135

    135 + white = 180

    white = 45

    In the triangle PSK, you apply the rule.

    α + β + brown + green = 180 → we've seen that: brown = 55

    α + β + 55 + green = 180 → given: green = 10

    α + β + 55 + 10 = 180

    α + β + 65 = 180

    α + β = 115

    α = 115 - β ← memorize this result as (1)

    In the triangle PSM, you apply the rule.

    yellow + θ + (purple - α) = 180 → given: yellow = 35

    35 + θ + (purple - α) = 180 → we've seen that: purple = 135

    35 + θ + (135 - α) = 180

    35 + θ + 135 - α = 180

    θ + 170 - α = 180

    θ - α = 10

    θ = 10 + α ← memorize this result as (2)

    In the triangle PSM, you apply the rule.

    yellow + γ + θ = 180 → given: yellow = 35

    35 + γ + θ = 180

    γ + θ = 145

    γ = 145 - θ → recall (2): θ = 10 + α

    γ = 145 - (10 + α)

    γ = 145 - 10 - α

    γ = 135 - α → recall (1): α = 115 - β

    γ = 135 - (115 - β)

    γ = 135 - 115 + β

    γ = 20 + β ← memorize this result as (3)

    …now you have all the conditions to find the angle PSM, i.e. θ → θ = 45

    Attachment image
  • I'm getting 147.5

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