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.

Side and Angle Congruence ABCD EFGH Sides: Angles: AB EF <A <E BC FG <B <F CD GH <C <G DA HE <D <H

4.2 Triangle 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.

Given: ABCD is a rhombus Prove: ABD DBC Statements Reasons ABCD is rhombusGiven AB BC CD DADefinition of Rhombus BD BDReflexive ABD DBCSSS

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.

Given: AB//CD AB CD Prove: ABD CBD StatementsReasons AB//CD AB CD Given <BDC <ABDAlternate Interior Angle DB DBReflexive ABD CBDSAS

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 StatementsReasons <A <E AC CEGiven <ACB <DCBVertical Angles ABC CDEASA

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.

Given: AD AE <C <B Prove: BAD CAE Statements Reasons AD AE <C <BGiven <DAB <EAC Reflexive BAD CAEAAS

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.

Given: ABC is isosceles BD perpendicular CA Prove: ABD CBD Statements Reasons ABC is isoscelesGiven BD perpendicular CAGiven AB BCDefinition of Isosceles <BDA= 90°Definition of Perpendicular <BDC=90° Definition of Perpendicular <BDA <BDCTransitive BD BDReflexive ABD CBD

4.4 Isosceles 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.

Given: AB BC Prove: <A <B StatementsReasons AB BCGiven DB is an angle bisector Construction <ABD <CBDDefinition of Angle Bisector DB DB Reflexive ABD CBDSAS <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°.

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