4.1 Classifying Triangles

Classifying Triangles By Sides Scalene Δ A triangle with no two sides congruent. Isosceles Δ A triangle with two or mores sides congruent. Equilateral Δ A triangle with all sides congruent. By Angles Acute Δ A triangle with all acute angles. Obtuse Δ A triangle with one obtuse angle. Right Δ A triangle with one right angle. EquiangularΔ A triangle with all angles congruent.

Combinations of Classifications Acute Δ Scalene Δ Obtuse Δ Isosceles Δ Right Δ Equilateral Δ Equiangular Δ

What Can You Determine? Distance Formula? Lengths of segments, thus you can determine if segments are congruent or not. Helps with classification by sides? Can it help with classification by angles? Pythagorean Theorem… If leg2 + leg2 = Hypt2, then it is a Rt Δ What can slopes help you with? Slopes can help determine if a Rt Δ

4.2 Angles of Triangles

Angle Sum Theorem Angle Sum Theorem – The sum of the measures of the angles of a triangle equals 180°. B C A Three adjacent angles – the sum of their measurements is 180

Third Angle Theorem The Third Angle Theorem (No Choice Theorem) states if two angles in one triangle are congruent to two angles in another triangle, then the third set of angles are also congruent. 90° 90° 60° 30° 60° 30°

Exterior Angles Exterior Angle – An angle made between one side of a triangle and the extension of the other side. 5 6 1 3 2 4 There six exterior angles in a triangle, two per vertex.

Exterior Angle Theorem Exterior Angle Theorem – The measure of the exterior angle of a triangle is equal to the sum of the two remote interior angles. C Triangle Sum Theorem m<2 + m<A + m<C = 180 <1 and <2 are LP, therefore Supplementary m<2 + m<1 = 180 1 2 A B Substitution: m<2 + m<1 = m<2 + m<A + m<C Add/Subt: m<1 = m<A +m<C

Triangle Corollaries Corollaries are just like theorems but are easily proved. Two Triangle Corollaries are: The acute angles of a right triangle are complementary. There can be at most one right or obtuse angle in a triangle (Euclidian Geometry) A m<A + m<B + m<C = 180° ΔSum Thrm m<A + 90° + m<C = 180° Subst. B C m<A + m<C = 90° Add/Sub

Flow Proofs Flow proofs are the 2nd and last formal proof that we will study. Similarities with the two column proof is that each element has a statement and reason. Two column proofs work well for “linear” type proofs – in other words, one step follows another, etc. Flow proofs work better for “non-linear” proofs – in other words, the order is less defined.

Old Proof BC = BC AB = CD Reflexive Given AB + BC = AC BC + CD = BD Given: AB = CD Prove AC = BD AB + BC = AC BC + CD = BD SAP AB + BC = BC + CD Add/Subt AC = BD Substitution

4.3 Congruent Triangles

Congruent Triangles Definition of Congruent Segments – Are Segments that have the same measurement. There is only one measurement for a segment. How many parts are in a triangle? Six? Three segments and three angles. So, Congruent Triangles are triangles where all SIX corresponding parts are congruent. CPCTC – Corresponding Parts of Congruent Triangles are Congruent.

Congruent Triangles A T C O G D Δ congruency statement

Important Concepts Unlike the order of the letters of an angle, the order of the letters of the triangles matters. <ABC is congruent to <CBA ΔABC may or may not be congruent to ΔCBA b/c <A may not be congruent to <C etc…. When you write a triangle congruency statement make sure the corresponding parts in fact are congruent.

Properties of Triangle Congruence Reflexive: ΔABC is congruent to ΔABC Symmetric: If ΔABC is congruent to ΔXYZ, then ΔXYZ is congruent to ΔABC. Transitive: If ΔABC is congruent to ΔXYZ, and ΔXYZ is congruent to ΔLMN, then ΔABC is congruent to ΔLMN.

4.4 Proving Congruence – SSS and SAS

Shortcuts Previously – in order to prove triangles were congruent to each other you needed to prove all three sets of angles and all three sets of sides were congruent. There are 8 shortcuts that can be used to prove triangles congruent. Today we’re going to use two of them: SSS (Side – Side – Side) SAS (Side – Angle – Side)

SSS SSS – If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. T N R S L M

SAS SAS – If two sides and one included angle of one triangle are congruent to the two sides and one included angle of the other triangle, then the triangles are congruent. T N R S L M

Important Concepts In the order of the proof you must have three sets of congruent marks (for sides and/or angles) BEFORE you can say that the triangles are congruent. Once you say that the triangles are congruent, then you can say that any other part of the triangle can be congruent by CPCTC.

Example N T R S L M Given SSS CPCTC

Flow Proof Hints: Make each piece of the given it’s own line down. Make the stuff you can get from the pictures (Vert Angles, LP, SAP, AAP, etc..) their own line down too. See example on next slide.

Example D Given: Prove: A B C

4.5 Proving Triangles Congruent by ASA and AAS.

ASA and AAS There are two more ways to prove triangles are congruent. ASA (Angle – Side – Angle) – If two angles and the included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent. AAS (Angle – Angle – Side) If two angles and a non included side of one triangle are congruent to two angles and a non included side in another triangle, then the triangles are congruent.

Example ASA Z X Y Here we have two sets of congruent angles that are congruent along with the included sides that are congruent, therefore the two triangles are congruent by ASA. N L M

Example AAS Z X Y Here we have two sets of congruent angles that are congruent along with the non included sides that are congruent, therefore the two triangles are congruent by AAS. N L M

Important Reminders If you’re trying to prove triangles congruent, you MUST have three sets of corresponding parts that are congruent BEFORE you can say that the triangles are congruent. (SSS, SAS, ASA and AAS) If you don’t have three sets of parts that are congruent, you can’t prove the triangles congruent. After you prove the triangles congruent, you can use CPCTC to prove any of the unused parts congruent.

General Flow Proof ? ? ? ASA CPCTC

Proving Right Triangles Congruent

Four Additional Ways I told you there were 8 short cuts to proving triangles congruent. Four ways that work for all triangles are SSS, SAS, ASA and AAS. The other four ways work for Right Triangles only. They are HA, LL, LA, and HL. S was for sides, and A was for angles. H is for Hypotenuse, L is for Leg and A is for ACUTE angle.

Process Notice that these four ways, HA, LL, LA, and HL only have two letters. That means you only need two sets of congruent marks to prove Right Triangles congruent. However, you need to tell me that they’re right triangles too. So, you still need three things…. Two sets of congruent marks on Right Triangles.

Examples HA – Hypotenuse Acute Angle LL – Leg Leg LA - Leg Acute Angle HL – Hypotenuse Leg

Similarities You will notice that SAS looks like LL if the sides are the legs. ASA looks like LA. AAS can look like HA or LA HL is the only Right Triangle Congruency Theorem that can not have a similar “all triangle” way to prove the triangles are congruent.

Right Triangle Flow Proof ? ? ? (HA, LL, LA HL) CPCTC

4.6 Isosceles Triangles

Parts of Isosceles Triangles Def – A triangle with two or more sides congruent. The parts have special names. C The Congruent Sides are called the Legs The included angle made by the legs is the Vertex Angle The angles opposite the legs are called the base angles A B The side opposite the vertex is the Base

Parts of Isosceles Triangles The key thing to remember is this: It doesn’t matter which way the triangle is oriented, the parts are all in relationship to the congruent sides. The Base is not on the bottom! The Vertex is not on the top!

Isosceles Triangle Theorem B ΔABC is Isosceles, with <B as the Vertex. Legs AB and BC are Congruent. Draw Auxiliary Line from B to D (D is MP of Segment AC. Segment AD is Congruent to DC (MP Thrm) A C D Segment BD is Congruent to itself (Ref) If Two sides of a triangle are congruent, then the angles opposite those sides are congruent ΔABD is Congruent to ΔCBD (SSS) <A is Congruent to <C (CPCTC) IF Δ then Δ

Converse of Isosceles Δ Thrm The Converse of the Isosceles Triangle Theorem is also true. If two angles of a triangle are congruent, then the sides opposite them are congruent. If.. then…

Triangle Corollaries A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60°

Equilateral Triangles Since an Equilateral Triangles are also Isosceles, each of the vertices of the triangle are Vertex angles. Each side is a Leg and a Base. All the properties of Isosceles Triangles exist for Equilateral Triangles as well.