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CHAPTER 4 Congruent Triangles •Congruent – when two geometric figures have the same size and shape SECTION 4-1 Congruent Figures •Congruent Segments – segments with the same length •Congruent Angles – angles with the same measure •Congruent Triangles – •Congruent Polygons – two triangles whose vertices can be paired in such a way so that corresponding parts (angles and sides) of the triangles are congruent. two polygons whose vertices can be paired in such a way so that corresponding parts (angles and sides) of the polygons are congruent. 1 SECTION 4-2 Some Ways to Prove Triangles Congruent •Included angle – angle between two sides of a triangle •Included side – the side common to two angles of a triangle Postulate 12 •Interior angles – angles determined by the sides of a triangle •Exterior angle – an angle that is both adjacent and supplementary to an interior angle Postulate 13 • If two sides and the included angle of one triangle are congruent to two sides and the included angles of another triangle, then the triangles are congruent (SAS) •If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent (SSS) Postulate 14 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 (ASA) 2 A Way to Prove Two Segments or Two Angles Congruent SECTION 4-3 Using Congruent Triangles SECTION 4-4 The Isosceles Triangle Theorems •Angles at the base are called base angles and the third angle is the vertex angle 1. Identify two triangles in which the two segments or angles are corresponding parts. 2. Prove that the triangles are congruent 3. State that the two parts are congruent, use the reason Corresponding parts of ≅ ∆ are ≅ •Isosceles Triangle – is a triangle with two legs of equal length and a third side called the base and THEOREM 4-1 If two sides of a triangle are congruent, then the angles opposite those sides are congruent 3 Corollary 1 An equilateral triangle is also equiangular. Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint . Corollary Corollary 2 An equilateral triangle has three 60° angles. THEOREM 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. SECTION 4-5 An equiangular triangle is also equilateral. Other Methods of Proving Triangles Congruent 4 THEOREM 4-3 • 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 (AAS) THEOREM 4-4 • 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. (HL) SECTION 4-6 Using More than One Pair of Congruent Triangles •Hypotenuse – is the side opposite the right angle of a right triangle. •Legs – the other two sides Ways to Prove Two Triangles Congruent •All triangles – SSS, SAS, ASA, AAS •Right Triangle - HL SECTION 4-7 Medians, Altitudes, and Perpendicular Bisectors 5 •Median – is the segment with endpoints that are a vertex of the triangle and the midpoint of the opposite side •Perpendicular bisector – is a line, ray, or segment that is perpendicular to a segment at its midpoint •Altitude – the perpendicular segment from a vertex to the line containing the opposite side •Distance from a point to a line – is the length of the perpendicular segment from the point to the line or plane THEOREM 4-5 THEOREM 4-6 •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 a segment. 6 THEOREM 4-7 THEOREM 4-8 •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. END 7