Math Chapter 5 Triangle Proofs
Isosceles triangle subarguments
-To show base angles of a triangle are congruent when you know two sides are congruent - To show two sides of a triangle are congruent when you know angles are congruent (the converse).
Key points to remember when writing proofs
1. A proof begins with the given statements and ends with the prove statement 2. Each conclusion (statement) is given with a justification 3. Each justification is the given, a definition, postulate, or theorem from geometry 4. Mathmatical Axioms, such as the reflexive property, can also be used as justifications. Other axioms you learned in Algebra are used sparingly in some of the more advanced proofs in some of the more advanced proofs An Axiom is a postulate
line of semmetry
A line that divides a figure into two congruent parts.
two-column proof
A two column proof, all the conclusions are written down in a single column on the left, and the corresponding justifications are written in the column to the right. On the left you have the statements On the right you have the reasons
Parallel line theorem converse
AIA theorem converse- If AIA are congruent, then the lines that form them are parallelCA theorem converse- If CA are congruent, then the lines that form them are parallel SSIA theorem converse- If SSIA are supplementary, then the lines that form them are parallel
Parallel Line Theorem
AIA theorem- If parallel lines are cut by a transversal then the alternate interior angles are congruent CA theorem- If parallel lines are cut by a transversal then the corresponding angles are congruent SSIA theorem- If parallel lines are cut by a transversal then the Same Side Interior angle is sedimentary aka equal to 180 degrees
Subargument
Based on the definitions of geometry words justification is always def of... You can write in the converse way
Converses
Because definitions are always biconditniol
Conditnial, converse, Inverse, and Contraposative example
Conditional: If you were born in Pennslyvania, then you are a U.S. citizen.Converse: If you are a U.S. citizen, then you were born in Pennslyvania. Inverse: If you were not born in Pennsylvania then you are not a U.S. citizen. Contrapostive: If you are not a U.S. citizen, then you were not born in Pennslyvania.
Contraposative
Contraposative Is the not statement of the converse example converse statement--If you are a US citizen, then you were born In Pennsylvania Contraposative statement-- If you are not a US citizen, then you were not born In Pennsylvania
AAA(Angle-Angle-Angle)
Does not prove triangle congruency but you can make a triangle with the three given anglesdoes prove they are similar though
Flow proof
Each statement/justification pair is given its own box and arrows are drawn so a reader can follow in logical order. The arrows demonstrate the dependence on previous statements
You need to include the Reflexive property in justifications
For example line RN would be congruent to line RN because of reflexive
CA(Corresponding Angles)
Forms F
AIA
Forms Z
Perpendicular Bisector theorem
If a line is a perpendicular bisecotor of a segment, then any point on that line is equidistant to the two endpoints of the segment
inverse
If not statement of originalorignal statement-- If p then q Inverse statement--If not p, then not q
Hypotenuse Leg Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent
Side Side Side triangle congruence postulate (SSS)
If three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent
Angle Angle Side triangle congruence theorem (AAS)
If two angles and a non included side of one triangle are congruent to two angles and a non included side of a second triangle then the triangles are congruent
Angle Side Angle triangle congruence postulate (ASA)
If two angles and an included side of one triangle are congruent to two angles and an included side of a second triangle then the triangles are congruent
Side Angle Side triangle congruence postulate (SAS)
If two sides and an included angle of one triangle are congruent to two sides and an included angle of a second triangle, then the traingles are congruent
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent and the other way.
You only use the def of isosceles triangle if you have to prove the angles or sides of a triangle are congruent if it is given that the triangles are congruent. You only use isosceles triangle theorem if you have two base angles that are equivalent and you that it is an isosceles triangle theorem. If you already know it is an isosceles triangle then it is def of isosceles triangle. If you are trying to prove it is an insoles triangle then use the isosceles triangle theorem.
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
SSIA (same side interior angles)
In the same angle very close to each other on top of each other Also known as consecutive interior angles
Triangle congruence statement
Indicates how congruent triangles correspond to each other. A congruence statement reads Triangle XYZ is congruent to triangle ABC. When naming the vertices of the first triangle may be written in any order but the vertices in the second triangle must be written so the corresponding parts math. AKA you match with the equivalent angles
congruence statement
Indicates how congruent triangles correspond to each other. A congruence statement reads Triangle XYZ is congruent to triangle ABC. When naming the vertices of the first triangle may be written in any order but the vertices in the second triangle must be written so the corresponding parts math. AKA you match with the equivalent angles
Paragraph proof
Is a proof written in complete sentences Example It is given that line BC is congruent to line AC, and also that C is the midpoint of line DE. We know CD is congruent to CE by the definition of midpoint. Then angle ACE is congruent to angle BCD by the VA theorem. Finnaly their is enough information to conclude That triangle ACE is congruent to triangle BCD because of SAS.
Corollary
Is a sort of an "add on" or extending relationship to a theorem Can be deduced from the theorem
conditniol statement
Is known as an "If then statement" We use them to create definitions and mathematical statements If...(Hypothesis), then... (conclusion)Example- If two angles sum to 90 degrees, then they are complements. Example- Two angles sum to 90 degrees arrow complementsHypothesis arrow conclusion can be true or false
biconditional statement
Is when the converse and the conditniol statement are both true. When combining it they use If and only if For example-- Today is Monday if and only if tomorrow is Tuesday. This if and only if can be replaced by double arrows. Mathematical definitions are biconditonal
Given
Means that a piece of information is supplied with the problem concluding from something that is given. For example Given line AC is parallel to line BD and line AC is congruent to line BD What is supplied with the problem
Counterexample
One specific example that proves that the conditnial statement is false. For example If a person lives in USA, then he lives in Pennslyvania. A counterexample would be is--- a resident of New York who DOES live in US, but DOES NOT live in Pennsylvania.
converse
The converse of the conditniol statement happens When the hypothesis and conclusion are switched. The converse of p ➝ q is q ➝ p.Example Conditniol statement--If a person lives in Pittsburgh, then he lives in Pennslyvania Converse--If a person lives in Pennslyvania, then he lives in Pittsburg
Justiication-reason when writing proofs
The reason for your conclusion--support The reason of a conclusion. There are three types - given, definitions, or true conditional statements
Triangle Angle Sum Theorem
The sum of the (interior) angles of a triangle equals 180 degrees
Def of perpendicular lines in subarguments
To prove that if two lines are perpendicular the angle is equivlent to 90 degrees a right angle. Example-MP is perpendicular to ZM given -- Angle PMZ is a right triangle because of defintion of perpendicular lines
supplementary angles
Two angles whose sum is 180 degrees
Def of Isosceles triangle subargument
When you are given two congruent Angles or two congruent sides and you find out the other side through Def of Isosceles triangle reltansionship. For example if you know that 2 angles in a triangle are the congruent the opposite two sides are congruent forming a isosceles triangle and the other way around. Example-- Line segment TR is congruent to line segment TS given-- Angle TRS is congruent to angle TRS because of Def of Isosceles triangle and you can have angles and find for the sides
Def of congruent triangles
When you are looking to prove that additniol parts are congruent after you have concluded that the two triangles are congruent. Because for a triangle to be congruent all of it's corresponding parts must be congruent to. Example-- Triangle MNZ is congruent to triangle MLZ because of AAS-- LZ is congruent to NZ because of def of congruent triangles
Def of isosles in subargument
When you are trying to prove that two things are equal either because you where told the triangle is isosles and you know the base. and you are trying to prove that two sides are two angles are equal. Example- Triangle BCD is isosles w base BC given-- line segment BD is congruent to line segment CD because of def of isosles.
AIA as a justification for having parallel lines and using parallel lines as a justification for AIA Subargument(Parallel line subarguments)
When you are trying to prove the lines are parallel through AIA or you know the lines are parallel and want to prove Angles are congruent using AIA converse. Example BD is parallel to JK Given-- angle 2 is congruent to angle 7 because of AIA. Example- Angle 6 is congruent to angle 3 given-- BD is parallel to JK because of AIA converse
All right angles are congruent
When you are trying to prove to right angles are congruent you say this. Example-PMZ is a right angle because of defintion of perpendicular lines--Angle QZM is a right angle because of defintion of perpendicular lines conclusion-- Angle PMZ is congruent to angle QZM because ALL RIGHT ANGLES ARE CONGRUENT.
Def of bisect in subargument
When you are trying to show that two angles are cut by something that makes them congruent and if two angles are cut by something that makes them congruent Example for angle - Ray PJ bisects angle XPN given-- Angle XPJ is congruent to angle NPJ because of def of bisect Example for Line- Line segment AD bisects line segment CB given-- Line segment BE is congruent to line segment CE because of Def. of bisect
deductive reasoning
When you combine general statements into specific conclusions mathmatians want to prove their results with deductive reasoning. Example Suppose you know that all drivers are legally bound to carry insurance. If you have a cousin who has become a licensed driver, you might reason "If my cousin drives a car, she will Carry insurance" Using prior knowledge to make an assumptionUsing known facts, defintions and accepted properties in logical order to reach conclusion or to show that that statement is true (proving. makes a rule)
inductive reasoning
When you examine a body of information and try to find patterns or relationships. An example of this is if you watch a football game and notice all the players wearing helmets. You might reason that "If you play football, you must wear a helmet" this is how we make conjectures in geometryMaking a general statement based on a number of observations(guessing, looking for a pattern)
Reflexive Property (Reflexive) in subarguments
When you find that two lines or two angles are shared by two triangles making those two angles or two sides congruent. Example-- RN is congruent to RN because of Reflexive. You can do with angles and line segments.
Definitions in proofs
When you use definitions allow you to draw conclusions You can either write Def. of... (e.g...Def of midpoint) or you can write out the definition (e.g., If a point is a midpoint than it divides a segment into two congruent segments).
Vertical angle(VA) subarguments
When you want to show two angles are congruent through vertical angles Example- Angle PQL is congruent to Angle MQN because of VA
Geometry statements (conditniol statements) in proofs
You can use geometry statements(theorems-postulates) as the basis of your conclusions. When you draw a conclusion by using a statement you either write the title (Corresponding Angles Theorem), or the abbreviation or the full statement (If parallel lines are cut by a transversal, then the corresponding angles are congruent. Always theorems or postulates
Triangle congruence postulate when you are trying to prove two triangles are congruent using (SSS, AAS, ASA, SAS)
You get enough information to make the conclusion that the two triangles are congruent using (SSS, AAS, ASA, SAS).
Parallel line subarguments
You use these subarguments in two cases -To show angles are congruent when you know lines are parallel -To show lines are parallel when you know angles are congruent. (The converse) note: the symbolic biconditional can be used as the justification for both
Triangle Congruence postulates
You use triangle congruence subarguments to show that two triangles are congruent. You always need three pieces of information. The subargument
complementary angles
angles that total to 90 degrees
Def. of midpoint in a subargument
when trying to prove that the midpoint cuts the two segment into two congruent parts Example-X is m.p. of AB given -- then you would say line segment AX is congruent to line segment BX.
