# How to solve this geometry problem, with proof?

Relevance

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°

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• 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 • Login to reply the answers
• I'm getting 147.5

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