Stuck on 4.1 – 4.4?

Katalina Urrea and Maddie Stein ;)

Vocabulary• Base angle- angles whose vertices are the endpoints

of the base• Base of an isosceles triangle- the angles whose

vertices are the endpoints of the base of an isosceles triangle

• CPCTC- Abbreviation for “corresponding parts of congruent triangles are congruent”

• Corollary- A theorem that follows directly from another theorem and that can easily be proved from that theorem

• Isosceles triangle- A triangle with at least two congruent sides

• Legs of an isosceles triangle- The two congruent sides of an isosceles triangle

• Vertex angle- The opposite angles formed by two intersecting lines.

4.1

Congruent Polygons

Polygon Congruence Postulate

• Two polygons are congruent IFF (if and only if) there is a correspondence between their sides and angles such that:

-Each pair of corresponding angles are congruent

-Each pair of corresponding sides are congruent

• (Converse is true as well)

Naming Polygons

• You must name polygons in order

• The name of this polygon

is ABCDEF

• You can also name it

BCDEFA, CDEFAB

and so on, but you MUST

keep it in order.

F

A

B

C

D

E

Side and Angle Congruence

D

A

B

C

H

E

G

F

ABCD EFGH

Sides: Angles:

AB EF <A <E

BC FG <B <F

CD GH <C <G

DA HE <D <H

4.2Triangle

Congruence

Side-Side-Side Postulate (SSS)

• If the sides of one triangle are congruent to the sides of another triangle then those triangles are congruent.

A

D

C

B

Given: ABCD is a rhombus

Prove: ABD DBC

Statements Reasons

ABCD is rhombus Given

AB BC CD DA Definition of Rhombus

BD BD Reflexive

ABD DBC SSS

Side-Angle-Side Postulate (SAS)

• If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then those two triangles are congruent.

A

D C

B

Given: AB//CD AB

CD

Prove: ABD CBD

Statements Reasons

AB//CD AB CD Given

<BDC <ABD Alternate Interior Angle

DB DB Reflexive

ABD CBD SAS

Angle-Side-Angle Postulate (ASA)

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

Given: <A <E AC CE

Prove: ABC CDE

C

D

B

A

E

Statements Reasons<A <E AC CE Given<ACB <DCB Vertical Angles ABC CDE ASA

4.3

Angle-Angle-Side Theorem (AAS)

• If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.

F

C

A

B

D E

Given: AD AE <C <B

Prove: BAD CAE

Statements ReasonsAD AE <C <B Given<DAB <EAC Reflexive BAD CAE AAS

HL (Hypotenuse-Leg) Congruence Theorem

• If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

B

A CD

Given: ABC is isosceles

BD perpendicular CA

Prove: ABD CBD

Statements Reasons ABC is isosceles GivenBD perpendicular CA GivenAB BC Definition of Isosceles<BDA= 90° Definition of Perpendicular<BDC=90° Definition of Perpendicular<BDA <BDC TransitiveBD BD Reflexive ABD CBD

4.4Isosceles Triangles

Isosceles Triangle Theorem (Base Angle Theorem)

• If two sides of the triangle are congruent, then the two angles opposite those sides are congruent.

• The converse is also true.

B

A CD

Given: AB BC

Prove: <A <B

Statements ReasonsAB BC GivenDB is an angle bisector Construction<ABD <CBD Definition of Angle BisectorDB DB Reflexive ABD CBD SAS<A <B CPCTC

Corollaries

1) The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

2) The measure of each angle in an equilateral triangle is 60°.

• http://www.washoe.k12.nv.us/ecollab/washoemath/dictionary/vmd/full/s/side-side-sidesss.htm

• http://www.ekacademy.org/mines/hspe/CreateHtm/htm/4-8-2_n-nevadan-4-2-3-1.htm