Classify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two classifications in Exercise 13.)

Transcription

1 hapter 4 ongruent Triangles 4.2 and 4.9 lassifying Triangles and Isosceles, and quilateral Triangles. Match the letter of the figure to the correct vocabulary word in xercises right triangle 2. obtuse triangle 3. acute triangle 4. equiangular triangle lassify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two classifications for xercise 7.) For xercises 8 10, fill in the blanks to complete each definition. 8. n isosceles triangle has congruent sides. 9. n triangle has three congruent sides. 10. triangle has no congruent sides. lassify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two classifications in xercise 13.)

2 Isosceles Triangles Remember Isosceles triangles are triangles with two congruent sides. The two congruent sides are called legs. The third side is the base. The two angles at the base are called base angles. Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent. onverse is true! 2

3 Isosceles Triangles; Proving Triangles ongruent Find the value of x x x 3x x x 72 x 3

4 4.3 ngle Relationships in Trinagles o The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. n interior angle is formed by two sides of a triangle. n exterior angle is formed by one side of the triangle and extension of an adjacent side. ach exterior angle has two remote interior angles. remote interior angle is an interior angle that is not adjacent to the exterior angle. xterior ngles: Find each angle measure. 37. m 38. m PR 39. In LMN, the measure of an exterior angle at N measures m L x 3 2 and m M x. Find m L, m M, and m LNM m and m G 41. m T and m V 42. In and F, m m and m m. Find m F if an exterior angle at measures 107, m (5x 2), and m (5x 5). 43. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle. 44. One of the acute angles in a right triangle measures 2x. What is the measure of the other acute angle? 4

5 45. The measure of one of the acute angles in a right triangle is What is the measure of the other acute angle? 46. The measure of one of the acute angles in a right triangle is x. What is the measure of the other acute angle? 47. Find m 48. Find m< 49. Find m K and m J 50. Find m<p and m<t Use the figure at the right for problems Find m 3 if m 5 = 130 and m 4 = Find m 1 if m 5 = 142 and m 4 = Find m 2 if m 3 = 125 and m 4 = Use the figure at the right for problems m 6 + m 7 + m 8 = If m 6 = x, m 7 = x 20, and m 11 = 80, then x =. 6. If m 8 = 4x, m 7 = 30, and m 9 = 6x -20, then x =. 7. m 9 + m 10 + m 11 = For 8 12, solve for x. 8. x x (5x)

6 4.4 ongruent Triangles ongruent Triangles: Two s are if their can be matched up so that corresponding angles and sides of the s are. ongruence tatement: \ R FOX List the corresponding s: corresponding sides: R R R xamples: 1. The two s shown are. a) O b) c) O d) O = O K 2. The pentagons shown are. 4 cm a) corresponds to b) LK c) = m d) K = cm e) If L, name two right s in the figures. R H L O 3. Given IG T, I = 14, IG = 18, G = 21, T = 2x + 7. Find x. The following s are, complete the congruence statement: 6

7 Use Graph paper to do the following: 7. Plot the given points. raw IG. Locate point P so that IG PIG. a) (1,2) I(4,7) G(4,2) b) (7,5) I(-2,2) G(5,2) Plot the given points on graph paper. raw and F. opy and complete the statement. 8. (-1,2) (4,2) (2,4) 9. (-7,-3) (-2,-3) (-2,0) (5,-1) (7,1) F(10,-1) (0,1) (5,1) F(0,-2) 10. (-3,1) (2,1) (2,3) 11. (1,1) (8,1) (4,3) (4,3) (6,3) F(6,8) (3,-7) (5,-3) F(3,0) Plot the given points on graph paper. raw and. Find two locations of point F uch that F. 12. (-1,0) (-5,4) (-6, 1) (1,0) (5,4) Parts of a Triangle in terms of their relative positions. 1. Name the opposite side to. 2. Name the included side between and. 3. Name the opposite angle to. 4. Name the included angle between and. Ways to Prove s : Proving Triangles ongruent Postulate: (side-side-side) Three sides of one are to three sides of a second, Given: bisects PW ; P W P W Postulate: (side-angle-side) Two sides and the included angle of one are to two sides and the included angle of another. X Given: PX bisects X; X X 7 P

8 Postulate: (angle-side-angle) Two angles and the included side of one are to two angles and the included side of another. M // TH Given: T // MH T M H Theorem: (angle-angle-side) Two angles and a non-included side of one are to two angles and a non-included side of another. Given: UZ bi sec ts UZ U ; UZ Z R U Z HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right are to the hypotenuse and leg of another right. Given: T F Isosceles F with legs ongruent Triangles xamples F, F T PT orresponding Parts of ongruent Triangles are ongruent tate which congruence method(s) can be used to prove the s. If no method applies, write none. ll markings must correspond to your answer H G I F 3. Q R 4. T 5. R V P Q 8 R T U

9 7. 8. tate which congruence method(s) can be used to prove the s. If no method applies, write none. ll markings must correspond to your answer Fill in the congruence statement and then name the postulate that proves the s are. If the s are not, write not possible in second blank. (Leave first blank empty)*markings must go along with your answer** ome may have more than one postulate F by F by F by by 9

10 5. 6. P Q by R by by by by by by by U by N by 10

11 #1 Given: R UT; R // UT; U Prove: T // UV 1. R UT; R // UT; U RT TUV T // UV 5. #2 Given: is the midpoint of ; Prove: bisects. 1. is the midpoint of ; bisects. 6. #3 Given: R Q; R QT Prove: T 1. R Q; R QT <R <Q R QT T 4. 11

12 Fill in Proofs: #1 3 4 Given: Prove: Δ Δ & 2 are right s Δ Δ 6. #2 Given: bisects 3 4 Prove: & 2 are right s bisects Δ Δ

13 ongruent Triangles Proofs R 1. Given: P ; O is the midpoint of P Prove: O is the midpoint of RQ O P Q 2. Given: ; is the midpoint of Prove: K 3. Given: K // NR; N // KR Prove: K NR; N KR R 4. Given: // M; M M is the midpoint Prove: M // N M 5. Given: Prove: 6. Given: MK Prove: is the midpoint of MK x y x y M K 13

14 7. Given: FM 1 2 Prove: bisects MF 1 F M 2 8. Given: P Q and PV QV Prove: x y x Y P V Q Given: 1 Prove: Given: Prove: 14