Geometry- Drawing inferences from givens common core geometry

CPCTC example

(make sure this is all correct) "=" represents a congruent sign ______G _____/-_ _T<-___>H _____\-A Given: TH bisects <GTA and GT=AT Prove: HT bisects <GHA S: 1)TH bisects <GTA 2)<GTH=ATH 3)GT=AT 4)TH=TH 5)Triangle GTH=triangle ATH 6)TG=AT 7)T is a midpoint of GA (change) R: 1) Given 2) An angle bisector divides on < into 2 = <'s 3) Given 4) reflexive 5) Hypotenuse Leg 6) CPCTC 7) A midpoint divides a segment into 2 = segments (change)

know once done with the chart when

-The conclusion will be the "prove" statement -also try to add as many vocab words in the chart -for theorems, conclude by telling what the diagram is ie. parallel lines cut to form two congruent alternate interior angles

Hypotenuse Leg (HL)

-Used only for right angles -used for (Angle Side Side or SSA (cannot list those two)

Order of the letters of the triangle must be correct

1st letter: Will have no A or S on it 2nd and 3rd letter: will be connected in any order

Rules to follow

Always write while saying: Parallel lines: "Parallel lines form AIA" __ bisects __: The first two letters will cut the last two letters, ie. AB bisects CD, meaning that CD has the side label, and not AB __ and __ bisects each other: Both segments will have the side label HL: cannot say it is HL unless it gives directions which say there is a right (90 degree) angle Supplementary angles: Two steps needed 1: Linear pairs form right angles 2: Supplements of congruent angles are congruent Addition property: ie, S: .)FE + BE=BD + BE R: .)Addition property Subraction proprty: ie. S: .) RA=SU R: .) Subtraction Perpendicular lines: write that perpendicular lines form right angles

CPCTC

Corresponding Parts of Congruent Triangles are Congruent (meaning that everything on both triangles are congruent, and side will also be congruent)

perpendicular symbol is

I --

parallel symbol is

II

SAS

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. -Must prove two sides and an angle are all congruent in the chart

What to remember (definitions and theorems)

REMEMBER THAT THERE MUST BE PROOF FOR EVERY STEP, OTHERWISE DON'T ADD IT -NO SHORTCUTS -REMEMBER AND TRY TO BREAK UP WORDS IN THE GIVEN, ie. PERPENDICULAR AND BISECTOR -explain with each question how I got to my answers -check off in the chart what I have already listed from the given -don't need to draw a chart if a given does not have any vocab words -what I write "=", it represents "=" with a squiggly line on top which means congruent-ency -when naming triangles, can be start off with any letter, but must be in the same order

SSS=SSS theorem

Side=S The triangle must be congruent Given _ is the midpoint of __=__ __ ie. _______^ _____/_I_\ ___/___I__\ -B is the upper vertex -A is right vertex -D is median -C is left vertex __________S_____________|_________R__________ 1)D is midp of AC_I_1)given 2) AD=DC___________I_2) midpoint divides... 3)AB=BC____________I_3.) given 4) BD=BD___________I_4) reflective property 5) Triangle__________I_5) SSS=SSS (know ABD=Triangle_____I they are the same) CBD_________________I

Altitude (definition)

Starts at a vertex and forms a perpendicular line with the opposite side (does not have to be at midpoint) ie.-________ (C)/__I(D)____\(B) -A is bottom left vertex CD is the altitude in triangle ABC CD (is perpendicular) is congruent to AB

median (definition)

Starts at a vertex and goes to the midpoint of the opposite side ie.___(B)_^ (A)/__(D)I____\(C) -B is the upper vertex BD is the median in triangle ABC D is the midpoint

How to set up a chart for answering questions (definitions)

Step 1: draw perpendicular line near at the upper end of the vertical line Step 2: write R (reason) on right and S (statement) on left. Step 3: write "1)" on both sides Step 4: for "1)" write in what is the given action in the work ie. in triangle ABC, AD is the MEDIAN. step 5: write in "given" for "1)", statement step 6: underline the vocab word, ie. median step 7: for "2)" reason, define the vocab word step 8: for "2)" statement, write the conclusion of the process, ie. <AD is congruent to <BDC IT CAN KEEP GOING: Step 9: write in for the next number, ie. "3)" reason the next vocab word, ie. midpoint step 10: define it on the reason "2)" step 11: then write in the conclusion on the same reason, ie. "3", IF THERE ARE MORE VOCAB WORDS TO ADD ON, REPEAT STEPS 5-11

bisects (definition)

To divide into two congruent parts (is at its midpoint)

supplementary angles (definition)

Two angles whose sum is 180 degrees

Subtraction property (theorem)

When segments can be subtracted and then be equal to one another. ie. AC=DB Prove: AB=CD _^ /I\\ ____ (overlap between the two lines) _______________S_______|___________R 1) AC=DC__________|_1) given 2)BC=BC___________|_2)reflexive 3) AB=CD_________|_3.) subtraction property -S2) must be stated when there is a reflexive (overlap)

The bonus question is

Which of the following theorms cannot prove a triangle is congruent

Segment bisector (definition)

a line or segment that divides a segment into 2 congruent segments at its midpoint ie. ______C ____A------M------B ___________________D -CD is the bisector -AM is congruent to MB

AAS

angle angle side -Cannot have a connection with the A's and S's and have an S in the middle

Alternate Interior angles (theorem)

angles between 2 lines and on opposite sides of a transversal ie. black is congruent to black -AIA are congruent

Theorms

are known (don't have to define it in the chart)

Perpendicular bisector (definition)

creates right angles AND divides a segment at its midpoint (into two congruent segments) (two processes occur) ie.__________I (C) (A)-------I(M)------(B) _________(D)I -angle AMC, CMB, AMD, and BMD are right angles and AM is congruent to MB

midpoint (definition)

divides a segment into two congruent segments ie. A------M-------B AM is congruent to MB -this is just a point for M

Angle bisector (definition)

divides an angle into two congruent angles ie._-A B<--D ___--C <ABD is congruent to DBC -BD is the angle bisector of <ABC

ASA

for one side and two angles to all be congruent -must prove that in the chart

Perpendicular lines (definition)

forms right angles (for definitions don't add the numbers/measurements) ie. _____I(C) A--D--B ie. <ADC+<BDC are right angles

parallel lines

lines in the same plane that never intersect

Corresponding angles (theorem)

shift down ie. black is congruent to black, brown is congruent to brown Know how to find x: 2x+50=x+100 x=50 know the interior, parallel and alternate angle processes. -corresponding angles are congruent

How to set up a chart for theorems

step 1: draw usual setup with Statement left and Reason right step 2: then draw the standard information , ie given info, ie given. step 3: but no definitions are required ie. _______^_______ ____/__I__\ ___--------___ -B is top vertex -A is left vertex -C is right vertex -D is the midpoint of AC Prove: DB=DB ________S_____I_______R______ 1)DB=DB___I_1) reflexive (overlaps (congruent to _______________I______itself)

two triangles are the same if...

the perimeters of both triangles are equivlent.

vertical angles (theorem)

two angles opposite each other that are congruent -vertical angles are congruent

linear pair (definition)

two angles that form a supplementary angle (180 degrees)

addition property (theorem)

when multiple segments equal each other by adding them ie. AB is congruent to CD BE is congruent to DF Prove: AE is congruent to CF S__________________I___R___ 1.) AB=CD______I1.) given 2.) BE=DF______I2.)given 3.) AB+BE=_____I3.) addition property CD+BF, AE=CFI

reflexive (theorem)

when two segments overlap