Polygons and Angles - Geometry in Two Dimensions (Pre-Algebra)

a The sum of the interior angles measures of a polygon with

n

vertices is given by the following formula.

(n - 2) 18 0^{\circ}

As Izabella pointed out, the home plate is a pentagon. This means that

n = 5 .

(n - 2) 18 0^{\circ}

Substitute

$n = 5$

(5 - 2) 18 0^{\circ}

SubTerm

Subtract term

(3) 18 0^{\circ}

Multiply

54 0^{\circ}

The sum of the interior angles measures of the home plate is

54 0^{\circ} .

Write an equation using this information.

m ∠ A + m ∠ B + m ∠ C + m ∠ D + m ∠ E = 54 0^{\circ}

Take a look at the given graph to find some information about the angles.

Angles

A

and

B

are denoted with square angle markers. This means that both are right angles, so they measure

9 0^{\circ}

each. Angles

C

and

E

have a measure of

13 5^{\circ}

each. Substitute all these values into the equation and solve it for

m ∠ D .

m ∠ A + m ∠ B + m ∠ C + m ∠ D + m ∠ E = 54 0^{\circ}

SubstituteValues

Substitute values

9 0^{\circ} + 9 0^{\circ} + 13 5^{\circ} + m ∠ D + 13 5^{\circ} = 54 0^{\circ}

AddTerms

Add terms

m ∠ D + 45 0^{\circ} = 54 0^{\circ}

SubEqn

$LHS - 45 0^{\circ} = RHS - 45 0^{\circ}$

m ∠ D = 9 0^{\circ}

The angle at the lower part of the home plate measures

9 0^{\circ} .

b The baseball field is a polygon with five sides. This means that the sum of the measures of the interior angles is

54 0^{\circ} .

Notice that the angle at the home plate is a right angle, so it measures

9 0^{\circ} .

Write an equation using all the given data. The degree symbol will be omitted for simplicity.

x + 5 + 2 x - 20 + 2 x - 20 + x + 5 + 90 = 540

Next, solve the equation for

x .

x + 5 + 2 x - 20 + 2 x - 20 + x + 5 + 90 = 540

AddSubTerms

Add and subtract terms

6 x + 60 = 540

SubEqn

$LHS - 60 = RHS - 60$

6 x = 480

DivEqn

$LHS / 6 = RHS / 6$

x = 80

Substitute

x = 80

into each of the expressions for the angle measures. Notice that two pairs of angles have the same measure.

Expression	$x = 80$	Measure
$x + 5$	$80 + 5$	$8 5^{\circ}$
$2 x - 20$	$2 (80) - 20$	$14 0^{\circ}$

Finally, write a list with all the interior angle measures of the baseball field. Here, the degree symbols will not be included.

{90, 85, 85, 140, 140}

Interior Angle	Corresponding Exterior Angles	Sum of Measures
$∠ 11$	$∠ 1$ and $∠ 2$	$m ∠ 1 + m ∠ 11 = 18 0^{\circ}$ $m ∠ 2 + m ∠ 11 = 18 0^{\circ}$
$∠ 12$	$∠ 3$ and $∠ 4$	$m ∠ 3 + m ∠ 12 = 18 0^{\circ}$ $m ∠ 4 + m ∠ 12 = 18 0^{\circ}$
$∠ 13$	$∠ 5$ and $∠ 6$	$m ∠ 5 + m ∠ 13 = 18 0^{\circ}$ $m ∠ 6 + m ∠ 13 = 18 0^{\circ}$
$∠ 14$	$∠ 7$ and $∠ 8$	$m ∠ 7 + m ∠ 14 = 18 0^{\circ}$ $m ∠ 8 + m ∠ 14 = 18 0^{\circ}$
$∠ 15$	$∠ 9$ and $∠ 10$	$m ∠ 9 + m ∠ 15 = 18 0^{\circ}$ $m ∠ 10 + m ∠ 15 = 18 0^{\circ}$

Interior Angle

Corresponding Exterior Angles

Sum of Measures

∠ 11

∠ 1

and

∠ 2

m ∠ 1 + m ∠ 11 = 18 0^{\circ}

m ∠ 2 + m ∠ 11 = 18 0^{\circ}

∠ 12

∠ 3

and

∠ 4

m ∠ 3 + m ∠ 12 = 18 0^{\circ}

m ∠ 4 + m ∠ 12 = 18 0^{\circ}

∠ 13

∠ 5

and

∠ 6

m ∠ 5 + m ∠ 13 = 18 0^{\circ}

m ∠ 6 + m ∠ 13 = 18 0^{\circ}

∠ 14

∠ 7

and

∠ 8

m ∠ 7 + m ∠ 14 = 18 0^{\circ}

m ∠ 8 + m ∠ 14 = 18 0^{\circ}

∠ 15

∠ 9

and

∠ 10

m ∠ 9 + m ∠ 15 = 18 0^{\circ}

m ∠ 10 + m ∠ 15 = 18 0^{\circ}

Expression	$x = 10$	Measure
$5 x$	$5 (10)$	$5 0^{\circ}$
$7 x + 5$	$7 (10) + 5$	$7 5^{\circ}$
$5 x + 2$	$5 (10) + 2$	$5 2^{\circ}$
$5 x + 3$	$5 (10) + 3$	$5 3^{\circ}$
$6 x + 1$	$6 (10) + 1$	$6 1^{\circ}$
$7 x - 1$	$7 (10) - 1$	$6 9^{\circ}$

Expression

x = 10

Measure

5 x

5 (10)

5 0^{\circ}

7 x + 5

7 (10) + 5

7 5^{\circ}

5 x + 2

5 (10) + 2

5 2^{\circ}

5 x + 3

5 (10) + 3

5 3^{\circ}

6 x + 1

6 (10) + 1

6 1^{\circ}

7 x - 1

7 (10) - 1

6 9^{\circ}

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Catch-Up and Review

Stop!

Figures With Three or More Angles

Polygon

Vertex of a Polygon

Connecting Non-consecutive Vertices

Diagonal of a Polygon

Classifying Polygons by Shape

Convex Polygon

Concave Polygon

Convex or Concave?

Interior Angles of a Polygon

Polygon Interior Angles Theorem

Play Ball!

Hint

Solution

Regular Polygons

Honeycombs

Hint

Solution

Finding the Missing Angle Measure

Exterior Angles of a Polygon

Sum of Measures of Exterior Angles

Polygon Exterior Angles Theorem

Designing a Logo

Hint

Solution

Finding an Exterior Angle Measure

Final Stop

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