CHAPTER 4 Congruent Triangles

SECTION 4-1 Congruent Figures

Congruent Triangles – two triangles whose vertices can be paired in such a way so that corresponding parts (angles and sides) of the triangles are congruent.

Congruent Polygons – two polygons whose vertices can be paired in such a way so that corresponding parts (angles and sides) of the polygons are congruent.

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 If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent (SSS)

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)

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)

SECTION 4-3 Using Congruent Triangles

A Way to Prove Two Segments or Two Angles Congruent 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 

SECTION 4-4 The Isosceles Triangle Theorems

THEOREM 4-1 If two sides of a triangle are congruent, then the angles opposite those sides are congruent

Corollary 1 An equilateral triangle is also equiangular.

Corollary 2 An equilateral triangle has three 60° angles.

Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

THEOREM 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollary An equiangular triangle is also equilateral.

SECTION 4-5 Other Methods of Proving Triangles Congruent

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)

Hypotenuse – is the side opposite the right angle of a right triangle. Legs – the other two sides

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)

Ways to Prove Two Triangles Congruent All triangles – SSS, SAS, ASA, AAS Right Triangle - HL

SECTION 4-6 Using More than One Pair of Congruent Triangles

SECTION 4-7 Medians, Altitudes, and Perpendicular Bisectors

Median – is the segment with endpoints that are a vertex of the triangle and the midpoint of the opposite side

Altitude – the perpendicular segment from a vertex to the line containing the opposite side

Perpendicular bisector – is a line, ray, or segment that is perpendicular to a segment at its midpoint

THEOREM 4-5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment

THEOREM 4-6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of a segment.

Distance from a point to a line – is the length of the perpendicular segment from the point to the line or plane

THEOREM 4-7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

THEOREM 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.

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