Congruent Triangles 4-1: Classifying Triangles 4-2: Angle Measure in Triangles 4-3: Congruent Triangles 4-4 & 4-5: Tests for Congruent Triangles 4-7: Isosceles Triangles Home Next

4-1: Classifying Triangles The Parts of a Triangle Triangle ABC can be written as ABC and is made up of the following parts. Sides: AB, BC, AC are line segments Vertices: A, B, and C are points Angles: BAC or A, ABC or B, and BCA or C Back Next

4-1: Classifying Triangles Classifying by Angles Acute Triangle Obtuse Triangle In an Acute Triangle all the angles are acute, less than 90. In an Obtuse Triangle there is only one obtuse angle, an angle greater than 90. Right Triangle A Right Triangle has one 90 degree angle. Back Next

4-1: Classifying Triangles Right Triangles In Right Triangle XYZ, the sides XY and YZ are called the legs, and side XZ is called the hypotenuse. Leg Hypotenuse Equiangular Triangles Leg In an Equiangular Triangle all three angles are congruent to one another. Back Next

4-1: Classifying Triangles Classifying by Sides Isosceles Triangle Scalene Triangle In an Isosceles Triangle at least two of the sides are congruent. In an Scalene Triangle all the sides have different lengths Equilateral Triangle In an Equilateral Triangle all the sides are congruent to each other Back Next

4-1: Classifying Triangles Isosceles Triangles Leg Base Vertex Angle Base Angles In an Isosceles Triangle the two congruent sides are called legs and form the vertex angle. The side opposite the vertex angle is called the base. The base and each leg form two base angles. * The base angles of an isosceles triangle are congruent. Back Section 4-2

4-2: Angle Measures in Triangles Theorem: The sum of the measures of the angles of a triangle is 180. Given: EFG mE + mF + mG = 180. Topic 4 Next

4-2: Angle Measures in Triangles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. Given EFG and PQR: If E  R, and F  Q, then G  P Back Next

4-2: Angle Measures in Triangles Definitions: An Exterior Angle is formed by one side of a triangle and another side extended. The angles of the triangle not adjacent to a given exterior angle are called Remote Interior Angles. Given: ZXY an interior angle of XYZ. WXZ is the corresponding exterior angle. Z and Y are the resulting remote interior angles Given: ZXY Exterior Angle Remote Interior Angles Back Next

4-2: Angle Measures in Triangles Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.  mWXZ = mZ + mY Back Next

4-2: Angle Measures in Triangles * Corollary Given Right Triangle ABC The acute angles of a right triangle are complementary. * Corollary There can be at most one right or obtuse angle in a triangle. mA + mB = 90. * A Corollary is a statement that follows directly from another theorem and that can be easily proved from that theorem. Back Next

4-3: Congruent Triangles * Definition: Two triangles are congruent if and only if their corresponding parts are congruent. ABC  XYZ if and only if the three corresponding sides are congruent and the three corresponding angles are congruent Sides AB  XY BC  YZ AC  XZ Angles A  X B  Y C  Z *This definition can be abbreviated CPCTC which means Corresponding Parts of Congruent Triangles are Congruent. Topic 4 Next

4-3: Congruent Triangles Theorem Congruence of Triangles is Reflexive, Symmetric and Transitive. Reflexive Property Symmetric Property Transitive Property ABC  ABC If ABC  DEF, then DEF  ABC If ABC  DEF, and DEF  GHI, then ABC  GHI Section 4-4 Back

4-4 & 4-5: Tests for Congruent Triangles Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. If AB  XY, BC  YZ, and AC  XZ, then ABC  XYZ. Topic 4 Next

4-4 & 4-5: Tests for Congruent Triangles Side-Angle-Side (SAS) Postulate If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent.   If PQ  XY, QR  YZ, and PQR  XYZ, then PQR  XYZ. Back Next

4-4 & 4-5: Tests for Congruent Triangles Angle-Side-Angle (ASA) Postulate If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent.     If DE  PQ, D  P, and E  Q, then DEF  PQR. Back Next

4-4 & 4-5: Tests for Congruent Triangles Angle-Angle-Side (AAS) Postulate If two angles and an non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.     If D  P, F  R, and DE  PQ, then DEF  PQR. Back Next

4-4 & 4-5: Tests for Congruent Triangles Sample Proof Given: R and T are Right Angles. 1  2 Prove: RV  TV RV  TV 1. R and T are Right Angles. 1  2 1. Given 2. R  T 2. Right Angles are Congruent 3. SV  SV 3. Reflexive 4. RSV  TSV 4. AAS 5. CPCTC 5. RV  TV Back Section 4-7

4-7: Isosceles Triangles Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. If AB  AC, then B  C  Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If B  C, then AB  AC Topic 4 Next

4-7: Isosceles Triangles Corollary A triangle is equilateral if and only if it is equiangular If XY  YZ  ZX, then X  Y  Z If X  Y  Z, then XY  YZ  ZX. Corollary Each angle of an equilateral triangle measures 60 degrees. If XYZ is equilateral then, mX = mY = mZ = 60. Back Applying Congruent Triangles