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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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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