Study Guide and Intervention

Transcription

1 4-5 NM T PIO tud Guide and Intervention Proving ongruence, Postulate The ngle-ide-ngle () Postulate lets ou show that two triangles are congruent. Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ind the missing congruent parts so that the triangles can be proved congruent b the Postulate. Then write the triangle congruence. a. ample b. Two pairs of corresponding angles are congruent, and. If the included sides and are congruent, then b the Postulate. X T W Y Y and X Y. If X, then T YXW b the Postulate. ercises What corresponding parts must be congruent in order to prove that the triangles are congruent b the Postulate? Write the triangle congruence statement X 3. W Y Z V 6. T U Lesson 4-5 Glencoe/McGraw-Hill 207 Glencoe Geometr

2 4-5 NM T PIO tud Guide and Intervention (continued) Proving ongruence, Theorem nother wa to show that two triangles are congruent is the ngle- ngle-ide () Theorem. Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. You now have five was to show that two triangles are congruent. definition of triangle congruence Postulate Postulate Theorem Postulate ample In the diagram,. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent b the Postulate? b the efleive Propert of congruence. The congruent angles cannot be 1 and 2, because would be the included side. If, then b the Theorem. 1 2 ercises In ercises 1 and 2, draw and label and. Indicate which additional pair of corresponding parts needs to be congruent for the triangles to be congruent b the Theorem. 1. ; 2. ; 3. Write a flow proof. Given: U; T bisects TU. Prove: T UT U T T bisects TU. Given T UT ef.of bisector U Given T T efl. Prop. of T UT T UT PT Glencoe/McGraw-Hill 208 Glencoe Geometr

3 NM T PIO 4-5 kills Practice Proving ongruence, Write a flow proof. 1. Given: N L J J K M K Prove: JKN MKL N K L M N L Given JK MK Given JKN MKL JKN MKL Vertical are. 2. Given: bisects. Prove: Given Given PT bisects. Given ef. of bisector 3. Write a paragraph proof. Given: G G G Prove: G Lesson 4-5 Glencoe/McGraw-Hill 209 Glencoe Geometr

4 4-5 NM T PIO Practice Proving ongruence, 1. Write a flow proof. Given: is the midpoint of Q T. Q T U Prove: Q TU Q U T is the midpoint of QT. Given Q TU Given Q T ef.of midpoint Q T lt. Int. are. Q TU Vertical are. Q TU 2. Write a paragraph proof. Given: G bisects. Prove: G G G HITTU or ercises 3 and 4, use the following information. n architect used the window design in the diagram when remodeling an art studio. and each measure 3 feet. 3. uppose is the midpoint of. etermine whether. Justif our answer. 4. uppose. etermine whether. Justif our answer. Glencoe/McGraw-Hill 210 Glencoe Geometr

5 NM T PIO 4-5 eading to Learn Mathematics Proving ongruence, Pre-ctivit How are congruent triangles used in construction? ead the introduction to Lesson 4-5 at the top of page 207 in our tetbook. Which of the triangles in the photograph in our tetbook appear to be congruent? eading the Lesson 1. plain in our own words the difference between how the Postulate and the Theorem are used to prove that two triangles are congruent. 2. Which of the following conditions are sufficient to prove that two triangles are congruent?. Two sides of one triangle are congruent to two sides of the other triangle.. The three sides of one triangles are congruent to the three sides of the other triangle.. The three angles of one triangle are congruent to the three angles of the other triangle.. ll si corresponding parts of two triangles are congruent.. Two angles and the included side of one triangle are congruent to two sides and the included angle of the other triangle.. Two sides and a nonincluded angle of one triangle are congruent to two sides and a nonincluded angle of the other triangle. G. Two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the other triangle. H. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. I. Two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of the other triangle. 3. etermine whether ou have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate or theorem that ou would use. If not, write not possible. a. b. T is the midpoint of U. U Helping You emember 4. good wa to remember mathematical ideas is to summarize them in a general statement. If ou want to prove triangles congruent b using three pairs of corresponding parts, what is a good wa to remember which combinations of parts will work? T V Lesson 4-5 Glencoe/McGraw-Hill 211 Glencoe Geometr

6 NM T PIO 4-5 nrichment ongruent Triangles in the oordinate Plane If ou know the coordinates of the vertices of two triangles in the coordinate plane, ou can often decide whether the two triangles are congruent. There ma be more than one wa to do this. 1. onsider and whose vertices have coordinates (0, 0), (2, 5), (9, 5), and (7, 0). riefl describe how ou can use what ou know about congruent triangles and the coordinate plane to show that.you ma wish to make a sketch to help get ou started. 2. onsider PQ and KLM whose vertices are the following points. P(1, 2) Q(3, 6) (6, 5) K( 2, 1) L( 6, 3) M( 5, 6) riefl describe how ou can show that PQ KLM. 3. If ou know the coordinates of all the vertices of two triangles, is it alwas possible to tell whether the triangles are congruent? plain. Glencoe/McGraw-Hill 212 Glencoe Geometr

7 4-6 NM T PIO tud Guide and Intervention Isosceles Triangles Properties of Isosceles Triangles n isosceles triangle has two congruent sides. The angle formed b these sides is called the verte angle. The other two angles are called base angles.you can prove a theorem and its converse about isosceles triangles. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem) If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If, then. If, then. Lesson 4-6 ample 1 ample 2 (4 5) (5 10), so m m ercises ind. ind. Isos. Triangle Theorem ubstitution ubtract 4 from each side. dd 10 to each side m m T, so T ind. T onverse of Isos. Thm. ubstitution dd 13 to each side. ubtract 2 from each side W P 40 2 Q T 3 6 V Y 3 Z P G K T Q (6 6) L T 3 7. Write a two-column proof. Given: 1 2 Prove: tatements easons Glencoe/McGraw-Hill 213 Glencoe Geometr

8 4-6 NM T PIO tud Guide and Intervention (continued) Isosceles Triangles Properties of quilateral Triangles n equilateral triangle has three congruent sides. The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles. 1. triangle is equilateral if and onl if it is equiangular. 2. ach angle of an equilateral triangle measures 60. ample Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle. Proof: tatements easons 1. is equilateral; P Q. 1. Given 2. m m m ach of an equilateral measures , 2 3. If lines, then corres. s are. 4. m 1 60, m ubstitution 5. PQ is equilateral. 5. If a is equiangular, then it is equilateral. ercises ind. P 1 2 Q G 3. L N 6 J H M 3 K KLM is equilateral. 4. P Q 5. X V L Z 4 4 Y M 4 O 60 H 7. Write a two-column proof. Given: is equilateral; 1 2. Prove: Proof: 1 2 tatements easons Glencoe/McGraw-Hill 214 Glencoe Geometr

9 NM T PIO 4-6 kills Practice Isosceles Triangles efer to the figure. 1. If, name two congruent angles. 2. If, name two congruent angles. Lesson If, name two congruent segments. 4. If, name two congruent segments. is isosceles, is equilateral, and m 150. ind each measure. 5. m 6. m m 8. m In the figure, P L L and L. 9. If m LP 100, find m L. L 10. If m LP 34, find m. P 11. Write a two-column proof. Given: G G Prove: G Glencoe/McGraw-Hill 215 Glencoe Geometr

10 NM T PIO 4-6 Practice Isosceles Triangles efer to the figure. 1. If V T, name two congruent angles. 2. If V, name two congruent angles. V U T 3. If T T, name two congruent segments. 4. If TV VT, name two congruent segments. Triangles GHM and HJM are isosceles, with G H M H and H J M J. Triangle KLM is equilateral, and m HMK 50. ind each measure. 5. m KML 6. m HMG 7. m GHM H J K L M G 8. If m HJM 145, find m MHJ. 9. If m G 67, find m GHM. 10. Write a two-column proof. Given: 1 2 Prove: POT pennant for the sports teams at Lincoln High chool is in the shape of an isosceles triangle. If the measure of the verte angle is 18, find the measure of each base angle. Lincoln Hawks Glencoe/McGraw-Hill 216 Glencoe Geometr

11 4-6 NM T PIO eading to Learn Mathematics Isosceles Triangles Pre-ctivit How are triangles used in art? ead the introduction to Lesson 4-6 at the top of page 216 in our tetbook. Wh do ou think that isosceles and equilateral triangles are used more often than scalene triangles in art? Lesson 4-6 Wh might isosceles right triangles be used in art? eading the Lesson 1. efer to the figure. a. What kind of triangle is Q? b. Name the legs of Q. c. Name the base of Q. d. Name the verte angle of Q. e. Name the base angles of Q. Q 2. etermine whether each statement is alwas, sometimes, or never true. a. If a triangle has three congruent sides, then it has three congruent angles. b. If a triangle is isosceles, then it is equilateral. c. If a right triangle is isosceles, then it is equilateral. d. The largest angle of an isosceles triangle is obtuse. e. If a right triangle has a 45 angle, then it is isosceles. f. If an isosceles triangle has three acute angles, then it is equilateral. g. The verte angle of an isosceles triangle is the largest angle of the triangle. 3. Give the measures of the three angles of each triangle. a. an equilateral triangle b. an isosceles right triangle c. an isosceles triangle in which the measure of the verte angle is 70 d. an isosceles triangle in which the measure of a base angle is 70 e. an isosceles triangle in which the measure of the verte angle is twice the measure of one of the base angles Helping You emember 4. If a theorem and its converse are both true, ou can often remember them most easil b combining them into an if-and-onl-if statement. Write such a statement for the Isosceles Triangle Theorem and its converse. Glencoe/McGraw-Hill 217 Glencoe Geometr

12 4-6 NM T PIO nrichment Triangle hallenges ome problems include diagrams. If ou are not sure how to solve the problem, begin b using the given information. ind the measures of as man angles as ou can, writing each measure on the diagram. This ma give ou more clues to the solution. 1. Given:, G, m 82 and G and each have opposite sides parallel and congruent. ind m. 2. Given:, and, m 44 and bisects. ind m. G 3. Given: m UZY 90, m ZWX 45, YZU VWX, UVXY is a square (all sides congruent, all angles right angles). ind m WZY. U V W 4. Given: m N 120, J N M N, JNM KLM. ind m JKM. J N M L K Z Y X Glencoe/McGraw-Hill 218 Glencoe Geometr

13 4-7 NM T PIO tud Guide and Intervention Triangles and oordinate Proof Position and Label Triangles coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines. 1. Use the origin as a verte or center of the figure. 2. Place at least one side of the polgon on an ais. 3. Keep the figure in the first quadrant if possible. 4. Use coordinates that make the computations as simple as possible. ample Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive -ais. tart with (0, 0). If T is a, then another verte is T(a, 0). a 2, b Lesson 4-7 or verte, the -coordinate is a. Use b for the -coordinate, 2 so the verte is a 2, b. (0, 0) T(a, 0) ercises ind the missing coordinates of each triangle (?, q) T(?,?) (?, b) (0, 0) (2p, 0) (0, 0) (2a, 0) (?,?) G(2g, 0) Position and label each triangle on the coordinate plane. 4. isosceles triangle 5. isosceles right 6. equilateral triangle QI T with base with legs e units long with verte Q(0, a) and 4a units long sides 2b units long T(2a, b) (e, e) Q(0, a) (0, 0) (4a, 0) (0, 0) (e, 0) ( b, 0) I(b, 0) Glencoe/McGraw-Hill 219 Glencoe Geometr

14 4-7 NM T PIO tud Guide and Intervention (continued) Triangles and oordinate Proof Write oordinate Proofs oordinate proofs can be used to prove theorems and to verif properties. Man coordinate proofs use the istance ormula, lope ormula, or Midpoint Theorem. ample Prove that a segment from the verte angle of an isosceles triangle to the midpoint of the base is perpendicular to the base. irst, position and label an isosceles triangle on the coordinate plane. One wa is to use T(a, 0), ( a, 0), and (0, c). Then U(0, 0) is the midpoint of T. Given: Isosceles T; U is the midpoint of base T. Prove: U T ( a, 0) (0, c) U(0, 0) T(a, 0) Proof: U is the midpoint of T so the coordinates of U are a a, (0, 0). Thus U lies on the -ais, and T was placed so T lies on the -ais. The aes are perpendicular, so U T. ercises Prove that the segments joining the midpoints of the sides of a right triangle form a right triangle. (0, 2b) P (0, 0) Q (2a, 0) Glencoe/McGraw-Hill 220 Glencoe Geometr

15 NM T PIO 4-7 kills Practice Triangles and oordinate Proof Position and label each triangle on the coordinate plane. 1. right GH with legs 2. isosceles KLP with 3. isosceles N with a units and b units base K P 6b units long base 5a long (0, a) L(3b, c) N 5 2 a, b G(0, 0) H(b, 0) K(0, 0) P(6b, 0) (0, 0) (5a, 0) ind the missing coordinates of each triangle (0,?) Z(?,?) M(?,?) Lesson 4-7 (0, 0) (2a, 0) X(0, 0) Y(2b, 0) O(0, 0) N(3b, 0) (2a, b) (?,?) T(?,?) P(0, 0) Q(?,?) N(0, 0) P(7b, 0) ( a, 0) U(a, 0) 10. Write a coordinate proof to prove that in an isosceles right triangle, the segment from the verte of the right angle to the midpoint of the hpotenuse is perpendicular to the hpotenuse. Given: isosceles right with the right angle and M the midpoint of Prove: M (0, 2a) M (0, 0) (2a, 0) Glencoe/McGraw-Hill 221 Glencoe Geometr

16 4-7 Position and label each triangle on the coordinate plane. 1. equilateral WY with 2. isosceles LP with 3. isosceles right GJ sides 1 a long base L 3b units long with hpotenuse J and 4 legs 2a units long NM T PIO Practice Triangles and oordinate Proof Y 1 8 a, b P 3 2 b, c (0, 2a) (0, 0) W 1 4 a, 0 (0, 0) L(3b, 0) G(0, 0) J(2a, 0) ind the missing coordinates of each triangle (?,?) (0,?) M(0,?) J(0, 0) 1 3 b, 0 ( 3a, 0) (?, 0) N(?, 0) P(2b, 0) NIGHOHOO or ercises 7 and 8, use the following information. Karina lives 6 miles east and 4 miles north of her high school. fter school she works part time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school. 7. Write a coordinate proof to prove that Karina s high school, her home, and the mall are at the vertices of a right triangle. Given: KM Prove: KM is a right triangle. M( 2, 3) K(6, 4) (0, 0) 8. ind the distance between the mall and Karina s home. Glencoe/McGraw-Hill 222 Glencoe Geometr

17 4-7 NM T PIO eading to Learn Mathematics Triangles and oordinate Proof Pre-ctivit How can the coordinate plane be useful in proofs? ead the introduction to Lesson 4-7 at the top of page 222 in our tetbook. rom the coordinates of,, and in the drawing in our tetbook, what do ou know about? eading the Lesson 1. ind the missing coordinates of each triangle. a. b. (?, b) T(a,?) (?, a) (?,?) Lesson 4-7 (?,?) (?,?) 2. efer to the figure. a. ind the slope of and the slope of T. b. ind the product of the slopes of and T. What does this tell ou about and T? c. What does our answer from part b tell ou about T? (0, a) ( a, 0) O(0, 0) T(a, 0) d. ind and T.What does this tell ou about and T? e. What does our answer from part d tell ou about T? f. ombine our answers from parts c and e to describe T as completel as possible. g. ind m T and m T. h. ind m O and m OT. Helping You emember 3. Man students find it easier to remember mathematical formulas if the can put them into words in a compact wa. How can ou use this approach to remember the slope and midpoint formulas easil? Glencoe/McGraw-Hill 223 Glencoe Geometr

18 4-7 NM T PIO nrichment How Man Triangles? ach puzzle below contains man triangles. ount them carefull. ome triangles overlap other triangles. How man triangles are there in each figure? How man triangles can ou form b joining points on each circle? List the vertices of each triangle I H G 8. J K 9. Q O P L N M V U T Glencoe/McGraw-Hill 224 Glencoe Geometr