Chapter 4 Notes 4.1 – Triangles and Angles B A C Scalene – No congruent sides A Triangle Three segments joining three noncollinear points. Each point is a VERTEX of the triangle. Segments are SIDES! Isosceles – At LEAST 2 congruent sides Equilateral – All sides congruent Acute – 3 acute angles Right – One right angle Obtuse – One obtuse angle Equiangular – all angles congruent B C A A is opposite the side BC Side AB is opposite C opposite means non - adjacent TRIANGLE SUM THEOREM The sum of the measures of the angles of a triangle is 180. 2 1 3 Given : ABC Prove : m1 m2 m3 180 Exterior Angles Theorem The measure of an exterior angle of a triangle equals the sum of the two remote interior angles. (remote means nonadjacent) Statement 3 1 2 4 Given : A triangle with exterior angle 1 Prove : m1 m3 m4 Reason Corollary to triangle sum theorem: Acute angles of a right triangle are complementary. All angles 180, if one is 90, the other two add up to 90, and are complementary 4.2 – Congruence and Triangles When TWO POLYGONS have the same size and shape, they are called CONGRUENT! Their vertices and sides must all match up to be congruent. When two figures are congruent, their corresponding sides and corresponding angles are congruent. Identical twins! Name all the corresponding parts and sides, then make a congruence statement. B AB FE A F BC ED B E CA DF C D ABC FED BCA EDF CAB DFE A D E C F If you notice, the way you name the triangle is important, all the CORRESPONDING SIDES must line up! 3rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the 3rd angles are congruent. B If A D, B E Then C F A E C D F E A G 80 5 y 10 M 45 15 H 3x Find x T Y 24 Find y O • Note, triangles also have the following properties of congruent: Reflexive, symmetric, and transitive. Reflexive : ABC ABC Symmetric : ABC DEF , DEF ABC Transitive : ABC DEF, DEF XYZ then ABC XYZ 4.3 – 4.4 Proving Triangles are Congruent A D If AB EF , BC DE , AC DF Then ABC FED B E C F SSS Congruence Postulate – If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. SAS Congruence Postulate – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. A D B E If BC DE , C D, AC DF Then ABC FED Included means IN BETWEEN C F A D If A F , AC DF , C D Then ABC FED B E C F ASA Congruence Postulate – If two angles and the included side of one triangles are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. AAS Congruence Theorem – If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two trianges are congruent. If B E, C D, AC DF Then ABC FED A D B E C F A D B E C F A B C E F Helpful things for the future! Reflexive sides Reflexive angles E B G A D BC BC H F C E E When you see shapes sharing a side, you state that fact using the reflexive property of congruence! A B D C Draw and write down if the triangles are congruent, and by what thrm\post Proofs! The way I like to think about it to look at all the angles and sides, and don’t be fooled by the picture. A D 1. C is the midpoint 1. Given C of AE and BD B E Given : C is the midpoint of AE and BD Prove : ABC EDC Tips, label the diagram as you go along. A B D Given : AD BE , AB DE Prove : ABD EDB E 1. AD BE AB DE 1. Given Given : EF EH , EB EC Prove : EBH ECF E B G What about the angle? F 1) EF EH , EB EC C 1) Given H Use SSS Congruence Postulate to show that ABC DEF (-5, 1) A B (-4, -3) AB BC DE AC DF EF (1, 3) E (5, 4) D F(2, 2) C (-3, -2) A B D E 1. AD || BE AB || DE Given : AD || BE , AB || DE Prove : ABD EDB Tips, label the diagram as you go along. 1. Given Given : DU || KC DU KC; D K Prove : DUC KCS K D U 1. DU || KC; DU KC; D K 1. Given C S Given : AC || BE , BC || DE , AC BE Prove : ABC BDE A B D C E Given : B E, BC CE Prove : ABC DEC B A C E D 4.5 – Using Congruent Triangles CPCTC Correspond ing Part of Congruent Triangles are Congruent Corresp Parts of ' s are A D 1. C is the midpoint 1. Given C B E Given : C is the midpoint of AE and BD Prove : AB || ED of AE and BD 2. AC EC 2. Def of mdpt BC DC 3. ACB ECD 3. VAT 4. ACB ECD 4. SAS post Some Ideas that may help you. If they want you to prove something, and you see triangles in the picture, proving triangles to be congruent may be helpful. If they want parallel lines, look to use parallel line theorems (CAP, AIAT, AEAT, CIAT) Know definitions (Definition of midpoint, definition of angle bisectors, etc.) Sometimes you prove one pair of triangles are congruent, and then use that info to prove another pair of triangles are congruent. A D Given : AD BE , AB DE Prove : BE || AD B E 1. AD BE AB DE 1. Given Given : DU || KC K D DU KC; D K Prove : C is the mdpt of US U 1. DU || KC; DU KC; D K 1. Given C S Given : B E, BC CE Prove : AC DC B A C E D Given : ALG ALN LGA LNA N L A E Prove : ELN ELG G ALG ALN, LGA LNA Given You try this classic proof! Given : 1 2, DE BE Prove : AC is the angle bisector of DCB D A 3 4 1 2 E B 5 6 C 4.6 – Isosceles, Equilateral, and Right Triangles. • Bring book Tuesday • We will go over what’s going to be on Wednesday’s Quiz at end of Tuesday lesson Vertex Angle LEGS BASE Base Angles Remember, definition of isosceles triangles is that AT LEAST two congruent sides. Base angles theorem – If two sides of a triangle are congruent, then the base angles are congruent. Converse of Base angles theorem – If base angles are congruent, then the two opposite sides are congruent. Corollary 1 – An equilateral triangle is also equiangular (Use isosceles triangle theorem multiple times with transitive) Corollary 2 – An equilateral triangle has three 60 degree angles (Use corollary 1 and angle of triangle equals 180) Hypotenuse Leg Theorem (HL) – If the hypotenuse and ONE of the legs of a RIGHT triangle are congruent, then the triangles are congruent. A B D C Draw and write down if the triangles are congruent, and by what thrm\post D Given : DUK is an isosceles triangle with DU and DK as legs. 1 and 2 are right angles. Pr ove : U K 1 2 U C K DU DK Def of isosceles triangle Given : AB AD, AC BD Prove : ABC ADC A B C D 4.7 – Triangles and Coordinate Proof Given a right triangle with one vertex (-20, -10), and legs of 30 and 40, find two other vertices, then find the length of the hypotenuse. Given a vertex of a rectangle at the origin, find three other possible vertices if the base is 15 and the height is 10 for a rectangle. Then find the area. Given the coordinates, prove that the AC is the angle bisector of BCD B A C D (d, k) (a, b) (__,__) (__,__) (__,__) (__, k) (h,k) (j,__) (a,__) (__,__) Picking convenient variable coordinates, prove that the diagonals of a rectangle are congruent.