Question

Algebra Find the value(s) of the variable(s) in each kite.

Answer 1

- The figure given is a Kite. Based on this, there are some properties we need to know in order to solve the question

1. The Opposite Obtuse angles are equal.

2. The longer diagonal divides the kite into two congruent triangles (That is, the triangles have equal angles and sides).

- With the above properties we can proceed to solve the question

Applying Property 1:

- The Opposite Obtuse angles are equal. This implies that the angles highlighted below are equal:

- Thus, we can write

`\begin{gathered} (4x+13)=(5x-15) \\ Remove\text{ the brackets} \\ 4x+13=5x-15 \\ \text{Collect like terms};\text{ Subtract 4x from both sides, and Add 15 to both sides} \\ \\ 5x-4x=15+13 \\ x=28\degree \end{gathered}`

- The value of x is 28°

Applying Property 2:

- The longer diagonal divides the kite into two congruent triangles.

- This implies that the angles on both triangles formed by that large diagonal are equal. That is,

- Now that we have established the following, we can simply pick any of the triangles and apply the "sum angle theorem" on any of them.

- This is done below:

`\begin{gathered} (5x-15)+(y-9)+y=180\degree\text{ (Sum of angles in a triangle is 180)} \\ \text{put }x=28 \\ (5(28)-15)+(y-9)+y=180 \\ 125+y-9+y=180 \\ \text{Collect like terms} \\ 2y=180-116 \\ 2y=64 \\ \text{Divide both sides by 2} \\ y=(64)/(2) \\ \\ \therefore y=32\degree \end{gathered}`

- The value of y is 32°

Final Answer

The values of x and y are:

`\begin{gathered} x=28\degree \\ y=32\degree \end{gathered}`