Question

what is the value of w?
(there is a triangle qrs with angle at q being 86 degrees, angle at r being 47 degrees, and two sides (qr and qs) marked as equal. we need to find the measure of angle w at s.)
w = \boxed{\circ}

Explanation:

Step1: Identify triangle type (isosceles)

The triangle has two equal sides (marked with red ticks), so it's isosceles. Thus, angles opposite equal sides are equal. Wait, no—wait, in triangle \( QRS \), sides \( QS \) and \( QR \) are equal? Wait, no, the marks: one on \( QR \), one on \( QS \). So \( QS = QR \), so the angles opposite them: angle \( R \) and angle \( S \)? Wait, no, angle opposite \( QS \) is angle \( R \), angle opposite \( QR \) is angle \( S \). Wait, but angle at \( Q \) is \( 85^\circ \), angle at \( R \) is \( 47^\circ \). Wait, no, sum of angles in a triangle is \( 180^\circ \). Wait, maybe I misread. Wait, the triangle: vertices \( Q \), \( R \), \( S \). Angle at \( Q \): \( 85^\circ \), angle at \( R \): \( 47^\circ \), angle at \( S \): \( w \). Wait, but if two sides are equal, then two angles are equal. Wait, the red ticks: one on \( QR \), one on \( QS \), so \( QS = QR \), so angles opposite: angle \( R \) (opposite \( QS \)) and angle \( S \) (opposite \( QR \))? Wait, no, side \( QS \) is opposite angle \( R \), side \( QR \) is opposite angle \( S \). So if \( QS = QR \), then angle \( R = \) angle \( S \)? But angle \( R \) is \( 47^\circ \), but then angle \( Q \) is \( 85^\circ \), sum would be \( 85 + 47 + 47 = 179 \), which is close but maybe typo? Wait, no, maybe I made a mistake. Wait, sum of angles in a triangle is \( 180^\circ \). So \( 85 + 47 + w = 180 \)? Wait, no, if it's isosceles, maybe the equal angles are different. Wait, let's recalculate. Sum of angles: \( 85 + 47 + w = 180 \). So \( 132 + w = 180 \), so \( w = 180 - 132 = 48 \)? Wait, no, 85 + 47 is 132, 180 - 132 is 48. Wait, but maybe the equal sides are \( RS \) and \( QS \)? No, the marks are on \( QR \) and \( QS \). Wait, maybe the triangle is isosceles with \( QS = QR \), so angles at \( R \) and \( S \) are equal? But angle at \( R \) is 47, so angle at \( S \) would be 47, but then angle at \( Q \) would be 180 - 47 - 47 = 86, but it's 85. Hmm, maybe the diagram is different. Wait, the user's diagram: angle at \( Q \) is 85, angle at \( R \) is 47, sides \( QS \) and \( QR \) are equal (red ticks). So \( QS = QR \), so angles opposite: angle \( R \) (opposite \( QS \)) and angle \( S \) (opposite \( QR \))? Wait, side \( QS \) is between \( Q \) and \( S \), so opposite angle \( R \). Side \( QR \) is between \( Q \) and \( R \), opposite angle \( S \). So if \( QS = QR \), then angle \( R = \) angle \( S \). But angle \( R \) is 47, so angle \( S \) is 47, then angle \( Q \) is 180 - 47 - 47 = 86, but the diagram says 85. Maybe the diagram has a typo, or I misread the angle at \( Q \). Wait, maybe the angle at \( Q \) is 85, angle at \( R \) is 47, and we just use sum of angles. Let's do that: sum of angles in triangle is \( 180^\circ \). So \( 85 + 47 + w = 180 \). So \( w = 180 - 85 - 47 \). Calculate: 85 + 47 = 132, 180 - 132 = 48. So \( w = 48 \).

Step2: Calculate \( w \)

Using the angle - sum property of a triangle (\( \text{Sum of angles} = 180^\circ \)):
\[
w=180 - 85 - 47
\]
First, \( 85 + 47 = 132 \), then \( 180-132 = 48 \).

Answer:

\( 48 \)