PERIMETERSA perimeter is the measure of the distance AROUND an object.

l1l2

l3

ww

l

l

Perimeter of a Scalene Triangle= l1 + l2 + l3

Perimeter of a Rectangle= w + l + w + l= 2w + 2l

S S

S

S

Perimeter of a Square= S +S+ S + S= 4S

Triangles

Scalene Triangle Isosceles Triangle(2 sides and 2 angles are equal)

A B

C

s s

tS=AC=BC

Perimeter of an Isoceles Triangle = s + s + t= 2s + t

A = B

A B

C

s = AC = BC =AB

Perimeter of an Equilateral Triangle = s + s + s= 3s

A = B = C

Equilateral Triangle(3 sides and 3 angles are equal)

s s

s

Example:

The perimeter of a rectangle is 26 ft. The length of the rectangle is 1 ft more than twice the width. Find the width and length of the rectangle.

Step 1) What are we trying to find? The width and length of the rectangle.Let w = width, and l = length.

Given info: Perimeter is 26 ft. It is a rectangle, so the formula for a rectangle’s perimeter is P = 2w + 2l.Also, length of the rectangle is 1 ft more than twice the width.

Step 2) Make an equation from given info.

Perimeter = 26 ft = 2w + 2lLength is 1 ft more than twice its width. l = 1 + 2w

We can combine these equations to solve for each variable, w and l.

26 = 2w + 2lSubstitute equation, l = 1 + 2w for l in the above equation.26 = 2w + 2(1 + 2w)

Step 3) Solve equation

Use distributive property to get rid of parentheses.26 = 2w + 2 + 4wCombine like terms

26 = 6w + 2

24 = 6w

4 = w What about l? l = 1 + 2w = 1 + 2(4) = 1+8=9Step 4) Check result. Perimeter with w=4 and l=9 should be 2626 = 2(4) + 2(9)= 8 + 18 = 26 Yes.

Step 5) State conclusion (Remember the measuring units!)The width of the rectangle is 4 ft and the length is 9 ft.

Example 1 The perimeter of an isosceles triangle is 25 ft.The length of the third side is 2 ft less than the length of one of the equal sides. Find the measures of the three sides of the triangle.

Now you try this one:A carpenter is designing a square patio with a perimeter of 52 ft. What is the length of each side?

Angles are formed by two rays, lines, or line segments with a common endpoint. The common endpoint is called the vertex. In the angle below, point A is the vertex. The angle can be denoted as

B

A

C

BAC or CABNote that the vertex, point A, must be in the middle.

Right angle90°

Acute angleLess than 90°

The unit of measurement of an angle is degrees.

Obtuse angleGreater than 90°

Straight angle 180°

vertex

Intersecting lines form angles at their intersection point.Angles that share a common side are called adjacent angles. The angles that are not adjacent are called vertical angles.

Intersection PointA

B

C

D

E

EAB and BAC are adjacent angles.

EAD and BAC are vertical angles.

Vertical angles have equal measures of degrees. We call this being “congruent.”

Two angles are supplementary angles when the sum of their measures is 180°.When two lines intersect, adjacent angles are supplementary because the sides that are not in common form a straight angle.

EAD and DAC are supplementary angles.

90°

A

B

Two angles are complementary angles when the sum of their measures is 90°.When two adjacent angles form a right angle with the sides that are not in common, these angles are complementary because the measures add up to 90°.

C

D BAC and CAD are complementary angles.

BAC and CAD are adjacent angles.

EAD + DAC = 180°

(3x+15)°

(4x-20)°

Example

The vertical angles property allows us to make an equation: 3x+15 = 4x -20 and solve for x.

Parallel lines are lines in the same plane that never intersect (that have the same slope). If two lines, l1 and l2 are parallel, we say l1 || l2.

Transversal – a line that intersects two or more lines on the same plane.

Alternate Interior Angle PropertyA transversal intersects parallel lines at congruent angles.Because of this and also because of the property of supplementary angles and the property of vertical angles, we can rewrite the diagram on the left as this:

A

E

B

DC

F

HG

A

A

180° - A

A180° - A

180° - A

A180° - A

The angles on the inside of the parallel lines that are congruent are called “Alternate Interior Angles.”

C = F C and F are alternate interior angles.

D = ED and E are alternate interior angles.

(3x-80)°

(3x+20)°

Example

These two angles are not equal, but we can still solve for x by using other properties.

Since these two lines are parallel, the transveral intersects them at congruent angles, so the angle adjacent to (3x+20)° is (3x-80)°.

The supplementary angle property says that if two adjacent angles form a straight angle with uncommon sides, they are supplementary, so (3x – 80) + (3x + 20) = 180.

Now that we have an equation, we can solve for x.

l1

l2

Given that l1|| l2, solve for x

(3x-80)°

(3x+20)°

l1

l2

(3x-80)°

The Sum of the Measures of the Interior Angles of a Triangle is 180°

The sum of angles in a triangle is always 180 degrees. A rigorous proof takes some work, but the statement can be made plausible by the following argument. In the triangle ABC, draw a line through point C that is parallel to AB. This creates two additional angles, A' and B'. The three angles (A',C,B') add up to 180 degrees, because they are adjacent to each other and backed off against a straight line. However, by the properties of parallel lines, Angle A = Angle A' and Angle B = Angle B'.

Therefore (A,C,B) also add up to 180 degrees.

l1

l2

l1 || l2

Example 5:Given that Angle a = 45°and Angle x = 100°, find the measure of angles b,c, and y.

a

b

x

cy

l

k

Angle b =

Angle c =

Angle y =

Try this one:

Given that Angle y = 55 °, and that lines m and k are perpendicular.find the measures of angles a, b, and d.

a

bd

y

m

k

Angle a =

Angle b =

Angle d =

Markup and Discount

$ Retail Price = $Cost + $ Markup

Markup = Markup Rate * $Cost

$ Sale Price= $ Original Price – $ Discount

$Discount = Discount Rate * $Original Price

Example

The manager of a clothing store buys a suit for $180 and sells that suit for $252. Find the markup rate.

$Markup = Markup Rate * $CostSoMarkup Rate = $Markup/ $Cost

We know the cost of the suit, that is $180.

What is the Markup?We can use the other formula, $Retail Price = $Cost + $Markupto find Markup.

$Markup = $Retail Price - $Cost = $252 - $180 = $72

Markup Rate = $Markup/ $Cost = $72/$180 = 0.4Converting 0.4 to percent, we get 40%.Conclusion: Markup Rate is 40%

You try this: The cost to a sporting goods store of a tennis racket is $120. The selling price of the racket is $180. What is the markup rate?

Amount the store paid for the item

Amount the store raises the price to make a profit

Selling price for an item

Markup rate must be converted from percent to decimal before multiplying .

Original Price is the “Regular Price” of the item.

HOMEWORK

EXTRA:p. 121-123 #143-167 EOO (Every Other Odd, e.g. #143, 147, 151, etc…)

p. 191-195 #6-57 ETP (Every Third Problem, e.g. 6, 9, 12, etc..)

p. 202-204 #6-42 ETP