THEOREMS WE KNOW PROJECT

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Shauna Caldwell
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1 1 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 Notes and 3 questions on proofs are due Wednesday The final project will be due Friday Theorems of Geometry Angles: If two are supplements of the same angle, then they are equal in measure. Statements Reasons <DAB <HEF B <DAB+<BAC=180 Definition of supplementary D A C <HEF+<FEG=180 Definition of F supplementary 180- <DAB=<BAC Property of subtraction H E G 180- <HEF=<FEG Property of subtraction 180- <HEF=<BAC Substitution <BAC and <FEG are equal to 180- <DAG therefore they are equal property Properties of equality <BAC=<FEG Equality If two are complements of the same angle, then they are equal in measure Statements Reasons <DAB <HEF <DAB+<BAC=90 Definition of complementary

2 2 <HEF+<FEG= <DAB=<BAC 90- <HEF=<FEG 180- <HEF=<BAC <BAC and <FEG are equal to 180- <DAG therefore they are equal Definition of complementary Property of subtraction Property of subtraction Substitution property Properties of equality Prove: <BOD = <DBA Know: <BOD+<COB=180 by supplementary <COA+<COB=180 by supplementary 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 of a triangle is 180. An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote interior.

3 3 If two sides of a triangle are equal in measure, then the opposite those sides are equal in measure. Statement Reasons 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 into two congruent 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 tri are congruent. C.P.C.T.C. If two of a triangle are equal in measure, then the sides opposite

of equilateral triangle <A <B;<B <C Isosceles triangle theorem <A <C Transitive property ΔABC is equal angular All are equal 360 3=60 Property

4 4 those are equal in measure If a triangle is equilateral, then it is also equiangular, with three 60 Statement Reason ΔABC is equilateral AC BC; AB AC Def. of equilateral triangle <A <B;<B <C Isosceles triangle theorem <A <C Transitive property ΔABC is equal angular All 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. A Statement Reason <A <B <B <C AB BC Two B C congruent, opposite sides are congruent BC AC Two are congruent opposite sides are congruent AB = AC Transitive property

5 5 The sum of the angle measures of an n- gon is given by the formula S(n)=(n- 2)180 The sum of the exterior angle measures of an n- gon, one angle at each vertex is 360. Lines If two parallel lines are intersected by a transversal, then alternate interior are equal in measure. : a b Prove: <1 <3 Angle 1 is equal to angle 4 because 6 4 corresponding angels are equal. Angle 3 is equal to Angle 4 because of vertical 3 5 theorem. Angle 1 is equal to angle because of transitive property. Therefor if a transversal intersects 2 parallel lines alternate interior are equal. If two parallel lines are intersected by a transversal, the co- interior angels are

6 6 supplementary. If two lines are intersected by a transversal and corresponding are equal in measure, then the lines are parallel. Statements Reasons <ACL <MAR S (Corresponding P C L Angles) <PCS <ACL Vertical Angles <MAR <QAC Vertical Angles Q <MAR+<QAM=180 Supplementary R A <MAR+<CAR=180 Supplementary M <QAM <CAR If two are supplementary to the same angle they are congruent <CAR <SCL Corresponding <PCA <SCL Vertical PL QR The transversal intersects the two lines with the same. If two lines are intersected by a transversal and alternate interior are equal in measure, then the lines are parallel.

7 7 If two lines are intersected by a transversal and co- interior are supplementary, then the lines are parallel. If two lines are perpendicular to the same transversal, then they are parallel. 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, 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 is, if two parallel lines are cut by a transversal, then the corresponding are congruent. The converse of that statement is, if the corresponding are congruent, the lines are parallel. Since all equal 90 degrees, all corresponding 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. What is given? r l,t r What do you need to prove? r l Statements Reason r l,t r <l is a right angle Def. of perpendicular lines M<l=90 Def. of right m<1 m<2 Corresponding

8 8 m<1=m<2 Def. of congruent 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. Statement Reason Tri: 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 In a triangle, a segment that connects the midpoints of two sides is parallel to the third side and half as long.

9 9 If two and the included side of one triangle are equal in measure to the corresponding and side of another triangle, then the tri are congruent. (ASA) <ABC=<ADC Line DC=Line BC Angle DCA= Angle ACB Line AC= Line AC Triangle ADC- Triangle ABC Reflexive property of equality SAS If two and a non- included side of one triangle are equal in measure to the corresponding and sides of another triangle, then the two tri are congruent. (AAS) 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 tri are congruent. (SAS)

the measure of the altitude is the geometric mean between the measures of the parts of the hypotenuse.

10 10 If the altitude is drawn to the hypotenuse of a right triangle, then the two tri formed are similar to the original triangle and to each other. 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 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. B Statements Reasons A h D C h=altitude ΔABD is similar to ΔBCD BD AD = DC BD h AD = DC h h 2 = AB DC By definition Altitude creates similar tri Properties of similar tri Properties of similar tri 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. A Statements BE is the shortest distance from vertex B to AE BA>BE. BA^2=AE^2+BE^2- AB>BC Reasons short distance theorem Pythagorean theorem

11 1 1 B C Triangle AEC= AC^2=AE^2+EC^2= AC>EC AB^2-BE^2= EC^2- AE^2 AB+AC>BE+BC AB+AC>BC Pythagorean theorem. AC^2= EC^AE^2 substitution property addition property segment addition postulate In an isosceles triangle, the medians drawn to the legs are equal in measure. b e a d c Statement ΔABC is isosceles Draw medians BD and CE Reasons Through any 2 points there is 1 line AB AC Properties of an isosceles triangle AB = AC Definition of congruence 1 2 AB = 1 2 AC Multiplication property BE = 1 2 AB;DC = 1 2 AC 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. Statement: A quadrilateral ABCD is a parallelogram if AB is

12 12 parallel to CD and BC is parallel to DA. AB ll CD In a kite, the diagonals are perpendicular to each other. L BAE is congruent to L DCE AB is congruent to CD L ABE is congruent to L CDE Triangle AEB is congruent to triangle DEC AE is congruent to EC BE is congruent to ED Definition of a parallelogram Alternate interior postulate Opposite sides in a parallelogram Alternate interior postulate ASA CPCTC CPCTC In a rectangle, the diagonals are equal in measure. In a parallelogram, opposite sides are equal in measure. B Statement <ABD <BDC <DBC <ADB Reason Alternate interior Alternate interior

1 3 C A D DB DB ΔADB ΔCBD AB DC; AD BC Reflexive property ASA C.P.C.T.C. If a quadrilateral is a parallelogram, then consecutive are supplementary. Notes Lets consider two consecutive DAB and ABC.

13 1 3 C A D DB DB ΔADB ΔCBD AB DC; AD BC Reflexive property ASA C.P.C.T.C. If a quadrilateral is a parallelogram, then consecutive are supplementary. Notes Lets consider two consecutive 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 are the corresponding at the parallel lines AC and BC and the transverse AE. The ABC and CBE are adjacent supplementary and make in sum the straight angle ABE of 180. Therefore, two consecutive DAB and ABC are non- adjacent supplementary and make in sum the straight angle of 180. Similarly, consider two other consecutive 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 are the corresponding at the parallel lines DC and AB and the transverse BF. The BCD and DCF are adjacent supplementary and make in sum the straight angle BCF of 180. Therefore, two consecutive ABC and BCD are non- adjacent supplementary and make in sum the straight angle of 180. You can repeat these steps for the other two sets of consecutive. Therefore if a quadrilateral is a parallelogram then all the of the consecutive are supplementary. If a quadrilateral is a parallelogram, then opposite are equal in measure. Statements Reasons E AD BC CD AB

14 14 F A D C <BCD <CDE <CDE <BAD <BCD <BAD <FAB <ABC <FAB <ADC <ABC <ADC Alternate int. Angles Corresponding Transitive property Alternate interior Corresponding Transitive property B The sum of the measures of the of a quadrilateral is 360. Quadrilaterals can be divided into two triangle The of tri are equal to 180 degrees Two tri add up to 360 degrees Quadrilaterals add up to 360 degrees Definition of a quadrilateral Triangle Angle Sum Theorem Additive property of addition Substitution property of addition Help from: a/additive_property_of_equality.htm If both pairs of opposite of a quadrilateral are equal in measure, then the quadrilateral is a parallelogram.

15 1 5 We need to prove the opposite 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. LCBE is congruent to LDAB LBCF is congruent to LADC LCBE is congruent to LBCD LBCF is congruent to LABC Hence, LDAB is congruent to LDCB Supplementary theorem Corresponding postulate Alternate interior postulate Steps 1,2, and 3 If the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. B Statements Reasons A Quadrilateral ABCD given Line AP is P congruent to PD. Line BP is congruent to PC C D Angle APB is congruent to angle CPD Triangle ABP is congruent to triangle CPD Angle BCD is congruent to angle CBA Angle BCD is congruent to angle CBA AB DC Diagonals bisect so diagonals bisect each other vertical angle theorem. SAS CPCTC alternate interior theorem converse of parallel transversal theorem

16 line AP is congruent to PD and line CP is congruent to BD Angle CAP is congruent to angle BDC Angle APC is congruent to angle DBP Triangle APC is congruent to Triangle BDP Angle ABC is congruent

16 16 line AP is congruent to PD and line CP is congruent to BD Angle CAP is congruent to angle BDC Angle APC is congruent to angle DBP Triangle APC is congruent to Triangle BDP Angle ABC is congruent to angle BCD Line AC BD alternate interior theorem vertical theorem SAS alternate interior angle theorem 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 at each base are equal in measure. a b Statement Reasons Trapezoid ABCD is isosceles c d <D and <C are base Definition of base <D <C Properties of an isosceles trapezoid AD BC 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

17 1 7 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 theorems have notes theorems have notes theorems have notes. Proofs (points taken off for each missing proof out of 10): 15-8 theorems have notes 8 or less theorems have notes.

Basic Quadrilateral Proofs

Basic Quadrilateral Proofs For each of the following, draw a diagram with labels, create the givens and proof statement to go with your diagram, then write a two-column proof. Make sure your work is neat