Chapter $ review

Chapter 4!!!

4-1 Classify Triangles: First there is classifying triangles by angles There is the acute triangle Where all of the angles are acute (M<90)

There is the obtuse triangle where one angle is obtuse(M>90)

There is the right triangle where one angle is right(M=90)

* An acute triangle with all angles congruent is an Equiangular Triangle

Classifying Triangles: Then theres classifying triangles by sides.

There is a scalene triangle where no two sides are congruent.

There is a isosceles triangle where at least two sides are congruent.

There is a equilateral triangle where all of the sides of the triangle are equal.

4-2

Angles of triangles: Angle sum theorem -The sum of the measures of the angles of a triangle is 180.

M< y + M< x+M<z=180

Proof: Angle sum theorem: X 1

Given: ABC Prove: M<C+M<2+M< B=180 c

statements: 1.

ABC 2. draw XY Through A Parallel to CB 3. <1 and <CAY form a linear pair 4.<1 and <CAY are supplementary 5.m<1+m<CAY=180 6.m<CAY=m<2+m<3 7.m<1+m<2+m<3+180 8.<1= <C, <3=<B 9.m<1=m<C, m<3=m<B 10.m<C+m<2+m<B=180

Third Angle Theorem:

A 2

Y 3 b

Reasons 1.Given 2.Parallel Postulate 3.Def. Of a linear pair 4.If 2<'s form a linear pair,they are supp 5. Def of supp. <'s 6. Angle Addition Postulate 7. substitution 8. Alt Int <'s Theorem 9. Def of == <'s 10.Substitution

If two Angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

C d c d B E B E A F A F ex:If <A= <F and <C = <D, then <B = <E

Exterior Angle Theorem: The Measure of an exterior angle of a triangle is equal to the measures of a triangle is equal to the sum of the measures of two remote interior angles.

Y

X ex: m<YZP= m<X+ m<Y

Z

P

4-3

Congruent Triangles: These are triangles that are the same size and shape

Definition of Congruent Triangles (CPCTC): Two triangles are congruent if and only if their corresponding parts are congruent.

Corresponding parts of congruent triangles are congruent

Properties of Triangle Congruence: Congruence of triangles is reflexive, symmetric, and transitive .

JKL =

Reflexive

JKL

Symmetric

Transitive

If tri. JKL= Tri. PQR,

If tri. JKL = tri. PQR, And tri PQR

then tri. PQR = tri. JKL

= tri. XYZ, then tri. JKL= tri XYZ

4-4

Proving Congruence-------SSS, SAS

Side-Side-Side Postulate- If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

ABC = DEF

Using SSS in proofs Given: AB = AC; BY = CY Prove: BYA = CYA Statements 1. AB = AC ; BY = CY 2. AY = AY 3. BYA = CYA

b

y

Reasons 1. Give.

2. Reflexive Property 3. SSS

c

a

Side-Angle-Side Congruence: 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.

Given: X is the midpt. Of BD; X is the midpt of AC Prove: DXC = BXA

d

a x

Flow Proof:

X is the midpt. of BD

Given X is the midpt. Of AC

Given <DXC = <BXA vert. <'s are cong.

DX = BX Midpt. theorem CX = AX midpt. Theorem

DXC = SAS

BXA

4-5

Proving Congruence ASA, AAS: Angle-Side-Angle Congruence- If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. c r Using ASA in a proof Given: CP bisects <BCR and <BPR Prove: BCP = RCP

b p

Proof: Since CP bisects <BCR and <BPR, <BCP is cong. To <RCP and <BPC is cong. To <RPC. CP is cong. To CP by the reflexive property. BY ASA tri. BCP is cong. To tri RCP

Angle-Angle-Side Congruence- If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle , then the two triangles are congruent.

Given: <EAD = <EBC, AD = BC Prove: AE = BE Flow Proof: <EAD = <EBC Given AD = BC Given <E = <E Reflx. Prop.

ADE = BCE AAS

AE = BE CPCTC

4-6

Isosceles Triangles: Isosceles Triangle Theorem- If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

R S P Using IST as a proof: Given: PQR, PQ = RQ Prove <P = <R Q Statements 1. Let s be the midpt. Of PR 2.Draw an auxiliary segment QS 3.PS = RS 4.QS = QS 5.PQ = RQ 6. PQS = RQS 7.<P = <R

Reasons 1. Every seg. Has exactly one midpt.

2. Two points determine a line 3. Midpt. theorem 4. Congruence of segments is reflexive 5. Given 6.SSS 7.CPCTC

IF two angles of a triangle are congruent, then the sides opposite those angles are congruent.

E

H P

C Name two congruent angles: <ICE = <IEC Name two congruent segments: PE = PC

I

Corollaries- A triangle is equilateral if and only if its equiangular Each angle of an equilateral triangle measures 60deg. â&#x20AC;˘ When you have to find the measures of an equilateral triangle all you do is solve for X by using what you learned in algebra 2.

4-7 Triangles and Coordinate Proof: When placing figures on the coordinate plane youst 1 . Use the origin as a vertex of the figure nd 2 . Place at least one side of a polygon on an axis rd 3 . Keep the figure within the first quadrant if possible th 4 . Use coordinates that make computations as simple as possible. y B(0,2b) p A(0,0) o Given: Right

C(2c,0)

x

ABD with right BAD Q is the midpt. Of BD

Prove: AP = 1/2BD Proof: By the midpt. Formula, the coordinates of Q are (0+2d/2, 2b+0/2) or (D, B).

Use the distance formula to find AP and BD. AP=

(d-0)^2 + (b-0)^2 = d^2+b^2

Therefore, AP = 1/2BC

BC= (2d-0)^2 + (0-2b)^2 BC= 4d^2 + 4b^2 or 2 1/2 BC= c^2 + b^2

c^2 + b^2