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4-1 Classifying Triangles Objective: Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. We can classify triangles by Angle Measures and by Side Lengths. By Angles: By Sides: Acute: If ALL angles are less than 90 Scalene: If NO side lengths are congruent Obtuse: If ONE angle is more than 90 Isosceles: If 2 sides or 3 lengths are congruent Equilateral: If ALL sides lengths are congruent Right: If ONE angle is equal to 90 Equiangular: If ALL angles are equal Use the figures to classify each triangle by its angle measures. 1. ∆ADC-Right Triangle ∆ADB-Acute Triangle ∆BDC-Obtuse Triangle 2. ∆EGH-Right ∆EFH-Obtuse ∆FGH-Equiangular Use the figures to classify each triangle by its side lengths. . 3. ∆EGH-Scalene ∆EFH-Isosceles ∆FGH-Scalene 4. ∆ABC-Equilateral ∆ADB-Scalene ∆ADC-Scalene Find the side lengths of ∆JKL. 5. X = 8.5 JK =23.3 JL=44.5 KL =23.3 Find the lengths side lengths of equilateral ∆FGH. 6. y= ___7_______ GF = ____17______ FH = ____17______ GH = __17_____ ___ 7. Total =18+18+18=54 420/54= 7 4-2 Angles of Triangles Find the measure of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Objective: Triangle Sum Theorem: The sum of the angle measures of a triangle is 180 Third Angles Theorem: If two angles of one triangle is congruent to two angles in another triangle, then the third pair of angles are congruent. Exterior Angle: An angle formed by 2 1 side of the triangle and the extension of an adjacent side. 5 1 Interior Angle: An angle formed by two sides of a triangle 3 4 Remote Interior Angle: An interior angle that is not adjacent to the exterior angle Ext. Angle = Remote int. angle + Remote int. angle COMPLEMENTARY 60 DEGREES Angle 5 has two Remote interior angles: 2 and 3; What is Angle 4 ? Remote inter angles? *The measure of each angle of an equiangular triangle is 60 degrees Exterior Angle Theorem: Ext. Angle = Remote int. angle + Remote int. angle *Acute angles of a right triangle are Complementary Exercises: Find the numbered angles in each picture. 1) = 120 *Triangle Sum Theorem 4) 109 29 71 2) = 115 3) *Exterior Angle Theorem 5) = 46 =60 =120 *Both Theorems = 46 = 84 MORE EXAMPLES FOR YOU TO TRY Exercises: Find the numbered angles in each picture. 6) 7) = 27 = 27 9) = 125 = 55 = 95 = 55 =70 = = = 10) 11) 60 A B A 12) C D B 80 80 J K F 141 G H I 4-3 Triangle Congruence Use properties of congruent triangles. Prove triangles are congruent by using the definition of congruence. C D Objective: Congruent Triangles: ORDER MATTERS!!! Example I: Name all the congruent, corresponding parts of CAT DOG . Example II: Polygon LMNP 5) Given 7) and , find . . . and . Find x and m<DBC x= m<DBC= BAC DOG, given polygon Name the congruent and corresponding parts of each triangle. 1) 2) 3) . . . . . 4. Given If CAT and 6) Given Find x and m<F x= m<F= 8) ACB . 9) JKM 10) PQR Proof: 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 4-4 Proving Congruence Apply SSS and SAS to construct triangles and to solve problems. Prove triangles are congruent by using SSS and SAS. Triangle Rigidity: SSS: Objective: SAS: Included Angle: 1) Refer to the figure. a) Name the sides of included angle. LMN for which L is the b) Name the sides of LMN for which N is the included angle. c) Name the sides of LMN for which M is the included angle. Examples: Determine what type of congruence for each triangle. If no congruence, write “no congruence.” 2) 3) 4) Examples: Determine what type of congruence for each triangle. If no congruence, write “no congruence.” 6) 5) 7) 9) 8) 11) 10) 12) 13) Show that the triangles are congruent for the given value of the variable 14) ∆MNO CPCTC - ∆PQR, when x = 5 15)∆STU ∆VWX, when y = 4 17) Given: T is the midpoint of SV 16) Given: BCAD , BCAD Prove: ABDCDB PROVE THE FOLLOWING! T is the midpoint of RU Prove: ∆RTS ∆UTV 17) 19) Given: ABCB DBbits sec Prove: ABC ADCD 4-5 Proving Congruence Objective: Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles are congruent by using ASA, AAS, and HL. AAS: ASA: Included Side: HL: Examples: Determine what type of congruence for each triangle. If no congruence, write not possible. Some Important Things You Should Remember: Things you can assume in a diagram collinear coplanar straight angle Angle Addition Postulate adjacent angles Things you cannot assume in a diagram congruent segments congruent angles angle bisector perpendicular or right angle midpoint parallel * You can draw conclusions about position but not size! * Draw and label ABC and DEF. Indicate which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem. 1) 2) 3) 5) 6) 8) 9) 4) 7) 10) 11) Directions: What corresponding parts must be congruent in order to prove that the triangles are congruent by the ASA Postulate? Write the congruence statement. 13) 12) 14) 15) 17) 16) 18) BCAD AD Statements Reasons Statements Reasons 19) 20) Statements RTEF Reasons Statements Reasons Statements Reasons 21) 22) 23) Prove: Statements Reasons 4-4 to 4-6 Overlapping Triangles Triangles sometimes overlap. You may find it helpful to redraw the two triangles separately to better visualize the congruent pieces. Examples: Draw out the two overlapping triangles. Determine what type of congruence for each pair of overlapping triangles. If no congruence, write not possible. Y and W are right angles. 1) Given: WXZ YZX Y W XZ 2) Given: A C D E B 3) Given: QP TP P Q R S T Examples: Name a pair of overlapping congruent triangles. State which congruence method can be used to prove the triangles congruent. If no triangles can be proved congruent, write none. It may help to draw out the two overlapping triangles separately. X W XZ YW XA YA 4) Given: 5) M X YX Y Z A Y A D 6) Given: AC DB AB BC DC BC ; ; BC J K 7) Given: Prove: JAB KBA KAB JBA J K Statements 1 AB Reasons 1) 2) 1) 3) 2) 3) 4) 4) JAB KBA KAB JBA JAB KBA AB AB CPCTC JK ASA Postulate Reflexive Property Given 4-7 Intro to Coordinate Proof Coordinate Proof: Tips for positioning figures in the coordinate plane! * Use the origin as a vertex, keeping the figure in Quad 1 * Center the figure at the origin * Center a side of the figure at the origin *Use one or both axes as sides of the figure Find the missing coordinates of each triangle. 1) 3) 2) Choose from the following statements and reasons: 6) 5) 7) 8) Example: Position each figure in the coordinate plane. 9) A square with a side length of 6 units 10) A right triangle with side lengths of 2 and 4 units Given the vertices, classify a triangle by its side lengths, and determine if it is a right triangle. F (3, 4) A (6, 6) N (7, -2) -To classify a by its sides, use the _____________________________ - To determine if a is a right triangle, calculate ________of the _______ Review: What are the five ways to prove triangles congruent? Does CPCTC go before or after two triangles are proved congruent? 4-8 Isosceles and Equilateral Triangles Objective: Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. A Legs: jk Fjf Base: Vertex Angle: Base Angles: B C Isosceles Triangle Theorem: Converse of Isosceles Triangle Theorem: Exercises: Refer to the figure. 11. 12. 13. 14. 15. 16. *A triangle is equiangular if and only if it is equilateral.* Examples: Find x. Statements Reasons 17.