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Geometry
Planning a Proof
Theorems about parallelograms
Find the indicated measure.
Second half of unit notes
Unit 3Test Study Guide

Homework help Unit 3.2
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 Planning a Proof
1.  Read the theorem (or given statement) carefully, being certain that you know the meaning of the words used.
If you do not understand every word, refer to your notes or to the definition section of the class website.

2.  If asked to prove a theorem, postulate, corollary, or definition, determine the hypothesis and the conclusion.  
The hypothesis of the theorem is the “if statement” and the conclusion is the “then statement.”  If the word “then” is not written, the statement after the comma is the then statement.  Sometimes “then” is left out because it is a “taken-for- granted” word.

3. Draw a figure which illustrates each point, angle, and line described in the hypothesis.  When working from a textbook, draw the figure even if it is provided for you.  Label the figure and mark the parts of the hypothesis that are given.  If working from a worksheet that provides the illustration, just mark the parts of the hypothesis that are given.

4.  If there is no “given” statement, write the given statement from the information provided by the hypothesis or the illustration.  Be sure to use each relation given in the hypothesis because it will be needed in the proof.  This includes the type of polygon.

5.  If the “prove” statement is not provided, write what there is to prove from the conclusion.

6. Plan the proof of the theorem.
In planning the proof, first search the assumptions, definitions, and the theorems, postulates and corollaries that have been proved in previous assignments or lessons.  Use your notes or your index “flash” cards. Then select the one most suitable to your proof.

7.  Write the proof, giving a reason for each statement used.

You can assume the following:
Straight lines (or straight angles) from an illustration.
Collinearity of points
Betweeness of points
Relative location of points
Two angles form a straight angle from an illustration
You cannot assume:
Right angles
congruent segments
congruent angles
relative size of segments and angles


  EACH NEW STATEMENT MUST LOGICALLY FOLLOW ONE OF THE STATEMENTS ALREADY PROVED.  The reason is the proof of the statement.





 Theorems about parallelograms:
A diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles of a parallelogram are supplementary.
Bitmap ImageGo to: The figure at the left is a parallelogram
The diagonals of a parallelogram bisect each other.
A rhombus is a parallelogram.
A rectangle is a parallelogram.
The diagonals of a rhombus are perpendicular (they measure 90 degrees).
The diagonals of a rectangle are congruent.
The diagonals of a kite are perpendicular.
A square is a rectangle.
A square is a rhombus.
The diagonals of a square are congruent and are the perpendicular bisectors of each other.

Theorems of quadrilaterals that are parallelograms.
If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorems of parallelograms that are rectangles or rhombuses.
If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle (This is known as the "House builder's Theorem".)
If one pair of adjacent sides of a parallelogram are congruent, then the parallelogram is a rhombus.
If The diagonals of a parallelogram bisect the angles of the parallelogram, then the parallelogram is a rhombus.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Click here to see *Theorems about parallelograms
Bitmap Image The figure at the left is a parallelogram.
First determine what you know about parallelograms.
<M @ <P; segment MS is congruent to segment NP; segment SP is congruent to segment MN.  Segment SN is the diagonal of the parallelogram; therefore, triangle MNS is congruent to triangle PNS.  Segment SP is also a transversal to the parallel line segments SP and MN and segments MS and NP.  Angle MSN is congruent to <PNS and <MNS is congruent to
< PSN (alternate interior angles formed by the transversal of parallel lines are congruent.).  

Bitmap Image
 Find the indicated measure.
1.  Given MN = 2t and SP = (t + 5), find MN
Bitmap Image
Statement
Reasons
MN = SP  
Given
2t = (t + 5)
Substitution property of equality
2t - 1t = t + 5 - t
Subtraction property of equality
t = 5
MN = 2t
Given
MN = 2(5)
Substitution
MN = 10

2.  Given m<M = 45 degrees, m<P = 3x degrees, and NP = x, find NP.
Bitmap Image
Statement
Reason
m<M = m<P
Opposite angles of parallelograms are (congruent) therefore, they are equal
45 = 3x
Substitution
45/3 = 3x/3
Division property of equality
15 = x
NP = x
Given
NP = 15
Substitution

3.  Given m<MSP = 5x degrees and m<P = x degrees, find m<MNP.
Bitmap Image
Statement
Reasons
SMNP is a parallelogram
m<MSP = 5x degrees;
m<P = x degrees
Given
m<MSP + m<P = 180 degrees  
Consecutive angles of a parallelogram are supplementary; therefore their measures equal 180 degrees
5x + x = 180
Substitution
6x = 180
Combine like terms
6/6x = 180/6
x = 30
Multiplication (division) property of equalities
m<MSP = 5(30 degrees)
m<MSP = 150 degrees
Substitution
m<MSP = m<MNP
Opposite angles of a parallelogram are congruent; therefore, they are equal.
m< MNP = 150 degrees
Transitive or Substitution Property of equality

4.  Given m<M = (x + 20) degrees and m<MNP = (2x + 10) degrees, find m<M.
Bitmap Image
Statement
Reasons
m<M = (x + 20) degrees and
m<MNP = (2x + 10) degrees
Given
m<M + m<MNP = 180 degrees
Consecutive angles of a parallelogram are supplementary.
(x + 20) + (2x + 10) = 180
3x + 30 = 180
Substitution
3x + 30 - 30 = 180 - 30
3x = 150
Subtraction property of equality
3/3 x = 150/3
x = 50
Division property of equality
m<M = x + 20 (from above)
m<M  = 50 + 20
m<M = 70
Substitution


5.  Given m<P = 55 degrees, m<M = (x + 5) degrees, and NS = (x - 15). Find NS.
Bitmap Image
Statement
Reason
m<P = 55 degrees,
m<M = (x + 5) degrees
Given
m<M = m<P
Opposite angles of a parallelogram are congruent; therefore they are equal.
x + 5 = 55
Substitution
x + 5 - 5 = 55 - 5
x = 50
Subtraction property of equality
NS = x - 15
Given
NS = 50 - 15
NS = 35
Substitution

6.  Given m<MNS = (5x + 10) degrees and ,<NSP = (x + 30) degrees, find m>MNS.
Bitmap Image
Statement
Reason
SMNP is a parallelogram
m<MNS = (5x + 10) degrees
m<NSP = (x + 30) degrees
given
m<NSP = m<MNS
Alternate interior angles are congruent; therefore, their measures are equal.
5x + 10 = x + 30
Substitution
4x = 20
Addition/subtraction property of equalities
x = 5
Multiplication/division property of equalities
m<MNS = 5(5) + 10 degrees
m<MNS = 35 degrees
substitution property of equalities
m<NSP = 35 degrees
substitution property of  equalities
You can check your work by determining the m<NSP through substitution:
m<NSP = x + 30
m<NSP = 5 +30
m<NSP = 35 degrees

7.  Given MN = 3x, SP = (40 - x) and MS = 2x, find NP.
Bitmap Image

Statement
Reasons
SMNP is a parallelogram
MN = 3x,
SP = (40 - x)
 MS = 2x
Given
MN = SP
The opposite sides of a parallelogram are congruent; therefore, by the segment congruence postulate, their measures are equal
3x = 40 -x
substitution
4x = 40
Addition property of equalities
x = 10
multiplication property of equalities
 MS = 2(10)
MS = 20
Substitution property of equalities
MP = MS
The opposite sides of a parallelogram are congruent; therefore, by the segment congruence postulate, their measures are equal
MP = 20
Substitution
8.  Given M<P = 80 degrees, find m<MNP.
Bitmap Image
SMNP is a parallelogram
m<P = 80 degrees
Given
<P and <MNP are supplementary
Consecutive angles of a parallelogram are supplementary
m<P + m<MNP = 180 degrees
Definition of supplementary angles
80 + m<MNP = 180 degrees
substitution
m<MNP = 100 degrees
addition property of equalities

9.  Given m<MNP = 120 degrees, m<MSP = 6x degrees, and NS = (x - 15), find NS
Bitmap Image

Statement
Reasons
SMNP is a parallelogram
m<MNP = 120 degrees, m<MSP = 6x degrees
Given
m<MNP = m<MSP
Opposite angles of a parallelogram are congruent; therefore, by the angle congruence theorem, their measures are equal.
120 = 6x
20 = x
substitution property of qualities
x = 20
Reflexive property of equalities
NS = (x - 15)
Given
NS = (20 - 15)
NS = 5
substitution property of equalities

10.  Given MS = 15, NP = (x - 5), and m<P = x degrees, find m<M
Bitmap Image
Statement
Reasons
SMNP is a parallelogram
MS = 15, NP = (x - 5), and m<P = x degrees
Given
NP = MS
Opposite sides of a parallelogram are congruent; therefore by the segment congruence theorem, their measures are equal
x - 5 = 15
substitution property of equalities
x = 20
Addition/subtraction property of equalities
m<P = 20 degrees
substitution
m<M = m<P
Opposite angles of a parallelogram are congruent; therefore by the angle congruence theorem, their measures are equal
m<M = 20 degrees
Substitution or transitive property of equalities


Bitmap Image
3.3 In the diagram above, lines m and n are parallel lines,
and the measure of angle1 = 135 degrees.

Find the measures of <2, <3, <4, and <5.  Explain in paragraph form how you determined your answers.  Be sure to include the theorems that helped you arrive at these answers.


Bitmap Image
3.4  Use the illustration above and the converses of the transversal theorems and postulates to prove that lines are parallel. Use a two-column proof:

Given: the diagram above with m<1 + m<4 = 180 degrees
Prove m is parallel to n.

Statement
Proof
Use the tab key to add more rows.

You can also submit your answers on the form at the bottom of the calendar page.
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 Second half of unit notes

Unit 3.5
The Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180 degrees.
Bitmap Image
m<1 + m<2 + 70 = 180              (see theorem above)
m<1 = 57 degrees                       Vertical angle theorem
  57 +x+ 70 = 180                          substitution
125 + x = 180  
x = 55                                              addition property of equalities

The Exterior Angle Theorem:
The measures of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles
Example 1:
Bitmap Image
<2 and <3 are remote interior angles of the exterior angle measuring107 degrees.
This is not enough information to determine the measures of each angle; however, we can determine the measure of angle 1 by using the definition of a linear pair and the supplementary angles theorem.

m<1 = 180 - 107                 (this is the result of the supplementary angles theorem)
m<1 = 73 degrees

Bitmap Image
By knowing the measure of one of the remote angles of the exterior angle that measures 107 degrees, we can determine the measure of the other remote angle.  We can also determine the measure of the unknown angle by using the triangle sum theorem.  

The measure of angle 1 is 73 degrees.  This was determined in example 1.  
Using the exterior angle theorem, we see that
62 + m<2 = 107
m<2 = 107 - 62
m<2 = 35 degrees

To check our work, we use the triangle sum theorem
m<1 + m<2 + m<3 = 180 degrees
73 + 35 + 62 = 180 degrees



 Unit 3 Test Study Guide

You will need to know the following definitions, theorems, and postulates:
The definitions, postulates, and theorems listed in the frame at the right.
Definition of slope:  The slope of a nonvertical line that contains point 1 (x, y) and point 2 (x,y) is equal to the ratio .
Parallel Lines Theorem:  Two nonvertical lines are parallel is and only if they have the same slope.  Any two vertical lines are parallel.
Perpendicular Lines Theorem:  Two nonvertical lines are perpendicular is and only if the product of their slopes is -1.  Any vertical line is perpendicular to any horizontal line.
Midpoint formula:  The midpoint of a segment with endpoints Bitmap Image and  Bitmap Imagehas the following coordinates:
Bitmap Image
Midsegment of a triangle:  A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides.
Midsegment of a trapezoid. A midsegment of a trapezoid is a segment whose endpoints are the midpoints of the two nonparallel sides.  
Notice, if the two non parallel sides of a trapezoid are extended in a direction toward the smaller base, the figure meets at a point.  If the lines stop at this point, a triangle is formed.