Question

The figure below is a kite. If m<1=74 and m<8=8, find the measures of the missing angles.

Answer 1

Since the quadrilateral ABCD is a kite, then the diagonals AC and BD are perpendicular.

Then, the angles formed in the intersection betwen AC and BD are all right angles.

Let O be the intersection between AC and BD. Then, the triangles ABO, BOC, COD and DOA are all right triangles.

Furthermore, the triangles ABO and BOC are congruent, as well as COD and DOA.

Since corresponding parts of congruent triangles are congruent, then:

`\begin{gathered} m\angle8=m\angle9 \\ m\angle6=m\angle7 \\ m\angle2=m\angle3 \\ m\angle1=m\angle5 \end{gathered}`

On the other hand, since each triangle is a right triangle, the acute angles are supplementary. since the measure of the angle 1 is 74, then the measure of the angle 2 must be 16. Since the measure of the angle 8 is 8, then the measure of the angle 6 must be 72.

Therefore, the measures of the angles are:

`\begin{gathered} m\angle1=74 \\ n\angle2=16 \\ m\angle3=16 \\ m\angle4=90 \\ m\angle5=74 \\ m\angle6=72 \\ m\angle7=72 \\ m\angle8=8 \\ m\angle9=8 \end{gathered}`