FINAL EXAM REVIEW Chapter 4 Key Concepts

Chapter 4 Vocabulary congruent figures corresponding parts equiangular Isosceles Δ legsbase vertex angle base angles medianaltitude perpendicular bisector CONGRUENCE METHODS: SSSSASASAAASHL

∆ ABC ∆ DEF Defn. of Congruent Triangles Two triangles are congruent ( ) if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangles are congruent. A BC D EF 7 A D 7 7 B E 7 7 C F 7 ABDE BCEF CAFD ORDER MATTERS!

SSS Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. A B C S R T ABC = RST by SSS Post. ~

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 triangles are congruent. E F G Q P R EFG = PQR by SAS Post. ~

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 triangles are congruent. X Y Z M L N XYZ = LMN by ASA Post. ~

The AAS (Angle-Angle-Side) Theorem X Y Z A C B ABC If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. XYZ

The HL (Hypotenuse - Leg) Theorem ABC If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. XYZ A BC X Y Z

Summary of Ways to Prove Triangles Congruent All triangles Right triangles SSS Post SAS Post ASA Post AAS Thm HL Thm

The Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Iso. Thm.

Converse to Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Converse to Iso. Thm.

Corollaries An equilateral triangle is also equiangular. An equilateral triangle has three 60 o angles. The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

Median A median of a triangle is a segment from a vertex to the midpoint of the opposite side. Each triangle has three medians. A C B A C B A C B...

Altitude The perpendicular segment from a vertex to the line that contains the opposite side. A B C A B C A B C Acute Triangles A B C A B C A B C Right Triangles A B C A B C A B C Obtuse Triangles

Perpendicular Bisector A line, ray, or segment that is perpendicular to a segment at its midpoint.

Theorem If a point lies on the perpendicular bisector of a segment, then… the point is equidistant from the endpoints of the segment.... If a point is equidistant from the endpoints of a segment, then… the point lies on the perpendicular bisector of the segment. CONVERSE:

Theorem If a point lies on the bisector of an angle then,… the point is equidistant from the sides of the angle.. If a point is equidistant from the sides of an angle, then…..the point lies on the bisector of the angle. CONVERSE:

Homework ► Chapter 3-4 Review Olympics W/S ► pg. 164 #1-9 (multiple choice)