Question

a. 206 b. 33 c. 25 d. 29
14 find ( m angle 1 ) and ( m angle 3 ) in the kite. the diagram is not to scale.
a. ( m angle 1=51, m angle 3=39 )
b. ( m angle 1=51, m angle 3=51 )
c. ( m angle 1=39, m angle 3=39 )
d. ( m angle 1=39, m angle 3=51 )
15 ( m angle r=140 ) and ( m angle s=80 ). find ( m angle t ). the diagram is not to scale.
a. 60 b. 30 c. 80 d. 70

Explanation:

Question 14

Step1: Use the property of kite

In a kite, the axis of symmetry bisects the angles at the vertices. So, \(m\angle1 = 39^{\circ}\) (by the property that the axis of symmetry of a kite bisects the vertex angle at \(A\)).

Step2: Use the right - triangle angle sum property

In the right - triangle (formed by the intersection of the diagonals of the kite), we know that the sum of angles in a triangle is \(180^{\circ}\). One angle is \(90^{\circ}\) (diagonals of a kite are perpendicular) and another angle is \(39^{\circ}\). Let \(\angle3\) be \(x\). Then \(x+39^{\circ}+90^{\circ}=180^{\circ}\). So \(x = 180^{\circ}-(90^{\circ}+39^{\circ})=51^{\circ}\)

Step1: Use the property of a kite (a quadrilateral with two pairs of adjacent sides equal)

The sum of the interior angles of a quadrilateral is \(360^{\circ}\). Let \(m\angle T=x\) and \(m\angle U = x\) (in a kite, non - vertex angles are equal). We know that \(m\angle R = 140^{\circ}\) and \(m\angle S=80^{\circ}\). Then \(x + x+140^{\circ}+80^{\circ}=360^{\circ}\)

Step2: Solve the equation

\(2x+220^{\circ}=360^{\circ}\). Subtract \(220^{\circ}\) from both sides: \(2x=360^{\circ}-220^{\circ}=140^{\circ}\). Divide both sides by \(2\): \(x = 70^{\circ}\)

Answer:

\(m\angle1 = 39\), \(m\angle3 = 51\) (Option D)

Question 15