Question

angle m has a measure of 47°. what is the measure of angle pnl? 43° 47° 86° 94°

Explanation:

Step1: Analyze triangle NPM

In triangle NPM, angle at P is \( 90^\circ \) (right angle), angle at M is \( 47^\circ \). The sum of angles in a triangle is \( 180^\circ \). So angle PNM \( = 180^\circ - 90^\circ - 47^\circ = 43^\circ \)? Wait, no, wait. Wait, the triangle LNM: sides LN and MN are equal (marked with same tick), so triangle LNM is isosceles with LN = MN. Also, LP = PM (marked with same tick), and NP is perpendicular to LM, so NP is the perpendicular bisector. So triangle LNM is isosceles with base LM, so angle at L equals angle at M? Wait, no, angle at M is \( 47^\circ \), so angle at L is also \( 47^\circ \)? Wait, no, wait. Wait, in triangle NPM, right-angled at P, angle at M is \( 47^\circ \), so angle PNM is \( 90^\circ - 47^\circ = 43^\circ \)? Wait, no, we need angle PNL. Wait, triangle LNM: since LN = MN (sides with same tick), so it's isosceles with LN = MN. Then, NP is the altitude, median, and angle bisector. So angle LNM: in triangle LNM, angles at L and M are equal? Wait, angle at M is \( 47^\circ \), so angle at L is also \( 47^\circ \), so angle at N (angle LNM) is \( 180^\circ - 47^\circ - 47^\circ = 86^\circ \). Then, since NP is the angle bisector (because it's the median and altitude in isosceles triangle), so angle PNL is half of angle LNM? Wait, no, wait. Wait, LP = PM, so P is the midpoint, and NP is perpendicular, so triangle LNP and MNP are congruent. So angle PNL = angle PNM? Wait, no, angle LNM is \( 86^\circ \), so if NP bisects it, then angle PNL = angle PNM = \( 43^\circ \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's re-examine. The triangle: LM is the base, with LP = PM (marked), LN = MN (marked), so triangle LNM is isosceles with LN = MN, so vertex at N. Then angle at M is \( 47^\circ \), so angle at L is also \( 47^\circ \), so angle at N (angle LNM) is \( 180 - 47 - 47 = 86^\circ \). Then NP is the altitude from N to LM, so it splits angle LNM into two equal angles? Wait, no, in an isosceles triangle, the altitude from the apex (N) to the base (LM) bisects the apex angle. So angle LNM is \( 86^\circ \), so angle PNL = angle PNM = \( 86^\circ / 2 = 43^\circ \)? But that's not one of the options? Wait, no, the options are 43, 47, 86, 94. Wait, maybe I messed up. Wait, wait, angle at M is \( 47^\circ \), NP is perpendicular to LM, so in triangle NPM, right-angled at P, angle at M is \( 47^\circ \), so angle PNM is \( 90 - 47 = 43^\circ \). But angle PNL: since triangle LNP is congruent to triangle MNP (SAS: LP = PM, angle at P is 90, NP common), so angle PNL = angle PNM? Wait, no, angle PNM is \( 43^\circ \), so angle PNL is also \( 43^\circ \)? But 43 is an option. Wait, but let's check again. Wait, the problem is asking for angle PNL. Let's see: in triangle NPM, right-angled at P, angle M = 47°, so angle PNM = 90° - 47° = 43°. But since LN = MN and LP = PM, triangle LNP ≅ triangle MNP (SAS), so angle PNL = angle PNM = 43°? Wait, but maybe I got the angle wrong. Wait, maybe angle at M is 47°, so in triangle NPM, angle at P is 90°, angle at M is 47°, so angle PNM is 43°, so angle PNL is equal to angle PNM? Wait, no, because LN = MN, so triangle LNM is isosceles, so angle at L is 47°, so angle LNM is 180 - 47 - 47 = 86°, then NP is the altitude, so it splits angle LNM into two equal parts? Wait, 86° split into two is 43°, so angle PNL is 43°, which is option A. Wait, but let's confirm. So angle PNL: in triangle LNP, right-angled at P (since NP ⊥ LM), angle at L is 47°, so angle PNL is 90° - 47° = 43°. Yes! Because in triangle LNP, right-angled at P, angle at L is 47°, so angle PNL = 90° - 47° = 43°. So that's the answer.

Step1: Identify triangle type and right angle

Triangle \( LNP \) is right - angled at \( P \) (since \( NP\perp LM \)), and \( \angle L = \angle M = 47^\circ \) (because \( LN = MN \), so \( \triangle LNM \) is isosceles with \( \angle L=\angle M \)).

Step2: Calculate \( \angle PNL \)

In right - triangle \( LNP \), the sum of the two non - right angles is \( 90^\circ \). Let \( \angle PNL=x \), we know that one non - right angle \( \angle L = 47^\circ \). Using the formula for the angles of a right - triangle (\( \angle PNL+\angle L = 90^\circ \)), we get \( x = 90^\circ- 47^\circ=43^\circ \).

Answer:

\( 43^\circ \) (corresponding to the option with \( 43^\circ \))