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Transcript
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. A B F C E D Side and Angle Congruence E A ABCD EFGH H D G B F C 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. A Given: ABCD is a rhombus Prove: ABD DBC B D C Statements Reasons ABCD is rhombus Given AB BC BD BD ABD CD DA Definition of Rhombus Reflexive 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 B Given: AB//CD AB CD Prove: D C Statements AB//CD AB <BDC DB ABD ABD CBD Reasons CD <ABD DB Given Alternate Interior Angle Reflexive 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. D Given: <A Prove: <E AC ABC E C B A Statements <A <E AC CE <ACB <DCB ABC CDE Reasons Given Vertical Angles ASA CE CDE 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. B C Given: AD Prove: BAD AE <C CAE F E D A Statements AD AE <C <DAB <EAC BAD CAE <B Reasons Given Reflexive AAS <B 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 Given: ABC is isosceles BD perpendicular CA Prove: A ABD CBD C D Statements ABC is isosceles BD perpendicular CA AB BC <BDA= 90° <BDC=90° <BDA <BDC BD BD ABD CBD Reasons Given Given Definition of Isosceles Definition of Perpendicular Definition of Perpendicular Transitive Reflexive 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. B A Given: AB BC Prove: <A <B C D Statements AB BC DB is an angle bisector <ABD <CBD DB DB ABD CBD <A <B Reasons Given Construction Definition of Angle Bisector Reflexive SAS 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/was hoemath/dictionary/vmd/full/s/side-sidesidesss.htm • http://www.ekacademy.org/mines/hspe/Cr eateHtm/htm/4-8-2_n-nevadan-4-2-31.htm