Chapter4 Geometry // + Indirect Proofs
Ask yourself 2 things when looking to prove something in two triangles...
1) What do I need to prove true? 2) What triangles do I use to prove true? --Can there by corresponding parts?
Under the Triangle Inequality theorem, there are two questions a math book might ask.
1. Given 3 lengths, can you form a triangle? 2. Given 2 lengths, determine the interval of possible lengths for one side. We have done exercises on question one, but what about question 2?
What is the included side of the two angles?
A side that is part of two angles of a triangle.
What is CPCTC? Of Corresponding Parts of Congruent Triangles are Congruent?
Congruent triangles is the hypothesis, which means that you need to prove that a triangle is congruent first before their parts of congruent. -You basically use this after you know the triangles are congruent, in all types of theorems in this section.
What is (15+9) and (15-9) represent from the last card? Or, (x-y) and (x+y)?
They are called conjugates
What is the technical umbrella term identifying AAS and HL
A theorem.
Overlapping Segments Theorem
Cool to prove that two overlapping triangles actually have legs that have congruent corresponding lengths.
Side Side Side Postulate [SSS Post]
If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent. - only applies to triangles
Remember, in the bow tie figure, you can always find an angle where?
In the middle, where there are vertical angles.
How do we know this past reasoning, based on an old postulate?
The intersection of two lines is a point, so the third vertex is easily formed in a triangle.
If you know the side measures lengths of a triangle, then you don't need to know the ______....
angle measures to construct the triangle.
How do you prove that a triangle inside a big isosceles one is isosceles itself?
1. PROVE THE BIG TRIANGLE IS ISOSCELES. Simply use the Isos triangle theorem to find the big triangles base angles are congruent, then conclude those 2 outer triangles are congruent. 2. PROVE THE BASE ANGLES of the large isosceles triangle ARE CONGRUENT. This uses the Isosceles Triangle theorem. 3. Then prove that two two small triangles on the outer edges are congruent. 4. Now you can say that the edge that they share with the inner triangles are congruent, due to CPCTC, and conclude that the inner triangle is thus isosceles by the Def. of isosceles.
What is the technical umbrella term identifying SSS, SAS, and AAS
A postulate
What does it mean for the angle measures if the triangle is equilateral?
Equal angle measures that measure 60d.
Easily support a picture with a SAS, or ASA, or SSS Post how?
Just look for what's inthe picture that both triangles have. Is it the same side lengths? a combiniation or two dsides and one angle measure? You look for it.
Isosceles Triangle Theorem
The base angles of an isosceles triangle are congruent.
Given 2 lengths, determine the interval of possible lengths for one side.
-We know that the smallest possible triangle can be made by pushing the top vertex as close into the base as possible. -The greatest triangle can be made if we pull out that top vertex out from the base.
How do you prove that the consecutive angles of a parallelogram are supplementary?
A parallelogram is a figure which contains two sets of parallel lines. For any two sides that are parallel connected by a segment, that segment is also a transversal. Therefore, according to the Consecutive Same Side theorem for parallel lines, consecutive angles add to 180d and thus are supplementary.
What is a congruence correspondence?
A statement that relates two congruent triangles to each other, in order of corresponding vertices. EX - triangle AMC is congruent to triangle LME, in THAT ORDER. The order is respective and each is congruent to their corresponding angle.
Why can't we use AAA? Use a mnemonic!
AAA is a battery name already! It's nonunique, unlike all the rest of the Triangle Congruence theorems.
Given triABC ≅ tri BCA, what can you say of triABC?
AB and BC are congruent sides, while BC and CA (line over top) are congruent, thus ABC and BCA are both equilateral .triangles.
If what the past card said is true, then what is true about segment length measures?
Absolute value I x-y I < z < x+y Ex - 15 and 9 are two segments of one triangle, the largest sum of lengths you can have is 6 units (15-9) and the greatest sum of lengths is 24 unis (15+9)
How to prove that the diagonals of a parallelogram bisect each other, given four triangles?
All you need to do is prove two opposite triangles are congruent, then that CPCTC is used to prove that the segments on either side on a bisecting point are congruent.
(12.1+12.4) Indirect Proofs
Also known as proof by contradiction. 1. A statement and its negation cannot both be true and is therefore a contradiction. 2. A conditional (if, then) and its contrapositive have the same truth value. 3. An assumption that leads to a contradiction must be rejected as false.
Two triangles are congruent if their corresponding congruent parts include two angles and a side... always true, sometimes true, or never true?
Always true! You have both the ASA and AAS postulate, so congruence can always be provide even if the side is included or not.
What is an included angle?
An angle formed by two sides of a triangle... If 2 sides and the included angle are fixed, then the size and shape of the triangle is fixed.
A triangle has 2 sides with length 8 and 10, what is the greatest perimeter possible for this triangle if the remaining side also has integer length? How would you solve?
Apply the rule that to find the greatest, add the two lengths together (but this would make the two segments overlap with the third, so you must subtract a unit). 8, 10, 17 is the combination! Still works under the Triangle Inequality Theorem = 35 units.
Why is it so important to have two elements of a triangle and the third element to be INCLUDED!!?
Because otherwise, a nonincluded side or angle is not enough to determine if two triangles are congruent.
What is the act of showing correspondence by naming angles in same order?
Congruence correspondence.
Chapter 4 - Congruence
Congruent Polygons. ex. Broken Windows, earrings.
Practice! Contrapositive this statement! Original conditional: If I am 100, then I am old.
Contrapositive: If I am NOT old, then I am NOT 100
If the diagonals of a parallelogram are perpendicular, then the pgram is a rhombus. PROVE
Diagonals of a pgram bisect each other. Reflexive property to prove that CPCTC. If one pair of consecutive siides of a pgram are congruent, then the pgram is a rhombus.
What is the LEAST perimeter for a triangle whose side lengths are consecutive odd integers? How would you solve?
First apply the general rule that two find the least side measure possible, find the difference of the 2 other sides (but add a unit to fit the rule ;). Then, apply the Triangle Inequality Theorem - sum of any two sides exceeds the last side. Therefore the combination cannot be 1, 3, 5 (4 does not exceed 5). The combination must be 3, 5, 7 = *15 units*
In the bow tie figure, how do you prove that two lines are parallel?
First prove that the triangles are congruent, and that alternate interior angles are congruent (because of CPCTC). Now you can use the converse of alt int angles theorem to say, if there are congruent AltIntAngles, then the lines must be parallel.
What information do you need to prove if a segment is an altitude?
First that the two angles are equal, then that they are supplementary through the LPP. (linear pair postulate). If two angles are congruent and supplementary, then they are both 90 degrees. -that's def. of perpendicular.
What is this exception called?
Hypotenuse Leg Congruence Theorem
Practice! Contrapositive this conditional! If I don't live in Florida, then I live in Los Angeles.
If I don't live in Los Angeles, then I live in Florida
You can use the ASA, SSS, and SAS postulates to prove triangles are congruent.
If opposite sides are parallel, and using alternative interior angle theorem, and using vertical angle theorem, you can use the ASA postulate.
Housebuilder theorem
If the diagonals of a parallelogram are congruent, then the pgram is a rectangle -USE CPCTC, and that consecutive interior angles of pgram are csupplmenetary, and then use substition and distributive. Just prove that one angle of a pgram is right, then pgram is a rectangle. -PICK THE RIGHT TRIANGLES (the ones that have a right angle as an angle
Hypotenuse Leg Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg or another right triangle, then the two triangles are congruent.
Angle Angle Side Congruence Theorem (AAS)
If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.
Angle Side Angle Postulate [ASA Post]
If two angles and their included side in one triangle are congruent to two angles and their included sides in another triangle, then the two triangles are congruent.
Side Angle Side Postulate [SAS Post]
If two sides and their included angle in one triangle are congruent to two sides and their included angle in another triangle, then the two triangles are congruent. (pair of sides and their angle)
Conversely to the polygon congruence definition,
If we are given congruence, then support the congruence with the 2 statements.
Why does the AAS Postulate work (two angles and a nonincluded side)
If we have an AAS combo, simply convert into a ASA combo by finding the 3rd angle, using the triangle sum theorem. Thus we now have 2 measures and an included side, fulfilling the ASA congruence postulate.
What happens if the two conjugates interact through all four operations?
If you multiply, you get difference of 2 squares (middle term cancels out) If you add, you get 2x If you subtract, you end up with twice the positive amount of the second value. If you subtract but flip the order, you get twice the negative amount of the second value. (x+5)(x-5) = x^2 - 25 (x+5) + (x-5) = 2x (x+5) - (x-5) = 10 (x-5) - (x+5) = -10
Is this info sufficient to determine triangle congruence? "Any two corresponding sides and one corresponding angle are congruent."
It can't be any angle. So no, the angle must be included by the two sides in order for a triangle to be congruent (SAS Postulate)
If you are asked, "Can I form a triangle if I know the 3 lengths?"
Just make sure the sum of the two least ones is greater than the last largest side.
Why can't we use SSA? Use a mnemonic!
Just spell SSA backwards and you understand why it's bad.
What information do you need to prove that a segment is an angle bisector?
Just that the two angles are equally split by that segment, that they have congruent measures. Do this by using the CPCTC.
Is SSA, side side angle, a possible postulate in proving two triangles are congruent?
NO, because according to the swinging door effect, two triangles can be congruent when a side can be moved, not affecting the other side or nonincluded angle.
(4.3) Is AAA, angle-angle-angle a possible postulate in proving two triangles are congruent?
NO, even if you know all angle measures, that only determines the shape of the triangle and not it's size. You need to know at least some other side measures too.
If 2 angles and the included side are fixed ina triangle, is the shape and size of the triangle fixed?
Only one angle is missing, so yes, the size and shape is fixed.
IF one angle of a pggram is a right angle, then the parallelogram is a rectangle. PROVE
Property is that consecutive sides are supplementary. Thus you can aply. If we know one side is right, it's a property that opp sides are congruent. Therefore all of the 4 angles are right, and we can conclude that the figure is a rectangle.
If one paifr of consecutive sides of a pgram are congruent, then the pgram is a rhombus. PROVE
Property of a pgram is that 2 pairs of sides are both congruent an dparallel. Use transitive property to relate oppsotie and consecutive sides. All sides are now congruent and that's what makes it a rhombus.
If 2 pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. PROVE
Prove CPCTC with SSS postulate, then use converse of opposite interior angles and the definition of pgram that lines are parallel.
ALWAYS ALWAYS, after you are done solving and checking a problem....
READ DIRECTIONS and make sure you answered everything!
What does "make a congruence relation, then support with a triangle postulate" mean?
Say, for example, triangle MNO ≅ triangle QPO Supported by the SAS postulate.
How to solve for segments using algebra?
Set lengths equal to each other, based on what you are given (ex, if you are told the triangle is isosceles). So set them equal to each other, and later equal to zero and factor out.
What if you were asked if the given combination of angles and sides determine a unique triangle?
Simply draw it out and apply your past knowledge of all the postulates, see if the postulates of included sides exist. WATCH OUT FOR - The one exception of the SSA Postulate!! If the triangle is a right triangle!!
Suppose Hexagon ABCDEF ≅ UVWXYZ. Why is segment BC ≅ segment VW?
Since B is the second vertex, and V is the second vertex, then that's the correspondence order. You're given congruence, and told to correspond the angles.
There are 2 factors of congruence... what are they?
Size and shape -shape is determined by comparing all angle measures. (ex; match up the angles of a small polygon to a bigger similar polygon)
Two triangles are congruent if their corresponding congruent parts include two sides an and angle... always true, sometimes true, or never true?
Sometimes true, because SAS is a postulate, and SOMETIMES SSA, if the theorem being applied is the Hypotenuse Leg Theorem.
What information do you need to prove if a segment is a median?
That the segment it intersects with is BISECTED, meaning equally cut in two. The two segments there must be congruent to each other.
Two important things to know very distinctly about Isosceles Traingle
The Median, angle bisector, and altitude are the same segment. (therefore the segment forms 90d, bisects the angle, and bisects the base) -- USEFUL for when proving triangle congruence. Base angles and sides of an isosceles triangle are congruent
What is a contrapositive?
The contrapositive switches the hypothesis and conclusion (it's the converse) but also uses two negatives/opposite of the statements.
What is the only exception to the SSA combination?
The special case is when the triangle is a right triangle. The swinging door side cannot pivot to touch a ray in two different places. It goes straight down and is the shortest possible segment.
Triangle Inequality Theorem (applies to side lengths, not angle measures!)
The sum of the lengths of any to sides of a triangle must exceed the length of the remaining side.
What was Justine's useful little poem?
Two S's in a row means no. If it ends in an S, that means yes. HL and ASA are okay.
The SAS post operates on a past postulate. What is it?
Two points form a line.
Polygon Congruence Definition (Biconditional)
Two polygons are congruent if and only if there is a correspondence between their sides and angles such that... 1. Each pair of corresponding angles is congruent 2. Each pair of corresponding sides is congruent
If the diagonals of a quadrilateral bisect each other, then the quad is a pgram. PROVE
Use SAS postulate, CPTC. Say, if one pair of opp sides are both parallel and congruent, then the quad is a parallelogram.
Say you have two triangles connected at the vertices (like a bowtie). How do you prove that the middle meeting point is the midpoint of that long segment?
Use a triangle congruence theorem (ASA, SAS, SSS, AAS, and HL) to prove that triangles are congruent. Then use CPCTC to prove that the corresponding parts of congruent, since the triangle is congruent. Then use the "definition of midpoint" to prove that that point really is a midpoint.
How do you know if there is only one unique triangle from a set of measures given to you?
Use all four postulates (SSS, ASA, SAS, and AAS) but not SSA or AAA to see if you can prove and determine one unique triangle. (look for "inclusion" of a side or angle)
How do you prove that the diagonals of a rhombus are perpendicular?
Use the definition of segment bisectors to prove that the two segments on either side of the bisecting point are congruent. Then two triangles sharing the side are proven congruent. Use the CPCTC that the two angles both equal each other. Then prove that according to the Linear Pair postulate, the two angles are supplementary. TWO ANGLES WHICH ARE CONGRUENT AND SUPPLEMENTARY form 90d. This is the definition of perpendicular.
How to prove that two lines are parallel to each other in a set of two triangles?
Use the given information to fulfill one of the Triangle Congruence postulates or theorems (SAS, ASA, SSS, AAS, or HL). Then use CPCTC to prove that angles are congruent, and finally a converse of Angles Theorem to prove lines are parallel.
How to prove that the opposite sides of a parallelogram are congruent given the two triangles inside?
Use the properties of a parallel to prove that sides are parallel, then you can use the various Angles Theorems, then reflexive property and use CPCTC.
How to prove that opposite angles of a parallelogram are congruent given the two triangles inside?
Use the properties of a parallelogram to prove that sides are parallel, then use the fact that angles are supplementary. Two angles supplementary to the same angle are congruent to each other. Thus opposite angles are congruent. You can also use SSS postulate and build on info from last proof, then use CPCTC
What are some congruence relations we can make about two triangles who share a vertex, who are surrounded by a square? Show congruence.
Verticle angles, opposite congruent parallel sides (due to being a square), Alternative interior angles if sides are parallel, and bisected segements which are congruent. All will be helpful to use in SSS, SAS, and ASA later on.
How do you prove that a Rhombus is a parallelogram when it's split into two triangles?
Well first prove that the triangles are congruent. It's definition of rhombus that all sides are congruent to each other. You can draw the split by saying that "two points determine a line." Use reflexive property, then SSS Post. The two Alt Int angles will be congruent to each other, by CPCTC. Then use the Converse of Alt Int Angles Theorem that the lines are parallel. This is the definition of a parallelogram; opposite sides are parallel.
When solving for measures of a triangle with algebra, what must you do?
Well first you may have to set two expressions equal to each other. For example, if you know the triangle is isosceles, then according to the Isosc Triangle Theorem, you know that the two base angles are also congruent. Next, set the expression equal to zero. Simplify if possible, factor out, then solve. You HAVE to substitute both values in to determine extraneous!!! (pls read all directions)
If triangle is in a regular polygon, how do you prove the inner triangle is isosceles?
Well, know that the polygon is equiangular, and equilateral, so prove that the two triangles are congruent by one of the postulates (ASA, SAS, SSS, AAS, and HL). Again use CPCTC, and then prove that df of isosceles is that the two lengths are congruent. ****DOn'T TRY TO PROVE THAT THE TRIANGLE IS CONGRUENT TO IT'S REFLECTION, because we're already given other triangles.
How do you write a proof proving if two triangles are congruent using all postulates
Write out a proof that states congruence of 'corresponding" sides and angles, and state your reasoning why. EX - (s) segment ES is congruent to segment ES because of the Reflexive Property or if you are given information , for example, EX - a rule that says a segment is bisected, you know that the two sides touching of the midpoints have congruent length...
If 2 sides and the included angle are fixed, then is the size and shape of the triangle fixed?
Yes, the last two angles are connected only by one unique segment. So they are always fixed, or rigid.
Is this info sufficient to determine triangle congruence? "Any two corresponding angles and one corresponding side are congruent."
Yes, we see this according to the ASA and AAS Postulate. That one side doesn't even have to be included by the angles.
Prove all three! ++++++++++++ If a segment is the angle bisector of the vertex angle of an isosceles triangle, then it is also the median and the altitude. If a segment is the altitude of the vertex angle of an isosceles triangle, then it is also the median and the angle bisector. If a segment is the median of the vertex angle of an isosceles triangle, then it is also the altitude and the angle bisector.
You have to first prove that two triangles are congruent, and then go from there to prove that the segment shows qualities of an angle bisector, median, or altitude!!
(4.4) Using triangle congruence to prove characteristics of a triangle, to prove triangle types, or various parts of a triangle.
You might prove midpoint, if a triangle is isosceles or not, if a segment/angle/triangle is congruent to another segment, if two lines in a diagram are parallel.
If two angles are congruent and supplementary, then they are...
both 90 degrees
Two angle measures might not be the same size,...
but if they are the same measure, they are just congruent.
(4.1-4.2) Reflection produces ________ figures
congruent figures.
If one pair of opposite sides of a quadrilateral are BOTH parallel and Congruent, then the quadrilateral is a parallelogram. PROVE
draw in diagonal, do SAS postulate to use CPCTC> Say "If two pairs of opposite sides are congruent on a quad, then the quad is a pgram."
An equiangular triangle means that the triangle is also...
equilateral, because same angles determine a rigid shape, and that's alll you need.
How is the Isosceles Triangle Theorem angle proven (because you usually are given 2 triangles, not 1!) ? Use the reflection of the triangle to prove it is congruent to itself.
if the two sides of the triangle are congruent, you have two sets of sides. the base segment is equal to itself by the reflexive property. Now that you have proved the triangle congruent, use the CPCTC.
In an isosceles Triangle, the altitude, median, and angle bisector of the vertex angle...
is the same segment.
(Chapter 4 Quiz Review) If pentagons FROWN ≅ SMILE, then segOW ≅ then ∠M
seg IL ∠R
First prove two CONGRUENT TRIANGLES, then prove that...
that Corresponding Parts are Congruent.
Just because two segments are parallel in a diagrma, doesn't mean
the triangles are congruent.
If we have the 2 pentagons, how many possible statements of congruence can we make?
there are 10 statements of congruence (sides + shape = 10)
How do you prove that the diagonals of a rectangle are congruent?
use properties of a parallelogram, since a rectangle is a parallelogram. Prove that two triangles are congruent, then CPCTC that the segments on either side of the bisecting point are congruent.
How do prove that a rectangle is a parallelogram?
use the property that rectangles have 90d angles and are perpendicular to each other. Then prove that two lines perpendicular to the same line are parallel to each other. This is the definition of a parallelogram; opposite sides are parallel.
