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Transcript
Sec. 4.1 Triangles and Angles Triangle – a figure formed by three segments joining three non-collinear points. vertex – each of the three points joining the sides adjacent sides – two sides sharing a common vertex legs – sides that form the right angle in a right triangle hypotenuse – side opposite the right angle in a right triangle legs (in an isosceles triangle) – two congruent sides in an isosceles triangle with only two sides congruent base – side that is not congruent to the two congruent sides in an isosceles triangle. interior angles – the three original angles in a triangle. exterior angles – When the sides of a triangle are extended, angles are formed. The angles adjacent to the interior angles are exterior. Thm. 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180. Thm. 4.2 Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. corollary to a theorem – a statement that can be proved easily using a theorem. Corollary to the triangle sum theorem : The acute angles of a right triangle are complementary. Sec. 4.2 Congruent Triangles Congruent figures have exactly the same shape and size. Corresponding angles – angles that match up with each other from one figure to the next. Corresponding angles must be congruent in congruent figures. Corresponding Sides – sides that match up with each other from one figure to the next. Corresponding sides must be congruent in congruent figures. Thm. 4.3 Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then their third angles are congruent. Examples 1. Name all of the congruent sides. 2. Name all of the congruent angles. NPLM EFGH 3. Find x. 4. Find y. 5. Find x. 6. Find mM. Sec. 4.3 Proving Triangles Congruent : SSS and SAS Q : How much do you need to know about two triangles to prove that they are congruent? Do you need to know that all corresponding sides are congruent and all corresponding angles are congruent in order to prove that the triangles are congruent? We have several theorems and postulates we use to prove triangles are congruent. Post. 19 : Side–Side–Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Once you prove the triangles are congruent using SSS, then you can state that the corresponding angles are congruent by ‘Corresponding Parts of Congruent Triangles Are Congruent’ or CPCTC. Post. 20 : Side-Angle-Side (SAS) Congruence Postulate If two sides and their included angle of one triangle are congruent to two sides and their included angle of another triangle, then the triangles are congruent by SAS. Once you prove the triangles are congruent using SAS, then you can state that the third pair of corresponding sides and the other two pairs of corresponding angles are congruent by CPCTC. ***When proving triangles congruent, always look for shared sides and vertical angles. Shared sides and vertical angles are automatically congruent because of reflexive property and the “vertical angles are congruent” theorem. Examples : 1. Prove PQW TSW 2. Prove AEB DEC 3. Prove PQR PSR Sec. 4.4 Proving Triangles are Congruent : AAS and ASA Post. 21 : Angle-Side-Angle (ASA) Congruence Post. If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. Thm. 4.5 Angle-Angle-Side (AAS) Congruence Thm. If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. Examples : Is it possible to prove the triangles congruent? If so, state the thm. or post. and write the congruence. Given : AD || EC ; BD BC Prove : ABD EBC Sec. 4.6 Isosceles, Equilateral, and Right Triangles base angles – two angles adjacent to the base in an isosceles triangles vertex angle – angle included by the two legs Thm. 4.6 Base angles theorem If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Thm. 4.7 Converse of the base angles theorem If two angles of a triangle are congruent, then the sides opposite those angles are also congruent. Corollary to Thm. 4.6 An equilateral triangle is also equiangular. Corollary to Thm. 4.7 An equiangular triangle is also equilateral. Thm. 4.8 Hypotenuse-Leg (HL) Congruence Thm. If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.