Question

what are the values of a and b?
k
47°
i
b
j
a

a = □°
b = □°

Explanation:

Step1: Identify the triangle type

The triangle has two equal sides (marked with red ticks), so it's isosceles. In an isosceles triangle, the angles opposite equal sides are equal. So angle \( a \) and angle \( b \)? Wait, no—wait, the sides \( KJ \) and \( IJ \)? Wait, no, the marks: one on \( KJ \) and one on \( IJ \)? Wait, looking at the triangle, vertices \( K \), \( J \), \( I \). The sides \( KJ \) and \( I J \)? Wait, no, the two sides with ticks: \( KJ \) and \( I J \)? Wait, no, the angle at \( K \) is \( 47^\circ \). Wait, in an isosceles triangle, the base angles are equal. Wait, maybe the equal sides are \( KJ \) and \( I K \)? No, the ticks: one on \( KJ \) (the left side) and one on \( I J \) (the bottom side). Wait, no, maybe the equal sides are \( KJ \) and \( I J \), so the angles opposite them: angle \( I \) (angle \( b \)) and angle \( K \)? No, angle \( K \) is \( 47^\circ \). Wait, no—wait, in triangle \( KJI \), sides \( KJ \) and \( I J \) are equal (ticks), so the angles opposite them: angle \( I \) (opposite \( KJ \)) and angle \( K \) (opposite \( I J \))? Wait, no, angle opposite \( KJ \) is angle \( I \) (angle \( b \)), and angle opposite \( I J \) is angle \( K \) ( \( 47^\circ \))? Wait, no, that can't be. Wait, maybe the equal sides are \( KJ \) and \( KI \)? No, the ticks are on \( KJ \) and \( I J \). Wait, maybe I got the sides wrong. Let's re-express: triangle \( KJI \), with \( K \) at top, \( J \) at bottom left, \( I \) at bottom right. The side \( KJ \) (left) and \( I J \) (bottom) have ticks, so they are equal. Therefore, the angles opposite them: angle \( I \) (at \( I \), opposite \( KJ \)) and angle \( K \) (at \( K \), opposite \( I J \))? Wait, angle at \( K \) is \( 47^\circ \), so angle at \( I \) (angle \( b \)) would be equal to angle at \( K \)? No, that would make it isosceles with \( KJ = I J \), so angles opposite: angle \( I \) (opposite \( KJ \)) and angle \( K \) (opposite \( I J \)) are equal. Wait, but then angle \( a \) is at \( J \). Then, sum of angles in a triangle is \( 180^\circ \). So if angle \( K = 47^\circ \), angle \( I = b \), and angle \( J = a \). If \( KJ = I J \), then angle \( K = angle I \), so \( b = 47^\circ \), and then \( a = 180 - 47 - 47 = 86^\circ \)? Wait, no, that doesn't make sense. Wait, maybe the equal sides are \( KJ \) and \( KI \), so angles at \( J \) and \( I \) are equal. Wait, the tick is on \( KJ \) (left side) and on \( KI \) (right side)? No, the right side is \( KI \), but the tick is on the bottom side \( I J \). Wait, I think I made a mistake. Let's start over: in a triangle, if two sides are equal, the angles opposite those sides are equal. So let's label the triangle: vertices \( K \) (top), \( J \) (bottom left), \( I \) (bottom right). Sides: \( KJ \) (left, from \( K \) to \( J \)), \( JI \) (bottom, from \( J \) to \( I \)), \( KI \) (right, from \( K \) to \( I \)). The ticks are on \( KJ \) and \( JI \), so \( KJ = JI \). Therefore, the angles opposite these sides: angle opposite \( KJ \) is angle \( I \) (at \( I \), between \( KI \) and \( JI \)), and angle opposite \( JI \) is angle \( K \) (at \( K \), between \( KJ \) and \( KI \)). So angle \( I \) ( \( b \)) and angle \( K \) ( \( 47^\circ \)) are equal? Wait, no, that would mean \( b = 47^\circ \), and then angle \( J \) ( \( a \)) is \( 180 - 47 - 47 = 86^\circ \)? Wait, but that would mean \( a = 86^\circ \) and \( b = 47^\circ \)? Wait, no, maybe the equal sides are \( KJ \) and \( KI \), so angles at \( J \) and \( I \) are equal. Wait, the tick is on \( KJ \) (left) and \( KI \) (right), so \( KJ = KI \), so angles at \( J \) ( \( a \)) and \( I \) ( \( b \)) are equal. Then angle \( K \) is \( 47^\circ \), so sum of angles: \( 47 + a + b = 180 \), and \( a = b \), so \( 47 + 2a = 180 \), so \( 2a = 133 \), \( a = 66.5 \)? No, that can't be. Wait, maybe I misread the ticks. Let me look again: the triangle has two sides with ticks: one on the left side ( \( KJ \)) and one on the bottom side ( \( JI \)). So \( KJ = JI \), so triangle is isosceles with \( KJ = JI \), so angles opposite: angle \( I \) (opposite \( KJ \)) and angle \( K \) (opposite \( JI \)) are equal. So angle \( I = angle K = 47^\circ \), so \( b = 47^\circ \). Then angle \( J \) ( \( a \)) is \( 180 - 47 - 47 = 86^\circ \). Yes, that makes sense. So \( a = 86^\circ \), \( b = 47^\circ \).

Step2: Calculate angle \( a \)

Sum of angles in a triangle is \( 180^\circ \). So \( a + b + 47^\circ = 180^\circ \). Since \( b = 47^\circ \) (from isosceles triangle, angles opposite equal sides), then \( a = 180 - 47 - 47 = 86^\circ \).

Answer:

\( a = 86^\circ \), \( b = 47^\circ \)