Arithmatic english 5 docx

■ A pair of angles is supplementary if the sum of the measures of the angles is 180 degrees.. Vertical angles are the nonadjacent angles formed, or the opposite angles.. If the two lines

 A n g l e s

Two rays with a common endpoint, called a vertex, form an angle The following figures show the different

types of angles:

The measure is between 0 and 90 degrees The measure is equal to 90 degrees

The measure is between 90 and 180 degrees The measure is equal to 180 degrees

Here are a few tips to use when determining the measure of the angles

■ A pair of angles is complementary if the sum of the measures of the angles is 90 degrees.

■ A pair of angles is supplementary if the sum of the measures of the angles is 180 degrees.

■ If two angles have the same measure, then they are congruent.

■ If an angle is bisected, it is divided into two congruent angles.

Lines and Angles

When two lines intersect, four angles are formed

1 2 3 4

vertex

Vertical angles are the nonadjacent angles formed, or the opposite angles These angles have the same

measure For example, m ∠ 1 = m ∠ 3 and m ∠ 2 = m ∠ 4

The sum of any two adjacent angles is 180 degrees For example, m ∠ 1  m ∠ 2 = 180 The sum of all

four of the angles formed is 360 degrees

If the two lines intersect and form four right angles, then the lines are perpendicular If line m is per-pendicular to line n, it is written m  n If the two lines are in the same plane and will never intersect, then the lines are parallel If line l is parallel to line p, it is written l || p.

Parallel Lines and Angles

Some special angle patterns appear when two parallel lines are cut by another nonparallel line, or a transversal When this happens, two different-sized angles are created: four angles of one size, and four of another size

■ Corresponding angles These are angle pairs 1 and 5, 2 and 6, 3 and 7, and 4 and 8 Within each pair,

the angles are congruent to each other

■ Alternate interior angles These are angle pairs 3 and 6, and 4 and 5 Within the pair, the angles are

congruent to each other

■ Alternate exterior angles These are angle pairs 1 and 8, and 2 and 7 Within the pair, the angles are

congruent to each other

■ As in the case of two intersecting lines, the adjacent angles are supplementary and the vertical angles

have the same measure

 P o l y g o n s

A polygon is a simple closed figure whose sides are line segments The places where the sides meet are called

the vertices of the polygon Polygons are named, or classified, according to the number of sides in the figure.

The number of sides also determines the sum of the number of degrees in the interior angles

3 4

5 6

7 8

t

l m

l  m

t is the transversal

The total number of degrees in the interior angles of a polygon can be determined by drawing the

non-intersecting diagonals in the polygon (the dashed lines in the previous figure) Each region formed is a tri-angle; there are always two fewer triangles than the number of sides Multiply 180 by the number of triangles

to find the total degrees in the interior vertex angles For example, in the pentagon, three triangles are formed Three times 180 equals 540; therefore, the interior vertex angles of a pentagon is made up of 540 degrees The

formula for this procedure is 180 (n – 2), where n is the number of sides in the polygon.

The sum of the measures of the exterior angles of any polygon is 360 degrees.

A regular polygon is a polygon with equal sides and equal angle measure.

Two polygons are congruent if their corresponding sides and angles are equal (same shape and same

size)

Two polygons are similar if their corresponding angles are equal and their corresponding sides are in

proportion (same shape, but different size)

 Tr i a n g l e s

Triangles can be classified according to their sides and the measure of their angles

All sides are congruent Two sides are congruent All sides have a different measure All angles are congruent Base angles are congruent All angles have a different measure

This is a regular polygon.

60°

3-SIDED

TRIANGLE

4-SIDED QUADRILATERAL

5-SIDED PENTAGON

6-SIDED HEXAGON

Acute Right Obtuse

The measure of each It contains one It contains one angle that is greater than 90 angle is less than 90 90-degree angle degrees

degrees

Triangle Inequality

The sum of the two smaller sides of any triangle must be larger than the third side For example, if the meas-ures 3, 4, and 7 were given, those lengths would not form a triangle because 3 + 4 = 7, and the sum must be greater than the third side If you know two sides of a triangle and want to find a third, an easy way to han-dle this is to find the sum and difference of the two known sides So, if the two sides were 3 and 7, the

meas-ure of the third side would be between 7 – 3 and 7 + 3 In other words, if x was the third side, x would have

to be between 4 and 10, but not including 4 or 10

Right Triangles

In a right triangle, the two sides that form the right angle are called the legs of the triangle The side oppo-site the right angle is called the hypotenuse and is always the longest side of the triangle.

Pythagorean Theorem

To find the length of a side of a right triangle, the Pythagorean theorem can be used This theorem states that the sum of the squares of the legs of the right triangle equal the square of the hypotenuse It can be expressed

as the equation a2+ b2= c2, where a and b are the legs and c is the hypotenuse This relationship is shown

geometrically in the following diagram

b2

2

c

2

a

b c a

angle greater than 90°

60°

50°

70°

Find the missing side of the right triangle ABC if the m∠ C = 90°, AC = 6, and AB = 9.

Begin by drawing a diagram to match the information given

By drawing a diagram, you can see that the figure is a right triangle, AC is a leg, and AB is the hypotenuse Use the formula a2+ b2= c2 by substituting a = 6 and c = 9.

a2+ b2= c2

62+ b2= 92

36 + b2= 81

36 – 36 + b2= 81 – 36

b2= 45

b =45 which is approximately 6.7

Special Right Triangles

Some patterns in right triangles often appear on the Quantitative section Knowing these patterns can often save you precious time when solving this type of question

45—45—90 R IGHT T RIANGLES

If the right triangle is isosceles, then the angles’ opposite congruent sides will be equal In a right triangle, this makes two of the angles 45 degrees and the third, of course, 90 degrees In this type of triangle, the measure

of the hypotenuse is always 2 times the length of a side For example, if the measure of one of the legs is

5, then the measure of the hypotenuse is 52

45°

°

A

B C

6

9

b

30—60—90 R IGHT T RIANGLES

In this type of right triangle, a different pattern occurs Begin with the smallest side of the triangle, which is the side opposite the 30-degree angle The smallest side multiplied by 3 is equal to the side opposite the 60-degree angle The smallest side doubled is equal to the longest side, which is the hypotenuse For exam-ple, if the measure of the hypotenuse is 8, then the measure of the smaller leg is 4 and the larger leg is 43

Pythagorean Triples

Another pattern that will help with right-triangle questions is Pythagorean triples These are sets of whole numbers that always satisfy the Pythagorean theorem Here are some examples those numbers:

3—4—5

5—12—13

8—15—17

7—24—25

Multiples of these numbers will also work For example, since 32+ 42= 52, then each number doubled (6—8—10) or each number tripled (9—12—15) also forms Pythagorean triples

 Q u a d r i l a t e r a l s

A quadrilateral is a four-sided polygon You should be familiar with a few special quadrilaterals

Parallelogram

This is a quadrilateral where both pairs of opposite sides are parallel In addition, the opposite sides are equal, the opposite angles are equal, and the diagonals bisect each other

30°

60°

8

4√¯¯¯3

4