Chapter 3: Congruent Triangles-2011

3.1 Congruent Triangle

If two triangles are congruent then corresponding parts of one triangle will be congruent to the corresponding parts of the second triangle.

The six congruence statements that can be made if the triangles are congruent.

When you name congruent triangles the corresponding congruent parts must be in the same position. Such as:

If two triangles are congruent and you need to give a reason for their parts being congruent write the letters CPCTC which stands for

Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Reflection over x axis Reflection over y axis Translation or slide Rotation

3.2 Three ways to prove congruent triangles

Proving Congruent Triangles

SSS Postulate

If three sides of a triangle are congruent to three sides of another triangle, then the two triangles are congruent. ABCDEF below.

ASA Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

SAS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

IMPOSSIBLE METHODS

Below are two methods that cannot be used and a description of why it is not possible to prove the triangles congruent

Two sides and an angle that is not between the two given sides of the triangle are congruent to the corresponding parts of the other triangle.

Three angles of one triangle are congruent to the corresponding parts of the other triangle. The triangles shown below have the same shape but are different sizes. Later we will use this method to prove the triangles are similar. (Same angle sizes but proportional sides)

3.3 CPCTC and Circles

A circle is the set of all points in a plane a given distance from a given point.

This distance is called the radius. (plural - radii)

All radii of a circle are congruent.

Proof example:

AC=AB=AD because all radii of a circle are congruent. BC = BD from the given information. The triangles will be congruent by SSS.

If 2 triangles are congruent and you need to state their parts are congruent. Use

CPCTC - corresponding parts of congruent triangles are congruent.

3.4Beyond CPCTC or Medians, Altitudes and Auxiliary lines.

Median of a triangle – The median of a triangle is the segment that joins a vertex of a triangle to the midpoint of the opposite side.

Every triangle has 3 medians.

The medians of a triangle meet in one point which is called the centroid of the triangle. It is also the center of mass.

The medians of a triangle always meet inside of the triangle.

Concurrent lines are lines that meet in exactly one point. The medians of a triangle are concurrent lines.

Altitudes of triangles – the altitude of a triangle is the segment drawn from a vertex perpendicular to the line containing the opposite side. An altitude is sometimes referred to as the height of the triangle.

The three altitudes of a triangle are concurrent and meet at a point that is referred to as the orthocenter.

Auxiliary lines – extra lines added in a diagram to aid in the solution of a proof.

An extra line can only be introduced if there is one and only one segment or line that can be drawn.

If you introduce a segment between two distinct points the reason given is the LINE POSTULATE. Through two points exactly one line can be drawn.

Below is a proof which needs an auxiliary line.

Given: AB = AC, BD = CD

Prove: mB = mC

Original diagram Introduce segment

Paragraph form of proof. Introduce AD by the line postulate. Since AB = AC , CD = BD and AD=AD by the reflexive property, the triangles are congruent by SSS. If the triangles are congruent then B=C by CPCTC

REVIEW for 2-8 through 3-4

Name the triangles that are congruent by the markings on the diagrams and which should be used to prove it. If it is not possible to prove with the information write not possible.method

1. 2.

methodmethod

ABDDCA –SAS JMNKML -ASA

3. 4.

methodmethod

not possible not possible

5. 6.

method method

A=A FHGFKG –SSS

AFCAED -ASA

7. What do the letters CPCTC stand for?

Corresponding parts of congruent triangles are congruent

Prove the following.

8. Given:

Prove:

1 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

10. 10.

9.Given:

Prove:

1.<1 and <2 are compl to <FPE 1.Given

2. 2.CCAC

3.AB=CD3.Given

4.AC=BD 4.Addition Property

5.PB= PC5.Given

6.P is the midpt of EB and CF6.Given

7.7.Multiplication

8.8.SAS Postulate

9.AF = DE 9.CPCTC

10.Given:

Prove:

1.CD is a median of BCE 1.Given

2.D is midpt of EB 2.Def. of median

3.ED =DB 3.Def. of midpt

4.D is midpt of AC 4.Given

5.DC=AD 5.Def. of midpt

6.CDE=ADB 6.Vert. <s are congruent

7. 7.SAS

11. Prove the Vertical Angle Theorem

If two angles are vertical then they are congruent.

See notes for proof.

3.5 Overlapping Triangles

Sometimes you are asked to prove segments or angles congruent but the triangles they are in that can be proven congruent overlap each other as below.

Suppose you are asked to prove AC = BD, the triangles they are in are ABC and DCB.

Note that side BC is the base of both triangles and would be a reflexive side in your proof

Example: Use diagram above.

3.6 Types of Triangles:

Triangles are classified by side or by angles.

Classification by sides:

Scalene triangles have no congruent sides.

Isosceles triangles have at least two congruent sides.

Parts of an isosceles triangle-

legs - refers to the congruent sides of the triangle

vertex angle - angle located at the intersection of the congruent sides

base - the third side of the triangle

base angles - angles located at the endpoints of the base

Equilateral Triangles have 3 congruent sides.

All equilateral triangles are isosceles

Classification of Triangles by angles -

Acute triangle - a triangle with 3 angles less than 90 degrees

Right Triangle- a triangle with one right angle

Obtuse Triangle - a triangle with one obtuse angle

3.7 Angle Side Theorems

ITT - If two sides of a triangle are congruent then the angles opposite those sides are congruent.

CITT - If two angles of a triangle are congruent then the sides opposite those angles are congruent

BABS - If two sides of a triangle are not congruent then the bigger angle is opposite the bigger side

BSBA - If two angles of a triangle are not congruent then the bigger side is opposite the bigger angle.

3.8 HL Postulate

If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of a 2nd right triangle then the triangles are congruent.

Review

1. If the perimeter of the triangle is 50, what is the value of x?6

2. Classify the triangle in question 1 by its sides.(Isos)

3. What is the name of a segment drawn from a vertex to the midpoint of the opposite side of a triangle? (median)

4. What is a centroid of a triangle?(pt of intersection of medians)

5. What are concurrent lines?(lines that intersect at one point)

6. Where are the altitudes of an obtuse triangle located? (2 outside the triangle - one inside)

7. Given isosceles triangle ABC with base BC. a) name the base angles(<B and < C)b) name the congruent sides (AB and AC)

8. In which type of triangle are the median and altitude the same segment?(Isosceles from the vertex angle)

Prove each of the following.

9.

1. < 1 = <2 and < 3 = < 4 1. Given

2. < 3 and < ABC are supplements 2. Linear pairs are supplements

< 4 and < AED are supplements

3. <ABC = < AED 3. SCAC

4. AB = AE 4. CITT - If two angles of one triangle are congruent then the sides opposite those angles are congruent.

5. Triangle ABC is congruent to Triangle AED 5. ASA

6. AC = AD 5. CPCTC

7. Triangle CDA is Isosceles 6. Definition of Isosceles

10.

1. < 1 = < 2, BH is perpendicular to FG 1. Given

2. <BFH = <1 and <BGH = <2 2. Vertical angles are congruent

3. <BFH = <BGH 3. Substitution or Transitive

4. BF = BG 4. CITT - if two angles of a triangle are congruent then the sides opposite those angles are congruent

5.<BHF and < BHG are right angles 5. Definition of perpendicular

6. Triangles BHF and BGH are right triangles 6. Definition of a right triangle

7 BH = BH 7. Reflexive Property.

8. Triangle BHF is congruent to Triangle BHG 8. HL

9. < 3 = < 4 9. CPCTC

11.

1. H is the midpoint of AG 1. Given

2. AH = HG 2. Definition of Midpoint

3. AB = FG and < HBF = < HFB 3. Given

4. BH = FH 4. CITT

5. Triangle ABH is congruent to Triangle GFH 5. SSS

6. < BAH = < FGH 6. CPCTC

7. AG = AG 7. Reflexive

8. Triangle BAG = Triangle FGA 8. SAS

9. AF = BG 9. CPCTC

12.

1. RX=SX=SY 1. Given

2. <SXY = < SYX 2. ITT

3. <SXY and < RXY are supplements 3. Linear Pairs are supplements

< SYX and < SYT are supplements

4. <RXY= < SYT 4. Supplements of congruent angles are congruent (SCAC)

5. XY = YT 5. Given

6. Triangle RXY is congruent to Triangle SYT 6. SAS

7. ST = RY 7. CPCTC

Draw an Isosceles right triangle