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Let M, N, and P be the midpoints of sides TU, US, and ST of triangle STU respectively. Let UZ be the altitude of the triangle. If angle TSU = 62 degrees and angle STU = 29 degrees, then what is angle NZM + angle NPM in degrees?

 Apr 13, 2024
 #1
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Angles in Triangle STU:

 

We are given that angle TSU = 62 degrees and angle STU = 29 degrees. Since the angles in a triangle add up to 180 degrees, we can find the third angle (angle SUT):

 

angle SUT = 180 degrees - angle TSU - angle STU = 180 degrees - 62 degrees - 29 degrees = 89 degrees.

 

Angles in Triangles MNU and MNP:

 

Since M, N, and P are midpoints of sides, segments MN, NP, and PM are parallel to TU, US, and ST respectively (corresponding sides theorem).

 

Due to alternate interior angles, angle MNU = angle TSU = 62 degrees and angle MNP = angle STU = 29 degrees (alternate interior angles theorem).

 

Angle NZM and Angle NPM:

 

Triangle NUZ is a right triangle (angle NUZ = 90 degrees) since UZ is an altitude.

 

Since M is the midpoint of TU, segment NZ is half of segment TU. Similarly, segment NP is half of segment US.

 

Therefore, triangles NUZ and MNP are similar (AA Similarity).

 

Corresponding angles in similar triangles are congruent. Thus, angle NZM = angle NMP (corresponding angles).

 

Finding Angle NZM + Angle NPM:

 

We know angle MNU = 62 degrees and angle NMP = angle NZM (from step 3). Since angles in triangle MNU add up to 180 degrees:

 

angle NZM + angle NMP + angle MNU = 180 degrees

 

Substitute the known values:

 

angle NZM + angle NZM + 62 degrees = 180 degrees

 

Combine like terms:

 

2 * angle NZM = 180 degrees - 62 degrees

 

Solve for angle NZM:

 

angle NZM = 118 degrees / 2 = 59 degrees

 

Therefore, angle NZM + angle NPM = angle NZM + angle NZM (since they are congruent) = 59 degrees + 59 degrees = 118 degrees.

 Apr 13, 2024
 #2
avatar+46 
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This ended up not being the answer, however, it did say as a hint "Remember that the midpoint of a hypotenuse is a special point." 

Ranger897  Apr 13, 2024

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