Solve for the missing angles of kite EKIT, where \(\angle K = 65^\circ\) and \(\angle E = 40^\circ\). Find the measure of angles EKS, KIS, STI, and KET.
Understand the Problem
The question asks us to find the measure of the missing angles in kite EKIT, which are angles EKS, KIS, STI, and KET, given the measure of (\angle K = 65^\circ) and (\angle E = 40^\circ). We will need to use the properties of kites and triangles to work this out.
Answer
$\angle EKS = 90^\circ$ $\angle KIS = 25^\circ$ $\angle STI = 90^\circ$ $\angle KET = 50^\circ$
Answer for screen readers
$\angle EKS = 90^\circ$ $\angle KIS = 25^\circ$ $\angle STI = 90^\circ$ $\angle KET = 50^\circ$
Steps to Solve
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Angle EKS Since the diagonals in a kite are perpendicular, $\angle EKS$ forms a right angle.
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Calculation of angle EKS Therefore, $\angle EKS = 90^\circ$.
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Angle STI Similarly, since the diagonals in a kite are perpendicular, $\angle STI$ forms a right angle.
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Calculation of angle STI Therefore, $\angle STI = 90^\circ$.
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Angle KET In triangle EKS, we know $\angle EKS = 90^\circ$ and $\angle KES = 40^\circ$. The sum of angles in a triangle is $180^\circ$, so we can find $\angle EKS$. Note that $\angle KES$ is the same as $\angle KET$.
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Calculation of angle KET $\angle KET = 90^\circ - 40^\circ = 50^\circ$
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Angle KIS In triangle KIS, we know $\angle STI = 90^\circ$ and $\angle SKI = 65^\circ$. The sum of angles in a triangle is $180^\circ$, so we can find $\angle KIS$.
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Calculation of angle KIS $\angle KIS = 90^\circ - 65^\circ = 25^\circ$
$\angle EKS = 90^\circ$ $\angle KIS = 25^\circ$ $\angle STI = 90^\circ$ $\angle KET = 50^\circ$
More Information
A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. The diagonals of a kite are perpendicular, and one of the diagonals bisects the other. Also, one of the diagonals bisects a pair of opposite angles.
Tips
A common mistake is assuming all angles in a kite are equal or that opposite angles are equal which is not generally true for kites (except for the angles between the unequal sides). Another mistake is not using the property that the diagonals are perpendicular. For angles, a common mistake is incorrectly applying the angle sum of a triangle.
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