Question

visit bitly/30hs7um or scan the qr - code to the right for an interactive tool to investigate this exploration (adapted from lesson 1.2). work with your teacher/tutor. the sum s of the angle measures of a polygon with n sides can be found using the formula s = 180(n - 2). write and solve an equation to find each value of x. justify the steps in your solution. a) this figure has 3 sides so the sum of angles is given by: s = 180(3 - 2)=180, 30+(30 + x)+9x = 180, 60+10x = 180, 10x = 180 - 60 = 120, x = 12. b) this figure has 4 sides so the sum of angles is given by: s = 180(4 - 2)=360, 50+(2x + 20)+(2x + 30)+x = 360, 100+5x = 360, 5x = 360 - 100 = 260, x = 52. use the angle - measurement tool to measure the angles of each polygon. do the angle measures agree with your results?

Explanation:

Step1: Identify the polygon type and sum - formula

For a triangle ($n = 3$), the sum of interior - angles formula is $S=180(n - 2)=180\times(3 - 2)=180^{\circ}$.

Step2: Set up an equation for figure A

In figure A, the sum of the interior angles is $30+(30 + x)+9x=180$. Combine like - terms: $60 + 10x=180$. Subtract 60 from both sides: $10x=180 - 60=120$. Divide both sides by 10: $x = 12$.

Step3: Identify the polygon type and sum - formula for figure B

For a quadrilateral ($n = 4$), the sum of interior angles is $S=180(n - 2)=180\times(4 - 2)=360^{\circ}$.

Step4: Set up an equation for figure B

In figure B, the sum of the interior angles is $50+(2x + 20)+x+(2x + 30)=360$. Combine like - terms: $50+2x+20+x + 2x+30=360$, which simplifies to $100 + 5x=360$. Subtract 100 from both sides: $5x=360 - 100=260$. Divide both sides by 5: $x = 52$.

Answer:

For figure A, $x = 12$. For figure B, $x = 52$.