Confidential2 1.Calculate 25% on 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

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

Confidential4 Percent What 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 %

Confidential5 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 Percent

Confidential6 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 numbers such as ones, tens, hundreds, and thousands. The numbers on the right side of the point are the decimal values. Those numbers represent values called tenths, hundredths, thousandths, and so on. What is a Decimal? Decimal and Percent

Confidential7  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

Confidential8 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 = Scale and Proportion drawing

Confidential9 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

Confidential10 What are perpendicular lines? Perpendicular lines are lines that form right angles (90°) where they intersect.

Confidential11 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.

Confidential12 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

Confidential13 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°

Confidential14 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”.

Confidential15 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.

Confidential16 Alternate Interior Angles Possible 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

Confidential17 Alternate Exterior Angles Possible pairs of alternate exterior angles: Angle 2 and angle 8 Angle 1 and angle 7

Confidential18 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° Rule If a transversal cuts two lines such that any one of the following conditions hold: Then the two lines are parallel.

Confidential19 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°

Confidential20 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°

Confidential21 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 || m 3)Find x Answer= 135° 4)Find y Answer= 135° 5)Find z Answer= 45° 6)Find w Answer= 45° Figure 1 Figure 2

Confidential22 Your Turn In the given figure, l || m 7)Name a pair of corresponding angles. Answer= angle 1 and 5 8)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

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Confidential24 Let us play a game Click here to play

Confidential25 1. In the given figure, AB || CD and CD || EF. Is AB || EF? Give reasons. AB || CD and PQ is a transversal So 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

Confidential26 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.

Confidential27 2. In the given figure, PQ || RS. Find the value of x. PQ || RS and BC is a transversal So 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)

Confidential28 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 °

Confidential29 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.

Confidential30  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.

Confidential31 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°

Confidential32 Great Lesson Today! Be sure to practice well