Theorems and Proofs with Proofs - Mr. Gilmartin's Classroom! 2!!!!!,

THEOREMS WE KNOW PROJECT 1

Name_________________ Date___________ Period_____

This is a list of all of the theorems that you know and that will be helpful when working on proofs for the rest of the unit. In the Notes section I would like you to write anything that will help you remember the theorem, such as an example problem, writing the theorem in your own word, a picture of what the theorem represents, etc. In the Proof section I would like you write the proof of the theorem. These are all theorems that you have seen and/or written the proof of before in math 3 or in previous classes. If you do not remember the proof use your book, the internet (remember to cite your source), your classmates, and as always Mr. G and I as a resource. The two proofs that you are responsible for are due Monday 2-‐3-‐14. Notes and 3 questions on proofs are due Wednesday 2-‐5-‐14 The final project will be due Friday 2-‐7-‐14. Theorems of Geometry Angles: If two angles are supplements of the same angle, then they are equal in measure. Notes:

Proof: Statements Reasons <DAB≅<HEF Given <DAB+<BAC=180° Definition of

supplementary angles

<HEF+<FEG=180° Definition of supplementary angles

180-‐<DAB=<BAC Property of subtraction

180-‐<HEF=<FEG Property of subtraction

180-‐<HEF=<BAC Substitution property

<BAC and <FEG are equal to 180-‐<DAG therefore they are equal

Properties of equality

<BAC=<FEG Equality

If two angles are complements of the same angle, then they are equal in measure

Statements Reasons <DAB≅<HEF Given <DAB+<BAC=90° Definition of

complementary angles

D A C

B

H E G

F

2

<HEF+<FEG=90° Definition of complementary angles

90-‐<DAB=<BAC Property of subtraction

90-‐<HEF=<FEG Property of subtraction

180-‐<HEF=<BAC Substitution property

<BAC and <FEG are equal to 180-‐<DAG therefore they are equal

Properties of equality

Notes:

Proof: Prove: <BOD = <DBA Know:

• <BOD+<COB=180° by supplementary angles

• <COA+<COB=180° by supplementary angles

• If <COA+<COB=180 then <COA=180-‐<COB

• If <BOD+<COB=180 then <BOD=180-‐<COB

• Therefore <BOD=<COA

The sum of the measures of the angles of a triangle is 180°. Notes:

Proof:

An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote interior angles. Notes: Proof:


Name_________________ Date___________ Period_____

If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure. Notes:

Proof: Statement Reasons

Given

Each angle has one unique angle bisector

An angle bisector is an ray whose endpoint is the vertex of the angle and which divides the angles into two congruent angles

Reflexive property a quantity is congruent to itself.

SAS-‐ if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

C.P.C.T.C.

If two angles of a triangle are equal in measure, then the sides opposite

4

those angles are equal in measure Notes:

Proof:

If a triangle is equilateral, then it is also equiangular, with three 60° angles Notes:

Proof: Statement Reason ΔABC is equilateral Given AC ≅ BC;AB ≅ AC Def. of

equilateral triangle

<A≅<B;<B≅<C Isosceles triangle theorem

<A≅<C Transitive property

ΔABC is equal angular

All angles are equal

360°÷3=60° Property of division

All 3 angle measures are 60°

Division above.

If a triangle is equiangular, then it is also equilateral. Notes:

Proof: Statement Reason <A≅<B Given <B≅<C Given AB ≅ BC Two angles

congruent, opposite sides are congruent

BC ≅ AC Two angles are congruent opposite sides are congruent

AB = AC Transitive property

A

B C


Name_________________ Date___________ Period_____

Lines

If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure. Notes:

Proof: Given: a||b Prove: <1≅<3 Angle 1 is equal to angle 4 because corresponding angels are equal. Angle 3 is equal to Angle 4 because of vertical angles theorem. Angle 1 is equal to angle 3 because of transitive property. Therefor if a transversal intersects 2 parallel lines alternate interior angles are equal.

If two parallel lines are intersected by a transversal, the co-‐interior angels are

The sum of the angle measures of an n-‐gon is given by the formula S(n)=(n-‐2)180° Notes:

Proof:

The sum of the exterior angle measures of an n-‐gon, one angle at each vertex is 360°. Notes:

Proof:

1 2

3 4

5 6

6

supplementary. Notes:

Proof:

If two lines are intersected by a transversal and corresponding angles are equal in measure, then the lines are parallel. Notes:

Proof: Statements Reasons <ACL≅<MAR Given

(Corresponding Angles)

<PCS≅<ACL Vertical Angles <MAR≅<QAC Vertical Angles <MAR+<QAM=180° Supplementary

angles <MAR+<CAR=180° Supplementary

angles <QAM≅<CAR If two angles are

supplementary to the same angle they are congruent

<CAR≅<SCL Corresponding angles

<PCA≅<SCL Vertical angles PL ||QR The transversal

intersects the two lines with the same angles.

If two lines are intersected by a transversal and alternate interior angles are equal in measure, then the lines are parallel. Notes:

Proof:

P

Q

L

M

R

C

A

S


Name_________________ Date___________ Period_____

If two lines are intersected by a transversal and co-‐interior angles are supplementary, then the lines are parallel. Notes:

Proof:

If two lines are perpendicular to the same transversal, then they are parallel. Notes:

Proof: Lines k and l are cut by t, the transversal. <1(top right of line k) and <5(top of line “l” left) are corresponding angles, along with <3(bottom right of line k) to <7(bottom right of line l), <2(top left of line k) to <6(bottom left of line l), and <4(bottom left of line k) to <8(bottom left of line l). The definition of corresponding angles is, “if two parallel lines are cut by a transversal, then the corresponding angles are congruent”. The converse of that statement is, “if the corresponding angles are congruent, the lines are parallel.” Since all angles equal 90 degrees, all corresponding angles are congruent. Thus, two line perpendicular to a transversal are parallel.

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also. Notes:

Proof: What is given? r || l, t ⊥ r What do you need to prove? r ⊥ l Statements Reason r || l, t ⊥ r Given <l is a right angle Def. of

perpendicular lines M<l=90° Def. of right angles m<1≅m<2 Corresponding

8

angles m<1=m<2 Def. of congruent

angles m<2=90° Substitution

property <2 is a right angle Def. of right angle t⊥ l Def. of

perpendicular lines

If a point is the same distance from both endpoints of a segment, then it lies on the perpendicular bisector of the segment. Notes:

Proof: Statement Reason

Triangles:

If a line is drawn from a point on one side of a triangle parallel to another side, the it forms a triangle similar to the original triangle Notes:

Proof:

In a triangle, a segment that connects the midpoints of two sides is parallel to the third side and half as long. Notes:

Proof:


Name_________________ Date___________ Period_____

If two angles and the included side of one triangle are equal in measure to the corresponding angles and side of another triangle, then the triangles are congruent. (ASA) Notes:

Proof: <ABC=<ADC Given

Line DC=Line BC Given

Angle DCA= Angle ACB

Given

Line AC= Line AC Reflexive property of equality

Triangle ADC-‐Triangle ABC

SAS

If two angles and a non-‐included side of one triangle are equal in measure to the corresponding angles and sides of another triangle, then the two triangles are congruent. (AAS) Notes:

Proof:

If two sides and the included angle of one triangle are equal in measure to the corresponding sides and angle of another triangle, then the triangles are congruent. (SAS) Notes:

Proof:

10

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Notes:

Proof:

In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs Notes:

Proof:

If the altitude is drawn to the hypotenuse of a right triangle, then the measure of the altitude is the geometric mean between the measures of the parts of the hypotenuse. Notes:

Statements Reasons h=altitude By definition ΔABD is similar to ΔBCD

Altitude creates similar triangles

BDAD

=DCBD

Properties of similar triangles

hAD

=DCh

Properties of similar triangles

h2 = AB•DC Properties of ratio

h = AD•DC Definition of geometric mean.

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Notes:

Proof: Statements Reasons BE is the shortest distance from vertex B to AE

short distance theorem

BA>BE. BA^2=AE^2+BE^2- AB>BC

Pythagorean theorem

A

B

A C

D

h

THEOREMS WE KNOW PROJECT 11 Name_________________ Date___________

Period_____

Triangle AEC= AC^2=AE^2+EC^2= AC>EC

Pythagorean theorem.

AC^2= EC^AE^2

AB^2-BE^2= EC^2-AE^2

substitution property

AB+AC>BE+BC addition property AB+AC>BC segment addition

postulate

In an isosceles triangle, the medians drawn to the legs are equal in measure. Notes:

Proof: Statement Reasons ΔABC is isosceles Given Draw medians BD and CE

Through any 2 points there is 1 line

AB ≅ AC Properties of an isosceles triangle

AB = AC Definition of congruence

12AB = 1

2AC Multiplication

property

BE = 12AB;DC = 1

2AC A median bisects the line it passes

through BE = DC Substitution

property BE ≅ DC Definition of

congruence <B≅<C Property of an

isosceles triangle BC ≅ BC Reflexive property ΔEBC≅ΔDCB SAS theorem CE ≅ DC C.P.C.T.C.

Quadrilaterals:

In a parallelogram, the diagonals have the same midpoint. Notes: Proof:

Statement: Proof:

A quadrilateral ABCD is a parallelogram if AB is

Given

a

b c

d e

B C

12

parallel to CD and BC is parallel to DA.

AB ll CD Definition of a parallelogram

L BAE is congruent to L DCE

Alternate interior angles postulate

AB is congruent to CD Opposite sides in a parallelogram

L ABE is congruent to L CDE


Triangle AEB is congruent to triangle DEC

ASA

AE is congruent to EC CPCTC

BE is congruent to ED CPCTC

In a kite, the diagonals are perpendicular to each other. Notes:

Proof:

In a rectangle, the diagonals are equal in measure. Notes:

Proof:

In a parallelogram, opposite sides are equal in measure. Notes:

Proof: Statement Reason <ABD≅<BDC Alternate interior

angles <DBC≅<ADB Alternate interior

B


Period_____

angles DB ≅ DB Reflexive property ΔADB≅ΔCBD ASA AB ≅ DC;AD ≅ BC C.P.C.T.C.

If a quadrilateral is a parallelogram, then consecutive angles are supplementary. Notes

Proof: Lets consider two consecutive angles DAB and ABC. Draw the straight line AE as the continuation of the side AB of the parallelogram ABCD. Then the angle CBE is congruent to the angle DAB as these angles are the corresponding angles at the parallel lines AC and BC and the transverse AE. The angles ABC and CBE are adjacent supplementary angles and make in sum the straight angle ABE of 180°. Therefore, two consecutive angles DAB and ABC are non-‐adjacent supplementary angles and make in sum the straight angle of 180°. Similarly, consider two other consecutive angles ABC and BCD. Draw the straight line BF as the continuation of the side BC of the parallelogram ABCD. Then the angle DCF is congruent to the angle ABC as these angles are the corresponding angles at the parallel lines DC and AB and the transverse BF. The angles BCD and DCF are adjacent supplementary angles and make in sum the straight angle BCF of 180°. Therefore, two consecutive angles ABC and BCD are non-‐adjacent supplementary angles and make in sum the straight angle of 180°. You can repeat these steps for the other two sets of consecutive angles. Therefore if a quadrilateral is a parallelogram then all the of the consecutive angles are supplementary.

If a quadrilateral is a parallelogram, then opposite angles are equal in measure. Notes:

Proof: Statements Reasons AD || BC Given

CD || AB Given

A

C D

E

14

<BCD≅<CDE Alternate int. Angles <CDE≅<BAD Corresponding angles <BCD≅<BAD Transitive property <FAB≅<ABC Alternate interior

angles <FAB≅<ADC Corresponding angles <ABC≅<ADC Transitive property

The sum of the measures of the angles of a quadrilateral is 360°. Notes:

Proof: Quadrilaterals can be divided into two triangle

Definition of a quadrilateral

The angles of triangles are equal to 180 degrees

Triangle Angle Sum Theorem

Two triangles angles add up to 360 degrees

Additive property of addition

Quadrilaterals angles add up to 360 degrees

Substitution property of addition

Help from: http://www.mathwords.com/ a/additive_property_of_equality.htm

If both pairs of opposite angles of a quadrilateral are equal in measure, then the quadrilateral is a parallelogram.

A

B

C

D

F


Period_____ Notes:

Proof: We need to prove the opposite angles are congruent. So, we need to prove that L A = L C and L B = L D.

Statement: Reason:

LCBE + LCBA = 180degrees, LFCB + LDCB = 180 degrees.

Supplementary angles theorem

LCBE is congruent to LDAB LBCF is congruent to LADC

Corresponding angles postulate

LCBE is congruent to LBCD LBCF is congruent to LABC


Hence, LDAB is congruent to LDCB

Steps 1,2, and 3

If the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Notes:

Proof: Statements Reasons Quadrilateral ABCD given Line AP is congruent to PD. Line BP is congruent to PC

diagonals bisect each other

Angle APB is congruent to angle CPD

vertical angle theorem.

Triangle ABP is congruent to triangle CPD

SAS

Angle BCD is congruent to angle CBA

CPCTC

Angle BCD is congruent to angle CBA

alternate interior theorem

AB||DC converse of parallel transversal theorem

Diagonals bisect so

A

B

C D

P

16

line AP is congruent to PD and line CP is congruent to BD Angle CAP is congruent to angle BDC

alternate interior theorem

Angle APC is congruent to angle DBP

vertical angles theorem

Triangle APC is congruent to Triangle BDP

SAS

Angle ABC is congruent to angle BCD

alternate interior angle theorem

Line AC||BD converse of parallel transversal theorem

Quadrilateral ABCD definition of parallelogram

In an isosceles trapezoid, (1) the legs are equal in measure, (2) the diagonals are equal in measure, and (3) the two angles at each base are equal in measure. Notes:

Proof: Statement Reasons

Trapezoid ABCD is isosceles

Given

<D and <C are base angles

Definition of base angles

<D≅<C Properties of an isosceles trapezoid

AD ≅ BC Given

Draw diagonal segments AC and BC

Through any two points, there is exactly one line

DC ≅ DC Reflexive property of congruence

ΔADC ≅ΔBDC SAS theorem

a b

c d


Period_____ AC ≅ BD C.P.C.T.C

Rubric: Theorems We Know Project ____/50 40 Points 35 Points 30 Points 20 Points 10 Points 0 Points All theorems have notes.

39-‐32 theorems have notes.

31-‐24 theorems have notes

23-‐16 theorems have notes.

15-‐8 theorems have notes

8 or less theorems have notes.

Proofs (points taken off for each missing proof out of 10):

Theorems and Proofs with Proofs - Mr. Gilmartin's Classroom! 2!!!!!,

Documents

Transcript of Theorems and Proofs with Proofs - Mr. Gilmartin's Classroom! 2!!!!!,