Question

point A is interior to angle GLD. given that angle GLA = n degrees, angle GLD= (5n + 12) degrees and angle DLA = (n^2). determine the measure of GLA

Answer 1

In order to determine the measure of GLA in triangle GLD, use the angle relationships. Substituting the given values into an equation will help find the value of n.

Step-by-step explanation:

In order to ascertain the measure of angle GLA within triangle GLD, the inherent angle relationships are harnessed.

The angle GLA is denoted as n degrees, angle GLD is expressed as 5n + 12 degrees, and angle DLA is represented by n^2 degrees.

Employing the fact that angle GLD is an interior angle to angle GLA, the relationship angle GLD + angle DLA = angle GLA is applied.

Substituting the provided values into this equation yields (5n + 12) + (n^2) = n.

Solving this quadratic equation reveals the value of n, subsequently allowing its substitution to determine the precise measure of GLA.

This analytical approach systematically utilizes angle relationships within the context of triangle GLD, providing a structured method for deducing the angle measures based on the given parameters.

Answer 2

The measure of angle GLA is 36 degrees, determined by solving the equation
`\(n^2 + 6n - 168 = 0\)` based on the given angle relationships in the triangle. The valid solution is n = 12, ensuring a positive angle measure.

Step-by-step explanation:

Let's denote the measure of angle GLA as n degrees. According to the given information, angle GLD is 5n + 12 degrees, and angle DLA is
`(n^2\)` degrees. The sum of the angles in a triangle is always 180 degrees. Therefore, we can set up an equation:

GLA + GLD + DLA = 180

Substitute the given expressions for GLD and DLA:

`\[ n + (5n + 12) + n^2 = 180 \]`

Combine like terms:

`\[ n^2 + 6n + 12 = 180 \]`

Subtract \(180\) from both sides:

`\[ n^2 + 6n - 168 = 0 \]`

Factor the quadratic equation:

`\[ (n - 12)(n + 14) = 0 \]`

This yields two potential solutions: n = 12or n = -14. However, since angles cannot have negative measures, we discard n = -14. Therefore, the measure of angle GLA is n = 12 degrees.