Parallel Lines Cut by a Transversal

The only angles that get added together and set equal to 180 are the following:  adjacent angles and consecutive interior/exterior angles. All other angle types are congruent, so set them equal to each

other. 

Angle pairs formed by parallel lines cut by a transversal

When two parallel lines are given in a figure, there are two main areas: the interior and the exterior. 

When two parallel lines are cut by a third line, the third line is called the transversal. In the example below, eight angles are formed when parallel lines m and n are cut by a transversal line, t. 

Vertical pairs:        1 and  4                             2 and  3                             5 and  8                             6 and  7 

Alternate   interior angles  two angles in the interior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate interior angles are non-adjacent and congruent.

Alternate exterior angles two angles in the exterior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate exterior angles are non-adjacent and congruent. 

Corresponding angles two angles, one in the interior and one in the exterior, that are on the same side of the transversal. Corresponding angles are non-adjacent and congruent. 

The pairs of angles on one side of the transversal but inside the two lines are called Consecutive Interior

Angles. These angles are also called same side interior angles. Consecutive Interior Angles must add up to 180

degrees. 

In this example, these are Consecutive Interior Angles:

d and f

c and e

The pairs of angles on one side of the transversal but outside the two lines are called Consecutive Exterior Angles. These angles are also called same side exterior angles. Consecutive Exterior Angles must add up to 180 degrees. 

In the above example, the following are consecutive exterior angles:

a and g b and h

Example

1:

 Since these angles are alternate exterior angles and alternate exterior angles are congruent, set the angles = to each other.

2x + 10 = 86 + x

Subtract 1x from both sides of the = sign and this will give 1x + 10 = 86

Subtract 10 from both sides of the = sign and this will give 1x = 76

Example 2:   

Since these angles are same side interior and same side interior angles have to add up to 180 degrees, add the angles together and set = to 180.

3x – 10 + 5x + 30 = 180

Combine like terms on the left side of the = sign: 3x + 5x = 8x and – 10 + 30 = 20 ; this gives:

8x + 20 = 180

Subtract 20 from both sides of the equal sign and this gives: 8x = 160

Divide both sides of the equal sign by 8 and this gives 1x = 20

Supplements and complements

Complementary Angles: Two angles that add up to 90 degrees (usually has a box or little square--makes a corner)

 

Supplementary Angles: Two angles that add up to 180 degrees (forms a straight line)

 http://techiemathteacher.com/2014/11/11/problem-solving-complementary-supplementary-angles-2/

Classifying Triangles based on the Sides:

Equilateral Triangle --- A triangle where all 3 sides have an equal measure. In addition, it is equiangular which means all 3 angles have the same measure of 60 degrees each (3 equal sides and 3 equal angles)

Isosceles Triangle ---A triangle where 2 sides have the same measure and the base angles have the same measure as well (2 equal sides and 2 equal angles)

Exterior Angle Theorem: This theorem states the following:  The sum of the two remote interior angles = exterior angle. Remember the definitions of the following terms:

sum means add

remote means the two inside angles that are far away from the exterior angle

interior means inside

exterior means outside

Students should combine like terms whenever possible, and then solve for x. Students may or may not have to plug in x to find the measure of a certain angle.

Example A:

If the measure of the exterior angle is (3x - 10) degrees, and the measure of the two remote interior angles are 25 degrees and (x + 15) degrees, find x.

exterior angle = interior angle + other interior angle (3x - 10) = (25) + (x + 15)     Combine the like terms of 25 and 15 and you get 403x - 10 = x + 40                     Add 10 to both sides of the equal sign3x = x + 50                            Subtract x from both sides of the equal sign2x = 50                                  Divide both sides by 2x = 25

Remember that "x" is not the answer here. We need the angles themselves, which are calculated as:

(3x-10) so plug in 25 for x:  3(25) – 10= 65 degrees

25 (This angle stays 25 degrees)

(x+15) so plug in 25 for x:  25 + 15 = 40 degrees.

Triangle Sum Theorem. Sum means add so this means the three interior (inside) angles should add up to 180.

EXAMPLE: What is the m∠T?

Since Angle T is unknown, label it as x

X + 27 + 82 = 180  (Combine like terms)

X + 109 = 180  (Subtract 109 from both sides)

X= 71 degrees for the measure of angle T

Triangle Midsegment theorem: https://www.shmoop.com/similar-triangles/triangle-midsegment-theorem-examples.html

 

 Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionality (the converse is also true)

See the following website for examples:

https://www.shmoop.com/similar-triangles/triangle-proportionality-theorem.html

Parallel Lines and Transversals Proportionality Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Parallel Lines and Proportionality

In the Triangle Proportionality Theorem , we have seen that parallel lines cut the sides of a triangle into proportional parts. Similarly, three or more parallel lines also separate transversals into proportional parts.

If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

In the figure shown, if AD←→∥BE←→∥CF←→

, then ABBC=DEEF,ACDF=BCEF , and ACBC=DFEF

.

  Example:

Find the value of x

.

If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

So write a proportion .

 

 

Triangle Angle Bisector Theorem:

 

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

 See the website below for examples:

 

http://www.mathwarehouse.com/geometry/similar/triangles/angle-bisector-theorem.php

Methods for Proving (Showing) Triangles to be Congruent

SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.(For this method, the sum of the lengths of any two sides must be greater than the length of the third side, to guarantee a triangle exists.)

SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The included angle is the angle formed by the sides being used.)

ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.  (The included side is the side between the angles being used.  It is the side where the rays of the angles would overlap.)

AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.  (The non-included side can be either of the two sides that are not between the two angles being used.)

HLRight

TrianglesOnly

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.  (Either leg of the right triangle may be used as long as the corresponding legs are used.)

 

BE CAREFUL!!!Only the combinations listed above will give

congruent triangles.

So, why do other combinations not work?

Methods that DO NOT Prove Triangles to be Congruent

AAA AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to also show they are the same size, thus congruent!

Consider the example at the right.

You can easily draw 2 equilateral triangles that are the same shape but are not congruent (the same

size).

SSAor

ASS

SSA (or ASS) is humorously referred to as the "Donkey Theorem".

This is NOT a universal method to prove triangles congruent because it cannot guarantee that one unique triangle will be drawn!!

Properties of a parallelogram:

If a quadrilateral is a parallelogram, then its opposite sides are parallel. Parallel means the two sides never intersect.

If a quadrilateral is a parallelogram, then its opposite sides are congruent. So students should set the two sides equal to each other.

If a quadrilateral is a parallelogram, then its opposite angles are congruent. So students should set the two angles equal to each other. Opposite angles live on different streets (different sides).

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. So students should add the two angles together and set equal to 180 degrees. Consecutive angles live on the same street.

If a quadrilateral is a parallelogram, then its diagonals bisect each other. Diagonals are inside the figure and connect opposite vertices (opposite angles). Bisect means to cut in half. Students should know that half the diagonal plus half the diagonal = whole diagonal. Each half will have the same measure.

A rhombus has the following properties:

Four congruent sides

Diagonals are perpendicular. Perpendicular means the lines intersect at 4 right angles.

Diagonal bisects a pair of opposite angles

A rectangle has the following properties:

Four congruent angles which are all right angles.

Diagonals are congruent

A square has the following properties:

Four congruent sides

Diagonals are perpendicular. Perpendicular means the lines intersect at 4 right angles.

Diagonal bisects a pair of opposite angles

Four congruent angles which are all right angles.

Diagonals are congruent

A rhombus, rectangle, and square also have all the properties of a parallelogram:

Reflections

There are two methods to reflect:

1) Point Method:  Use the rule to reflect the points of the pre-image, then plot the points of your image. The rules are as follows:

Reflecting Over the X-Axis:   (x, y) reflects to (x, - y). This means that x stays the same  but change the sign of y

Reflecting Over the Y-Axis:   (x, y) reflects to ( - x, y). This means that the sign of x changes but y stays the same.

Reflecting over the line y = x:   (x, y) reflects to (y, x). This means switch x and y.

Reflecting over the line y = -x:   (x, y) reflects to ( - y, - x ). This means to switch x and y AND   change the signs of x and y.

2)  The point method cannot be used when reflecting over y = # or x = #.  Instead, the following method must be applied:

 Draw the reflection line as a dotted line:

If the reflection line is y = #, then draw a horizontal dotted line at that y value. 

If the reflection line is x = #, then draw a vertical dotted line at that x value.

Count how far the point is from the dotted reflection line.

Draw the point of the new image on the OPPOSITE   side of the line. 

Method # 2 does not work for y =x or y = - x

Translation:  A transformation that "slides" a shape to another location ( will slide up, down, left, right and/or combination)

A translation to the x-values will move the graph:

(x - #) will move the graph left

(x + #) will move the graph right

A translation to the y-values will move the graph:

(y - #) will move the graph down 

(y + #) will move the graph up

Rotation Basics

Rotate 90 degrees clockwise about the origin (same as 270 degrees counterclockwise)   (x, y) -----> ( y, - x ) ; which means change the sign of x and switch places with the letters

Rotate 90 degrees counterclockwise about the origin (same as 270 degrees clockwise)

  (x, y) ----->  ( - y, x );  which means change the sign of y and switch places with the letters

Rotate 180 degrees about the origin: (x, y) -----> ( - x , - y); which means change the sign of both x and y

Rotational Symmetry. If a figure can be rotated (turned) around a center point by fewer than 360 degrees and the figure appears unchanged, then the figure has rotational symmetry. The point around which the figure is rotated is called the center of rotation, and the smallest angle

needed to turn is called the angle of rotation. The angle of rotation is given by 360n

where n = number of vertices.

Example 1: So, a triangle has 3 vertices: 360

3 = 120 degrees

Example 2: So, a regular hexagon has 6 vertices: : 360

6 = 60 degrees

A dilation is an enlargement or reduction of a figure. The scale factor indicates how much the figure will enlarge or reduce. The variable for the scale factor is k. When k > 1, the dilation is an enlargement. When k < 1, the dilation is a reduction. So, both x and y values in the ordered pair

should be multiplied by the scale factor to get the points of the new image. Then label the new points with an apostrophe after the corresponding letter.

When deciding the scale factor given a graph use the formula: New image

pre−image(original)

All you need is one ordered pair from both the pre-image (original) and the image (new figure). It must be the same letter from both images. Then choose either x or y from both figures (must choose x for both or y for both). Using the calculator, reduce the fraction. This will be the scale factor.

30º-60º-90º Triangle Pattern Formulas

        

Labeling:

H = hypotenuseLL = long leg (across from 60º)SL = short leg (across from 30º)

 

Short Cut Pattern Formulas:

short leg:

SL = Hypotenuse

2You must remember that these formula patterns can be used

ONLY in a 30º-60º-90º triangle.

long leg:LL= SL * √3Hypotenuse:

H = SL * 2

 

 

EASY:

Find x and y.

x is the short leg

SL = Hypotenuse

2

SL = 142

SL = 7  Answer

y is the long leg

LL= SL * √3LL = 7 * √3

LL = 7√3  Answer

 HARDER:

Find x and y.

6 is the short leg andx  is the hypotenuse

(start with what you have given)

Hypot=Short Leg∗2

Hypotenuse = 6 * 2

  Answer

y is the long leg

LL= Short leg * √3LL= 6 * √3

  Answer

 

HARDER:(requires more algebraic manipulation)

Find x and y.

8 is the long leg and x  is the hypotenuse

(start with what you have given)

Hyp = Short leg * 2

8√33

* 2 = 16√3

3

Hypotenuse = 9.2

y  is the short leg

Short leg = Longleg

√3

SL = 8√3

= 8√3

3

  Answer

30-60-90 Triangles:

The short leg of a 30-60-90 will always be opposite the 30 degree angle.

 The long leg of a 30-60-90 will always be opposite the 60 degree angle.

 The hypotenuse of a 30-60-90 will always be opposite the 90 degree angle.

 When given the short leg of a 30-60-90:

hypotenuse = short leg * 2 long leg = short leg * √3

When given the hypotenuse of a 30-60-90:

short leg = Hypotenuse

2 long leg = short leg * √3

When given the long leg of a 30-60-90:

short leg = Long Leg

√3 hypotenuse = short leg * 2

45º-45º-90º (Isosceles Right Triangle)Pattern Formulas

(you do not need to memorize these formulas as such, but you do need to memorize the patterns)

H = hypotenuseL = leg

Leg = Hypotenuse

√2

Trig Ratio: A ratio of the lengths of two sides in a right triangle.

Things to Remember:

Hypotenuse: across from the right angle and is always the longest side

Opposite:  the side farthest from or across from the angle

Adjacent:  the side next to (touching) the angle

Students must choose which trig ratio to use based on the information given. If x is on top of the fraction, then multiply. If x is on the bottom of the fraction, then divide (the trig ratio will go on the bottom or in the calculator last).

Students must choose which trig ratio to use based on the information given. See the below example:

List the values of sin(α), cos(α),sin(β), and tan(β) for the triangle below, accurate to three decimal places:

For either angle, the hypotenuse has length 9.7 since it is opposite the right angle and the longest side.

For the angle α:

"opposite" is 6.5 and "adjacent" is 7.2

So, the sine of α will be:  opposite

hypotenuse=¿ 

6.59.7 = 0.6701030928… = 0.670

So, the cosine of α will be: adjacent

hypotenuse=¿ 

7.29.7 = 0.7422680412.... = 0.742

For the angle β;

"opposite" is 7.2 and "adjacent" is 6.5

So the sine of β will be: :  opposite

hypotenuse=7.2

9.7  = 0.7422680412... = 0.7423

So, the tangent of β will be oppositeadjacent  =

7.26.5= 1.107692308....  = 1.1077

Students must know that the sin of one angle is the same as the cos of its complement angle. The  tan of one angle is the reciprocal of the tan of its complement angle (so just flip the fraction).

Inverse of trig functions:

When finding the missing angle, the x will always be on the left of the equal sign. Students will then press twice the appropriate trig function button on the calculator and then enter the fraction. Students must remember to put degrees next to the number since it is an angle. 

If the hypotenuse is blank, then the trig ratio  tan inverse must be used. 

If the hypotenuse is given, then either sin inverse or cos inverse will be used. If the side given is across from (opposite) the angle, then sin inverse should be used. If the side given is next to (adjacent) the angle, then cos inverse should be used. 

Example: Find the measure of angle a

We know:

The distance down is 18.88 m

The cable’s length is 30 m

And we want to know the angle "a"

 

sin a° = opposite

hypotenuse = 18.88

30

Use the calculator to find the inverse of sin (hit the sin button twice on the TI-36 XPRO calculator to find the inverse of sin)

sin−1 ( 18.88

30 ) = angle

Angle is 39 degrees

If finding a missing side:

Students must choose which trig ratio to use based on the information given. If x is on top of the fraction, then multiply. If x is on the bottom of the fraction, then divide (the trig ratio will go on the bottom or in the calculator last).

Students must know that the sin of one angle is the same as the cos of its complement angle. The  tan of one angle is the reciprocal of the tan of its complement angle (so just flip the fraction).

Sine= opposite

hyptoenuse If the hypotenuse is a letter or number, then either sin or cos will

be used. If the side given is across from (opposite) the angle, then sin should be used

Use sine, cosine or tangent to find the value of side x in the triangle below.

Step 1Based on your givens and unknowns, determine which soh cah toa ratio to use

Since we know the 53 angle, the hypotenuse, and we want to find length of the opposite side, we should use sine

Step 2Set up an equation based on the ratio you chose in the step 1

Sin (53)= oppositehypotenuse

Sin (53)= x15

Step 3Cross multiply and solve the equation for the side length. (round to the nearest hundredth)

X = 15 * sin (53)x≈11.98

Cosine= adjacent

hypotenuse If the hypotenuse is a letter or number, then either sin or cos will be

used. If the side given is next to (adjacent) the angle, then cos should be used. 

Use sine, cosine or tangent to find the length of side k in the triangle below.

Step 1Based on your givens and unknowns, determine which soh cah toa ratio to use

Since we know the 53 angle, the adjacent side, and we want to find length of the hypotenuse, we should use cosine

Step 2

http://www.mathw

Set up an equation based on the ratio you chose in the step 1

Cos (53)= adjacenthypotenuse

Cos (53)= 45x

Step 3Cross multiply and solve the equation for the side length. (round to the nearest hundredth)

X = 45 / cos (53)x≈74.8

Tangent= oppositeadjacent

If the hypotenuse is blank, then the trig ratio tan must be used. 

Directions: Use soh cah toa to find the given side length.

Step 1Based on your givens and unknowns, determine which soh cah toa ratio to use

Since we know the 67 angle, its adjacent side length and we want to know length of the opposite side, we should use tangent

Step 2Set up an equation based on the ratio you chose in the step 1

Tan (67)= oppositeadjacent

Tan (67)= x14

Step 3Cross multiply and solve the equation for the side length. (Let's round to the nearest hundredth)

X = tan (67) * 14

x≈32.98

Angle of elevation/depression which is an angle created by a horizontal surface and the hypotenuse. Angle of elevation is when looking up and angle of depression is when looking down. 

Angles of Elevation and DepressionThe angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (the line of sight).

If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.

Example 1

From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34º.  How far is the object from the base of the cliff?

Solution:

Let x m be the distance of the object from the base of the cliff.

Tan (34) = oppositeadjacent

Tan (34) = 40x

Since x is on the bottom, then divide (what is on top, stays on top)

X = 40tan 34

X = 59.30 m

So, the object is 59.30 m from the base of the cliff.

Example 2:

The height of a building is 250 ft. What is the angle of elevation from a point on the level ground 200 ft away from the base of the building?

Solution: The lengths of opposite and the adjacent legs are known for angle θ which is the angle of elevation.

θ =  tan-1  oppositeadjacent =

tan-1 (250200  )  ≈ 51.3º

 

Example 3:  A girl is flying her kite with 42° angle of elevation. If she knows the length of the string of her kite which is 300 m, how high is the kite? The triangle formed is shown on the left.

Sin (42) = x

300 Since x is on top, then multiply.

X = 300 * sin (42) = 200.74 m

Complementary means the angles add up to 90 degrees. So, if you have the measure of one angle, then to find the complement of that angle just subtract from 90.