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Confidential3 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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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
Confidential 4
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 %
Confidential 5
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
Confidential 10
• What are perpendicular lines?
• Perpendicular lines are lines that form right angles (90°) where they intersect.
Confidential 11
• 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.
Confidential 12
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
Confidential 13
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°
Confidential 14
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”.
Confidential 15
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
Confidential 18
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.
Confidential 19
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°
Confidential 20
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
Confidential 22
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
Confidential 26
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.
Confidential 27
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)
Confidential 28
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