Confidential 2

1.Calculate 25% on 3500. 875

2. Convert 4.78 to percent 47.8%

3. Write 7/8 as a percent 87.5%

4. Write 90% as a fraction 9/10

5. If 2inches = 5 feet,4 inches = ___ feet 10

Warm up

Confidential 3

1. Ratio of two numbers X & Y(< >0) is X/Y (proper or improper fraction) written as X:Y.

A ratio should be expressed in its lowest form

2. In the ratio X : Y, the term X is called the antecedent and the term Y is called the consequent

3. All equivalent ratios can be expressed in one common ratio in its lowest form

4. Ratios are said to be in proportion when they are equivalent

Ratios

Lets review what we have learned in the last lesson

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PercentWhat is a percent?

• The word percent comes from the Latin phrase per centum, which means per hundred.

• A percent is a ratio whose second term is 100.• Percent means parts per hundred. • we use the symbol % for percent. • Example: 60 %, 75 %, 90 %

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Converting fractions to percent For converting fractions to percent follow the

following steps:

• Divide the numerator of the fraction by its denominator

• Multiply the result by 100 (for that move the decimal point two places to the right)

• Put the % sign at the end of the result

Fractions and PercentFractions and Percent

Confidential 6

The decimal point separates the whole units on the

left from the decimal units on the right. • The numbers on the left side represent whole number

s such as ones, tens, hundreds, and thousands.

• The numbers on the right side of the point are the deci mal values. Those numbers represent values called

tenths, hundredths, thousandths, and so on.

What is a Decimal?

Decimal and Percent

Confidential 7

A rate is a ratio that is used to compare different kinds of quantities.

A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

A unit price is a rate comparing the price of an item to its unit of measure.

To find the unit price divide the total cost by the number of items bought.

Ratio

Confidential 8

• A proportion is an equation that states that two ratios are equal .

• We can write proportions in two ways.

or 2:3 = 6:9 In a proportion, the product of the means

equals to the product of the extremes.

2 63 9=

Scale and Proportion drawing

Confidential 9

• What are parallel lines?

• Parallel lines are two lines in a plane which do not meet even when produced indefinitely in either direction.

• Examples of parallel lines:

• opposite edges of a scale, railway lines, etc.

Lets Get Started

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• What are perpendicular lines?

• Perpendicular lines are lines that form right angles (90°) where they intersect.

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• Let us draw two parallel lines l and m and a transversal t which intersecting them at the points L and M respectively. Then eight angles are formed.

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Angles made by a transversal in parallel lines

• Interior angles• Exterior angles• Corresponding angles• Alternate Exterior angles• Alternate Interior angles

Note: Transversal is a line which intersects two or more lines in a plane at different points

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Interior Angles• The angles whose arms include

the line segment LM are called interior angles.

• Angle 3, angle 4, angle 5 and angle 6 are “Interior angles”.

• When a transversal cuts two parallel lines pairs, the sum of the interior angles on the same side of transversal is 180°

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Exterior Angles• The angles whose arms do not include

the line segment LM are called exterior angles.

• Angle 1, angle 2, angle 7 and angle 8 are “Exterior angles”.

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Corresponding Angles

Possible pairs of corresponding angles:

• Angle 1 and angle 5• Angle 2 and angle 6• Angle 4 and angle 8• Angle 3 and angle 7

When a transversal cuts two parallel lines pairs of corresponding angles are equal.

Confidential 16

Alternate Interior AnglesPossible pairs of alternate interior

angles:

• Angle 3 and angle 5

• Angle 4 and angle 6

When a transversal cuts two parallel lines pairs of alternate interior angles are equal

Confidential 17

Alternate Exterior AnglesPossible pairs of alternate exterior

angles:

• Angle 2 and angle 8

• Angle 1 and angle 7

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Pairs of alternate interior angles are equal Pairs of corresponding angles are equal The sum of the interior angles on the same

side of transversal is 180°

RuleIf a transversal cuts two lines such that any one of the following conditions hold:

Then the two lines are parallel.

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Example 1• In the given figure, l || m, find the values of x and y.

• When a transversal cuts two parallel lines pairs of alternate interior angles are equal.

So x = 40° ( x and 40° are alternate interior angles) y + 40° = 180° ( t is a straight line) y = 180° - 40° (subtract 40 on both the sides) y = 140°

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Example 2• In the given figure, l || m and p || q, find the values of a, b, c and

d.

c = 80° ( Vertically opposite angles) c = d (l || m so Corresponding angles are equal) d = 80° ( c =d and c = 80° proved above) b = c (p || q so Corresponding angles are equal) b = 80° ( b =c and c = 80° proved above) similarly we can prove a = b and then a = 80°

Confidential 21

Your Turn In the given figure 1, AB || CD

and AC || BD

1) Find x Answer= 40°2) Find y Answer= 35° In the given figure 2, l || m3) Find x Answer= 135°4) Find y Answer= 135°5) Find z Answer= 45°6) Find w Answer= 45°

Figure 1

Figure 2

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Your TurnIn the given figure, l || m

7) Name a pair of corresponding angles.

Answer= angle 1 and 58) Name a pair of alternate interior

angles. Answer= angle 3 and 5 9) What is the sum of angles 4 and 5? Answer= 180° 10) If angle 4 = 100° , find angle 5. Answer= 80°

Figure 1

Confidential 23

Confidential 24

Let us play a game

Click here to play

Confidential 25

1. In the given figure, AB || CD and CD || EF.

Is AB || EF? Give reasons.

AB || CD and PQ is a transversalSo angle 1 = angle 2 (corresponding

angles are equal) CD || EF and PQ is a transversal So angle 2 = angle 3 (corresponding

angles are equal)It follows that angle 1 = angle 3

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Now AB and EF are two lines, PQ is a transversal and a pair of corresponding angle (angle 1 and angle 3) is equal so lines are parallel.

That is AB || EF.

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2. In the given figure, PQ || RS. Find the value of x.

PQ || RS and BC is a transversalSo angle PBC = 130° (corresponding angles are equal) angle PBC = angle PBA + 55 ° So angle PBA + 55 ° = 130° which gives angle PBA = 75 °(by subtracting 55 on both

the sides)

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Now PQ || RS and AB is a transversal

So angle PBA = x (corresponding angles are equal)

But angle PBA = 75 °(Proved before)

We can say that x =75 °

Confidential 29

3. In the given figure, a transversal t cuts two lines l and m such that angle 2 = 65° and angle 8 = 65°. Are the lines parallel?

Angle 2 = 65° (Given)But Angle 2 = angle 4 (vertically opposite angles)Therefore angle 4 = 65° Also, Angle 8 = 65°(Given) Therefore Angle 4 = angle 8 But these are corresponding angles are equal. Therefore l || m.

Confidential 30

Parallel lines are two lines in a plane which do not meet even when produced indefinitely.

When a transversal cuts two parallel lines pairs, the sum of the interior angles on the same side of transversal is 180°.

When a transversal cuts two parallel lines pairs of corresponding angles are equal.

When a transversal cuts two parallel lines pairs of alternate interior angles are equal.

Confidential 31

Review Two lines are parallel if any one of the

following conditions hold:

• Pairs of alternate interior angles are equal.

• Pairs of corresponding angles are equal.• The sum of the interior angles on the

same side of transversal is 180°

Confidential 32

Great Lesson Today!

Be sure to practice well