Question

date:
ch kite.
2
m∠1 =
m∠2 =
m∠3 =

Explanation:

Step1: Find \( m\angle1 \)

In a kite, one of the diagonals is the perpendicular bisector? No, actually, the diagonals of a kite are perpendicular? Wait, no, in a kite, one diagonal is bisected by the other, and one pair of opposite angles are equal. Wait, the given angle is \( 72^\circ \), and the diagonal divides the kite into two triangles. Wait, \( \angle1 \) is a straight line? No, wait, the diagonal that is the axis of symmetry. Wait, actually, in a kite, the diagonals are perpendicular? Wait, no, the diagonals intersect at right angles? Wait, no, let's recall: in a kite, one diagonal is the perpendicular bisector of the other. Wait, maybe \( \angle1 \) is a straight angle? No, wait, the angle marked \( 72^\circ \), and the triangle with \( 72^\circ \): wait, the two triangles on the top are isosceles (since the sides are marked equal). So in the top right triangle, the two sides are equal, so it's isosceles. So the base angles are equal? Wait, no, the angle at the top is \( 72^\circ \), so the other two angles in that triangle: sum of angles in a triangle is \( 180^\circ \), so \( 180 - 72 = 108 \), divided by 2 is \( 54^\circ \). But maybe \( \angle1 \) is \( 90^\circ \)? Wait, no, maybe the diagonals are perpendicular. Wait, the problem is about a kite, so diagonals are perpendicular? Wait, no, in a kite, one diagonal is perpendicular to the other? Wait, actually, in a kite, one diagonal is the perpendicular bisector of the other. Wait, maybe I made a mistake. Wait, let's look at \( \angle1 \): the diagonal that is horizontal and vertical. Wait, the vertical diagonal and horizontal diagonal intersect. If the kite has two pairs of adjacent sides equal, then the diagonals are perpendicular. So \( \angle1 = 90^\circ \)? Wait, no, maybe not. Wait, the angle given is \( 43^\circ \) on the left. Wait, let's start with \( m\angle1 \). Wait, the horizontal diagonal and vertical diagonal: in a kite, the diagonals are perpendicular, so \( \angle1 = 90^\circ \)? Wait, no, maybe \( \angle1 \) is \( 180 - 72 = 108 \)? No, that doesn't make sense. Wait, maybe the vertical diagonal is the axis of symmetry, so the angle \( 72^\circ \) and the angle on the left: wait, no, let's check the triangle with \( 43^\circ \). Wait, the left triangle: angle \( 43^\circ \), and the sides are equal (marked with two lines), so it's isosceles? Wait, no, the left triangle has two sides marked with one line? Wait, the diagram: the top two sides are marked with one line, bottom two with two lines. So the kite has two pairs of adjacent sides equal: top left and top right (one line), bottom left and bottom right (two lines). So the diagonal that connects the bottom vertices is the axis of symmetry. So the vertical diagonal is the axis of symmetry, so it bisects the angles at the bottom and top. Wait, the top angle is \( 72^\circ \), so the vertical diagonal bisects it? No, the horizontal diagonal? Wait, maybe \( \angle1 \) is \( 90^\circ \) because diagonals in a kite are perpendicular. So \( m\angle1 = 90^\circ \).

Step2: Find \( m\angle2 \)

In the left triangle, we have angle \( 43^\circ \), angle \( \angle1 = 90^\circ \), so the third angle \( \angle2 \): sum of angles in a triangle is \( 180^\circ \), so \( 180 - 90 - 43 = 47^\circ \)? Wait, no, \( 90 + 43 + \angle2 = 180 \), so \( \angle2 = 180 - 90 - 43 = 47^\circ \)? Wait, no, maybe \( \angle2 \) is in the top left triangle. Wait, the top left triangle: sides are equal (one line), so it's isosceles. The angle at the top is \( 72^\circ \), so the base angles: \( (180 - 72)/2 = 54^\circ \). But the horizontal diagonal and vertical diagonal intersect at \( \angle1 \), which is \( 90^\circ \). Wait, maybe \( \angle2 = 54 - 43 = 11^\circ \)? No, that doesn't make sense. Wait, let's re-examine. The left angle is \( 43^\circ \), and the horizontal diagonal splits the angle into \( 43^\circ \) and \( \angle2 \). Wait, the vertical diagonal is the axis of symmetry, so the angle at the left vertex: the total angle is \( 43^\circ + \angle2 \), and the angle at the right vertex is equal? Wait, no, the kite has two pairs of adjacent sides equal, so the angles between the unequal sides are equal. Wait, maybe \( \angle2 \) is \( 180 - 90 - 72/2 = 180 - 90 - 36 = 54 \)? No, I'm confused. Wait, let's start over.

Wait, the kite has two pairs of adjacent sides equal: let's call the vertices A (top), B (right), C (bottom), D (left). So AB = AD (one line), CB = CD (two lines). Diagonals AC (vertical) and BD (horizontal) intersect at point O (angle 1). In a kite, diagonal AC is the axis of symmetry, so it bisects angles A and C, and diagonal BD is bisected by AC (so BO = OD). Also, diagonals are perpendicular (angle AOB = 90°), so \( \angle1 = 90^\circ \).

Now, angle at A is \( 72^\circ \), so AC bisects it, so angle OAB = \( 72/2 = 36^\circ \). In triangle AOB, angle AOB = 90°, angle OAB = 36°, so angle OBA = \( 180 - 90 - 36 = 54^\circ \). Now, angle at D: angle ODA = 43°? Wait, no, the diagram shows angle 43° at D, between side CD and diagonal BD. Wait, CD = CB (two lines), so triangle CDB is isosceles with CD = CB, so angle CDB = angle CBD = 43°? No, that can't be. Wait, maybe the angle 43° is angle ODB = 43°, and in triangle ODB, angle DOB = 90° (since diagonals are perpendicular), so angle OBD = \( 180 - 90 - 43 = 47^\circ \). But angle OBA is 54° (from triangle AOB), so angle 2 (angle ODA? No, angle 2 is angle OAD? Wait, no, the diagram: angle 2 is between diagonal AD and BD, at vertex D. Wait, maybe angle 2 is angle ODA. Wait, AD = AB (one line), so triangle ABD is isosceles with AD = AB, so angle ADB = angle ABD. Angle ABD is angle OBA = 54°, so angle ADB = 54°, but angle ODB is 43°, so angle 2 = 54 - 43 = 11°? No, that's not right.

Wait, maybe the angle 43° is angle ODC = 43°, and since CD = CB, triangle CDB is isosceles, so angle OBC = 43°, so angle OBA = angle OBC? No, because AB = AD and CB = CD. Wait, I think I made a mistake with \( \angle1 \). Maybe \( \angle1 \) is not 90°. Wait, the sum of angles in a triangle: the top triangle (AOB) has angle at A = 72°, sides AB = AD, so it's isosceles, so angle OBA = angle ODA. Wait, the left triangle (DOA) has angle at D = 43°, side AD = AB, and side DO = OB (since diagonals bisect each other? No, in a kite, only one diagonal is bisected). Wait, I'm stuck. Maybe the answer is:

\( m\angle1 = 90^\circ \) (diagonals perpendicular), \( m\angle2 = 180 - 90 - 72 = 18 \)? No, that's not. Wait, maybe the angle at the top is 72°, so the triangle with 72°: sum of angles is 180, so the other two angles are (180 - 72)/2 = 54°, so angle 2 is 54 - 43 = 11°, and angle 3 is 90 - 43 = 47°? No, I think I need to recall that in a kite, diagonals are perpendicular, so \( \angle1 = 90^\circ \). Then, in the left triangle with 43°, angle 2 = 90 - 43 = 47°? No, 90 + 43 + 47 = 180. Then angle 3: in the bottom triangle, sides are equal, so it's isosceles, angle 3 = 90 - (something). Wait, maybe the correct answers are \( m\angle1 = 90^\circ \), \( m\angle2 = 11^\circ \), \( m\angle3 = 47^\circ \). But I'm not sure. Wait, let's check:

If \( \angle1 = 90^\circ \) (diagonals perpendicular), then in triangle with 72°, the two base angles are (180 - 72)/2 = 54°, so angle between horizontal diagonal and top side is 54°. The angle at the left is 43°, so angle 2 = 54 - 43 = 11°. Then, in the bottom triangle, angle 3: since the vertical diagonal is perpendicular, angle 3 = 90 - 43 = 47°? No, 90 - 11 = 79? I'm really confused. Wait, maybe the answer is:

\( m\angle1 = 90^\circ \), \( m\angle2 = 11^\circ \), \( m\angle3 = 47^\circ \). But I think the key is that diagonals in a kite are perpendicular, so \( \angle1 = 90^\circ \). Then, in the top triangle, angles are 72°, 54°, 54°, so the angle between horizontal diagonal and top side is 54°. The left angle is 43°, so angle 2 is 54 - 43 = 11°. Then, in the bottom triangle, angle 3: since the vertical diagonal is perpendicular, and the bottom sides are equal, angle 3 = 90 - 43 = 47°? No, 90 - 11 = 79. I'm not sure. Maybe the correct answers are \( m\angle1 = 90^\circ \), \( m\angle2 = 11^\circ \), \( m\angle3 = 47^\circ \).

Answer:

\( m\angle1 = 90^\circ \), \( m\angle2 = 11^\circ \), \( m\angle3 = 47^\circ \) (Note: The exact values may depend on the correct properties of the kite, but based on diagonals being perpendicular and angle sum in triangles, these are the likely values.)