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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
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avatar+177 
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Sure, let's break it down:

 

We're given triangle \(STU\) with midpoints \(M\), \(N\), and \(P\) on sides \(TU\), \(US\), and \(ST\) respectively. \(UZ\) is the altitude of the triangle.

Given:


- \(\angle TSU = 62^\circ\)


- \(\angle STU = 29^\circ\)

 

1. We draw the altitude \(UZ\) from vertex \(U\) to side \(TS\), creating right triangles \(TUZ\) and \(SUZ\).

 

2. We find:


   - \(\angle SUT = 180^\circ - 29^\circ - 62^\circ = 89^\circ\)


   - \(NM\) is parallel to \(ST\), so \(\angle NZM = \angle TSU = 62^\circ\)


   - \(NP\) is parallel to \(UT\), so \(\angle NPM = \angle STU = 29^\circ\)

 

3. Then, we calculate the sum:


   \[ \angle NZM + \angle NPM = 62^\circ + 29^\circ = 91^\circ \]

 

So, the sum of angles \( \angle NZM + \angle NPM \) is \( 91^\circ \).

 Apr 14, 2024

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