Question

how many diagonals can be drawn from each vertex of a 16 - gon?
14
13
9
18
none of these answers are correct.
question 13
1 pts
in the convex quadrilateral ( a b c d ), if ( angle a cong angle d ), and ( m angle b = 112^{circ} ), ( m angle c = 80^{circ} ), find ( m angle a ).
degrees

Explanation:

Step1: Recall the formula for the number of diagonals from a single vertex

The formula for the number of diagonals that can be drawn from a single vertex of an \(n -\)gon is \(n - 3\). This is because a vertex cannot form a diagonal with itself (\(1\) vertex) and the two adjacent vertices (\(2\) vertices). So in total, \(1+2 = 3\) vertices are excluded from forming diagonals with the given vertex.

Step2: Substitute \(n = 16\) into the formula

Given \(n=16\), we substitute into the formula \(n - 3\). So we have \(16-3\).

$$16 - 3=13$$

for the second question (quadrilateral angle sum):

Step1: Recall the sum of interior angles of a quadrilateral

The sum of the interior angles of a quadrilateral is \((4 - 2)\times180^{\circ}=360^{\circ}\).

Step2: Set up an equation using the given angles

Let \(m\angle A=m\angle D = x\). We know that \(m\angle B = 112^{\circ}\) and \(m\angle C=80^{\circ}\). Then the equation for the sum of angles is \(x + x+112^{\circ}+80^{\circ}=360^{\circ}\).

Step3: Simplify the equation

Combine like - terms: \(2x+192^{\circ}=360^{\circ}\).
Subtract \(192^{\circ}\) from both sides: \(2x=360^{\circ}- 192^{\circ}\).

$$2x = 168^{\circ}$$

Step4: Solve for \(x\)

Divide both sides by \(2\): \(x=\frac{168^{\circ}}{2}\).

$$x = 84^{\circ}$$

Answer:

13