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Transcript
Lesson 4.1 Classifying Triangles Today, you will learn to… * classify triangles by their sides and angles * find measures in triangles ABC A B C Equilateral Triangle 3 congruent sides Isosceles Triangle 2 congruent sides Scalene Triangle no congruent sides Equiangular Triangle 3 congruent angles Acute Triangle 70° 60° 50° 3 acute angles Obtuse Triangle 95° 60° 25° 1 obtuse angle Right Triangle 30° 60° 1 right angle We classify triangles by their sides and angles. SIDES Equilateral Isosceles Scalene ANGLES Equiangular Acute Obtuse Right Identify the side opposite the given angle. A C B CB is opposite _____ A. AC is opposite _____ B. AB is opposite _____ C. Leg? leg hypotenuse ? leg Leg? leg ? base leg Theorem 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is ________ 180° 1. Find m X. Y mX = 44˚ 61º Z 75º X If the sum of the interior angles is 180º, what do you know about 1 and 2? 1 2 Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary _________________. 2. Find m F. D 54˚ + mF = 90˚ 54˚ E mF = 36˚ F 3. Find m 1 and m 2. 50º 70º m1 = 60˚ m2 = 120˚ exterior angle 1 2 adjacent angles Theorem 4.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles. 1 2 m1 + m2 = m4 3 4 m1 + m2 + m3 = 180˚ m4 + m3 = 180˚ m1 + m2 = m4 1 2 Sum of nonadjacent interior s 3 4 = ext. 4. Find mE. E m E + 60˚ = 110˚ m E = 50˚ 60˚ D 110˚ F G 5. Find x. E x˚ 60˚ D x + 60 = 3x + 10 x = 25 (3x + 10)˚ F G Lesson 4.2 Congruence and Triangles Today, you will learn to… * identify congruent figures and corresponding parts * prove that 2 triangles are congruent Def. of Congruent Figures Figures are congruent if and only if all pairs of corresponding angles and sides are congruent. Statement of Congruence Δ ABC Δ XYZ vertices are written in corresponding order AX BY CZ XY AB XY BC YZ AC XZ XZ 1. Mark ΔDEF to show that Δ ABC Δ DEF. B C A F E D 2. Find all missing measures. ABC DEF B B ? 8.2 A D 10 ? 8.2 ? ? 5.7 ? 55˚ C 5.7 ? ? 55˚ F E 35˚ 10 3. In the diagram, ABCD KJHL. Find x and y. K A B 93˚ 85˚ D 9 cm 75˚ J L (3y)˚ (4x – 3) cm H C 4x-3 = 9 x=3 3y = 75˚ y = 25 4. ΔABC ΔDEF. Find x. A 93˚ 57 30˚ 4x + 15 = 57 x = 10.5 B C F (4x + 15)˚ D E Theorem 4.3 Third Angles Theorem If 2 angles of one triangle are congruent to 2 angles of another D triangle, then… O 60˚ 70˚ ? C G A 60˚ 70˚ ? the third angles are also congruent. T 5. Decide if the triangles are congruent. Justify your reasoning. Vertical Angles Theorem H E 58° G 58° F J Third Angles Theorem HG ΔEFG Δ______ J 6. W 1 2 M Z 5 X 3 4 6 Y 1) WX YZ , WX | | YZ, 1) Given M is the midpoint of WY and XZ 2) 1 6 and 2 5 2) Alt. Int. s Theorem 3) 3 4 3) Vertical Angles Th. 4) WM MY and ZM MX 4) Def. of midpoint 5) Def. of figures 5) ΔWXM ΔYZM 7. Identify any figures you can prove congruent & write a congruence statement. B A D C Alt. Int. Th. Reflexive Property ACD C AB Third Angle Th. Theorem 4.4 Properties of Congruent Triangles ABC ABC Reflexive If ABC XYZ, Symmetric then XYZ ABC If ABC XYZ Transitive and XYZ MNO then ABC MNO Lesson 4.3 Proving Triangles are Congruent Today, you will learn to… * prove that triangles are congruent * use congruence postulates to solve problems SSS Experiment Using 3 segments, can you ONLY create 2 triangles that are congruent? Side-Side-Side Congruence Postulate B If Side AB XY C A Y Side AC XZ Side BC YZ, then ΔABC ΔXYZ by SSS X Z If 3 pairs of sides are congruent, then the two triangles are congruent. 1. Does the diagram give enough info to use SSS Congruence? K L C A no J B 2. Given: LN NP and M is the midpoint of LP Prove: ΔNLM ΔNPM N L P M Def of midpoint Reflexive Property LM MP NM NM NLM NPM SSS Congruence 3. Show that ΔNPM ΔDFE by SSS if N(-5,1), P (-1,6), M (-1,1), D (6,1), F (2,6), and E (2,1). F P N D M E (- (6 5 –– -2) 1)22++(1 (1––6) 6)22 NM = 4 MP = 5 NP = 41 DE = 4 EF = 5 DF = 41 SAS Experiment Using 2 congruent segments and 1 included angle, can you ONLY create 2 triangles that are congruent? B Side-Angle-Side Congruence Postulate If Side AB XY C A Y Angle B Y Side BC YZ, then ΔABC ΔXYZ by SAS X Z If 2 pairs of sides and their included angle are congruent, then the two triangles are congruent. SAS? 5. 4. NO! SAS 6. 7. NO! SAS 8. Does the diagram give enough info to use SAS Congruence? A B D C ABD AC _ _D _ by SAS 9. Does the diagram give enough info to use SAS Congruence? W V no Y X Z 10. Does the diagram give enough info to use SAS Congruence? E A no B C D 11. Given: W is the midpoint of VY and the midpoint of ZX Prove: ΔVWZ ΔYWX VW WY and ZW WX Def. of midpoint VWZ YWX Vertical Angles Th VWZ YWX SAS Congruence 12. Given: AB PB , MB AP Prove: ΔMBA ΔMBP M A B P ABM & PBM are right s Def of lines ABM PBM MB MB MBA MBP All right s are Reflexive Property SAS Congruence What is the best way to get better at proofs? Lesson 4.4 Proving Triangles are Congruent Today, you will learn to… * prove that triangles are congruent * use congruence postulates to solve problems ASA Experiment Using 2 angles connected by 1 segment, can you ONLY create two triangles that are congruent? Angle-Side-Angle Congruence Postulate B If Angle B Y, C A Y Side BC YZ, Angle C Z then ΔABC ΔXYZ X Z by ASA If 2 pairs of angles and the included sides are congruent, then the two triangles are congruent. Included side? A B C The included side between AB A and B is _____ Included side? A B C The included side between CB B and C is _____ Included side? A B C The included side between A and C is _____ AC ASA? 2. 1. NO! ASA 3. 4. NO! ASA 5. Does the diagram give enough info to use ASA Congruence? B A C Reflexive Property Third Angles Theorem D Δ ABD Δ ACD by ASA 6. Does the diagram give enough info to use ASA Congruence? A D Reflexive Property B Alt. Int. Angles Theorem C yes, ΔACB ______ ΔC A D by ASA 7. Does the diagram give enough info to use ASA Congruence? A D Reflexive Property B C no 8. Does the diagram give enough info to use ASA Congruence? K L J C B A KLJ ACB _ _ _ by ASA 9. Determine whether the triangles are congruent by ASA. L K Vertical Angles Theorem H J Alt. Int. Angles Theorem G HJG K _ _J L_ by ASA Angle-Angle-Side Congruence Theorem B If Angle B Y C A Angle C Z Y Side AB XY then ΔABC ΔXYZ by AAS Z X If 2 pairs of angles and a pair of nonincluded sides are congruent, then the two triangles are congruent. AAS? 11. 10. AAS NO! 12. 13. AAS NO! 14. Does the diagram give enough info to use AAS Congruence? B A C Reflexive Property D ABD ACD _ _ _ by AAS 15. Does the diagram give enough info to use AAS Congruence? K L J C A B KLJ ACB _ _ _ by AAS 16. Determine whether the triangles are congruent by AAS. L K Vertical Angles Theorem H J Alt. Int. Angles Theorem G HJG K _ _JL_ by AAS SSA Experiment Using 2 sides and 1 angle that is NOT included, can you ONLY create two triangles that are congruent? NO AAA Experiment Using 3 angles, can you ONLY create two triangles that are congruent? NO All of the angles are , but the s are NOT Triangle Congruence? SSS SSA ASA AAA SAS AAS Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ? 17. CB ZY , AC XZ SAS Congruence X A C B Z C Z Y Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ? 18. CB ZY , AC XZ SSS Congruence X A C B Z AB XY Y Mark the given information on the triangles. What additional congruence would you need to show ABC XYZ? 19. CB ZY , C Z SAA Congruence X A C B Z A X Y What is the best way to get better at proofs? Lesson 4.5 Corresponding Parts of Congruent Triangles are Congruent Today, you will learn to… * use congruence postulates to solve problems CPCTC 1. Given: AB || CD , BC || DA Prove: AB CD B 3 1 A 1 2 , 3 4 BD BD ABD C D B AB CD C 4 2 D Alt. Int. Angles Theorem Reflexive Property ASA CPCTC 2. Given: 1 2 , 3 4 D Prove: CD CB C CA CA ABC A D C CD CB 1 2 4 3 B Reflexive Property ASA CPCTC A 3. Given: AC AD , BC BD A Prove: C D C B AB AB ABC AB D CD Reflexive Property SSS CPCTC 4. Given: A is the midpoint of MT M A is the midpoint of SR R A Prove: MS || TR S T Def. of midpoint MA AT SA AR Vertical Angles Theorem SAM RAT SAS SAM R AT CPCTC M T Alt. Int. Angles Converse MS | | TR Triangle Congruence? 2 angles & 1 side? 2 sides & 1 angle? AAS ASA SAS SSA 3 sides or 3 angles? SSS AAA You can ONLY use CPCTC after you use one of these! Does the quilt design have vertical, horizontal, or diagonal symmetry? Does the quilt design have vertical, horizontal, or diagonal symmetry? Lesson 4.6 Isosceles, Equilateral, and Right Triangles Students need rulers and protractors. Today, you will learn to… * use properties of isosceles, equilateral, and right triangles Use a ruler to draw two congruent segments that share one endpoint. Connect the endpoints to create a triangle. Measure each interior angle. What do you notice? leg leg base base angles Theorem 4.6 Base Angles Theorem If 2 sides of a triangle are the congruent, then … angles opposite them are congruent. Theorem 4.7 Base Angles Converse If 2 angles of a triangle are congruent, then the sides opposite them are congruent. 1. What angles are congruent? A C by the Base Angles Theorem A B D C 2. What sides are congruent? B AB BC by the Base Angles A Converse D C 3. Find mB. C mB = 75˚ A 75˚ B 4. Find mB. C mB = 44˚ A 68˚ ? 68˚ B 5. Find x. C 2x + 4 = 18 2x = 14 18 2x + 4 x=7 A B 6. Find x. C 6x – 10 = 5 B 6x = 15 x = 2.5 A 7. Find x and y. y˚ ? 50˚ y˚ y = 32.5 115˚ ? 65˚ x˚ x˚ ? x = 65 Corollaries to Theorem 4.6/4.7 (hint: don’t write these yet) IfAatriangle triangle is equilateral, equilateral thenifit and is equiangular. only if AND it is equiangular. If a triangle is equiangular, then it is equilateral. 8. Find x. 7x + 3 = 24 7x = 21 x=3 C 24 A 10x – 6 = 7x + 3 3x = 9 7x + 3 B 10x - 6 = 24 10x = 30 9. Find x. C What is the measure of each angle? 2x˚ 2x = 60˚ x = 30 B A 10. Find x and y. y˚ 50˚ ? y = 80 x˚ 70˚ ? 60˚ 70˚ ? 60˚ 60˚ ? 50˚ x = 40 Experiment Using a right angle, a hypotenuse, and a leg, can you ONLY create 2 triangles that are congruent? Hypotenuse-Leg Congruence Theorem Y If Hyp BC YZ Leg AB XY then ΔABC ΔXYZ X by HL B A C Z The triangles MUST be right triangles. If the hypotenuse and a leg of two right triangles are congruent, then the two triangles are congruent. 11. Does the diagram give enough info to use HL Congruence? Reflexive Property W X Y NO Z 12. X is a midpoint. Does the diagram give enough info to use HL? V Def. of midpoint Y W X Z VWX YZ _ _X _ by HL 13. Does the diagram give enough info to use HL Congruence? W Reflexive Property Y X Base Angles Converse Z YWX YZ _ _X _ by HL