4.6+Isosceles,+Equilateral,+Right+Triangles

In lesson 4.6 you learn how to find if angles and sides are congruent without having the numbers. 4.6 also teaches the hypotenuse leg theorem. That is when the hypotenuse and a leg are congruent to the ones on another triangle, so the triangles are congruent. The goals of this section are to understand the properties of there triangles. These theorems can be used to help solve proofs, find algebraic variables, lengths of sides, degrees of angles, and prove congruence.

Base angles- the angles on the side of the base. Vertex Angle- angle where the two legs meet. -

Theorem 4.6- Base Angles Theorem Theorem 4.7- Converse of the Base Angles Theorem If a triangle is equiangular, then it is equilateral.
 * If two sides of a triangle are congruent, then the angles opposite them are congruent. (textbook page 236)
 * [[image:http://images.classwell.com/ebooks/images/mcd_geo/mcd_ma_geo_lsn_0395937779_p236_f05.gif align="top"]]
 * If a triangle is equilateral, then it is equiangular.
 * If two angles of a triangle are congruent, then the two sides oppisite to them are congruent.(textbook page 236)[[image:http://images.classwell.com/ebooks/images/mcd_geo/mcd_ma_geo_lsn_0395937779_p236_f06.gif width="114" height="69" align="top"]]

Theorem 4.8- Hypotenuse leg Congruence Theorem
 * If the hypotenuse and one of the legs are congruent to the hypotenuse and one of the legs of another triangle, then the triangles are congruent.(textbook page 238)[[image:http://images.classwell.com/ebooks/images/mcd_geo/mcd_ma_geo_lsn_0395937779_p236_f13.gif align="top"]]
 * If [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_bc.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/congruent.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_ef.gif align="absmiddle"]] and [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_ac.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/congruent.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_df.gif align="absmiddle"]], then Δ//ABC// [[image:http://images.classwell.com/ebooks/images/mcd_geo/congruent.gif align="absmiddle"]] Δ//DEF//.

Example problems: > > 3. Find the value of X. Links to help: http://www.mathwarehouse.com/geometry/congruent_triangles/isosceles-triangle-theorems-proofs.php http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenuse-leg-theorem.php
 * 1) [[image:http://images.classwell.com/ebooks/images/mcd_geo/mcd_ma_geo_lsn_0395937779_p236_f23.gif align="top"]]Solve for X and Y and say what theorems you used.
 * || [[image:http://images.classwell.com/ebooks/images/mcd_geo/given_arrow.gif align="absmiddle"]] || //D// is the midpoint of [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_ce.gif align="absmiddle"]], [[image:http://images.classwell.com/ebooks/images/mcd_geo/angle.gif align="absmiddle"]]//BCD// and [[image:http://images.classwell.com/ebooks/images/mcd_geo/angle.gif align="absmiddle"]]//FED// are right angles, and [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_bd.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/congruent.gif align="absmiddle"]] [[image:http://images.classwell.com/ebooks/images/mcd_geo/linesegment_fd.gif align="absmiddle"]]. ||
 * [[image:http://images.classwell.com/ebooks/images/mcd_geo/prove_arrow.gif align="absmiddle"]] || Δ//BCD// [[image:http://images.classwell.com/ebooks/images/mcd_geo/congruent.gif align="absmiddle"]]Δ//FED// ||  ||