Geometry hapter 4 esource Masters
NME E PEI 4 eading to Learn Mathematics Vocabulary uilder his is an alphabetical list of the key vocabulary terms you will learn in hapter 4. s you study the chapter, complete each term s definition or description. emember to add the page number where you found the term. dd these pages to your Geometry tudy Notebook to review vocabulary at the end of the chapter. acute triangle Vocabulary erm Found on Page efinition/escription/eample Vocabulary uilder base angles congruence transformation kuhn G uhns congruent triangles coordinate proof corollary equiangular triangle equilateral triangle eterior angle (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Geometry
NME E PEI 4 eading to Learn Mathematics Vocabulary uilder (continued) flow proof Vocabulary erm Found on Page efinition/escription/eample included angle included side isosceles triangle obtuse triangle remote interior angles right triangle scalene triangle KY leen verte angle Glencoe/McGraw-Hill viii Glencoe Geometry
4 NME E PEI Learning to ead Mathematics Proof uilder his is a list of key theorems and postulates you will learn in hapter 4. s you study the chapter, write each theorem or postulate in your own words. Include illustrations as appropriate. emember to include the page number where you found the theorem or postulate. dd this page to your Geometry tudy Notebook so you can review the theorems and postulates at the end of the chapter. heorem or Postulate heorem 4. ngle um heorem Found on Page escription/illustration/bbreviation Proof uilder heorem 4.2 hird ngle heorem heorem 4.3 Eterior ngle heorem heorem 4.4 heorem 4.5 ngle-ngle-ide ongruence () heorem 4.6 Leg-Leg ongruence (LL) heorem 4.7 Hypotenuse-ngle ongruence (H) (continued on the net page) Glencoe/McGraw-Hill i Glencoe Geometry
4 NME E PEI Learning to ead Mathematics Proof uilder (continued) heorem or Postulate heorem 4.8 Leg-ngle ongruence (L) Found on Page escription/illustration/bbreviation heorem 4.9 Isosceles riangle heorem heorem 4.0 Postulate 4. ide-ide-ide ongruence () Postulate 4.2 ide-ngle-ide ongruence () Postulate 4.3 ngle-ide-ngle ongruence () Postulate 3.4 Hypotenuse-Leg ongruence (HL) Glencoe/McGraw-Hill Glencoe Geometry
4- NME E PEI tudy Guide and Intervention lassifying riangles lassify riangles by ngles ne way to classify a triangle is by the measures of its angles. If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle. If one of the angles of a triangle is a right angle, then the triangle is a right triangle. If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle. If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle. a. b. c. Eample lassify each triangle. ll three angles are congruent, so all three angles have measure 60. he triangle is an equiangular triangle. 60 35 he triangle has one angle that is obtuse. It is an obtuse triangle. G 90 E 20 25 F Lesson 4-60 30 H J he triangle has one right angle. It is a right triangle. Eercises lassify each triangle as acute, equiangular, obtuse, or right.. K 2. N 3. 67 90 L 30 20 23 M P 30 Q 60 60 60 4. 5. W 6. 45 50 60 65 U 65 V 90 X 45 Y F 28 92 Glencoe/McGraw-Hill 83 Glencoe Geometry
4- NME E PEI tudy Guide and Intervention (continued) lassifying riangles lassify riangles by ides You can classify a triangle by the measures of its sides. Equal numbers of hash marks indicate congruent sides. If all three sides of a triangle are congruent, then the triangle is an equilateral triangle. If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle. If no two sides of a triangle are congruent, then the triangle is a scalene triangle. Eample lassify each triangle. a. H b. N c. 23 2 L J P wo sides are congruent. ll three sides are he triangle has no pair he triangle is an congruent. he triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle. X 5 V Eercises lassify each triangle as equilateral, isosceles, or scalene.. 2. G 3. G 2 3 K 8 8 8 I M 2 7 Q 9 4. 5. 32 6. 8 32 W U F E 7. Find the measure of each side of equilateral with 2 2, 3, and 5 4. 8. Find the measure of each side of isosceles with if 4y, 3y 2, and 3y. 9. Find the measure of each side of with vertices (, 5), (6, ), and (2, 6). lassify the triangle. Glencoe/McGraw-Hill 84 Glencoe Geometry
4- NME E PEI kills Practice lassifying riangles Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.. 2. 3. 4. 5. 6. Lesson 4- Identify the indicated type of triangles. 7. right 8. isosceles E 9. scalene 0. obtuse LGE Find and the measure of each side of the triangle.. is equilateral with 3 2, 2 4, and 0. 2. EF is isosceles, is the verte angle, E 7, F 3, and EF 2 5. Find the measures of the sides of and classify each triangle by its sides. 3. (0, 2), (2, 5), (4, 2) 4. (, 3), (4, 7), (5, 4) Glencoe/McGraw-Hill 85 Glencoe Geometry
4- NME E PEI Practice lassifying riangles Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.. 2. 3. Identify the indicated type of triangles if, E E,, and E. E 4. right 5. obtuse 6. scalene 7. isosceles LGE Find and the measure of each side of the triangle. 8. FGH is equilateral with FG 5, GH 3 9, and FH 2 2. 9. LMN is isosceles, L is the verte angle, LM 3 2, LN 2, and MN 5 2. Find the measures of the sides of KPL and classify each triangle by its sides. 0. K( 3, 2) P(2, ), L( 2, 3). K(5, 3), P(3, 4), L(, ) 2. K( 2, 6), P( 4, 0), L(3, ) 3. EIGN iana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there? Glencoe/McGraw-Hill 86 Glencoe Geometry
4- NME E PEI eading to Learn Mathematics lassifying riangles Pre-ctivity Why are triangles important in construction? ead the introduction to Lesson 4- at the top of page 78 in your tetbook. Why are triangles used for braces in construction rather than other shapes? Why do you think that isosceles triangles are used more often than scalene triangles in construction? eading the Lesson. upply the correct numbers to complete each sentence. a. In an obtuse triangle, there are acute angle(s), right angle(s), and obtuse angle(s). b. In an acute triangle, there are acute angle(s), right angle(s), and obtuse angle(s). c. In a right triangle, there are acute angle(s), right angle(s), and obtuse angle(s). Lesson 4-2. etermine whether each statement is always, sometimes, or never true. a. right triangle is scalene. b. n obtuse triangle is isosceles. c. n equilateral triangle is a right triangle. d. n equilateral triangle is isosceles. e. n acute triangle is isosceles. f. scalene triangle is obtuse. 3. escribe each triangle by as many of the following words as apply: acute, obtuse, right, scalene, isosceles, or equilateral. a. b. c. 70 80 30 35 4 5 3 Helping You emember 4. good way to remember a new mathematical term is to relate it to a nonmathematical definition of the same word. How is the use of the word acute, when used to describe acute pain, related to the use of the word acute when used to describe an acute angle or an acute triangle? Glencoe/McGraw-Hill 87 Glencoe Geometry
4- NME E PEI Enrichment eading Mathematics When you read geometry, you may need to draw a diagram to make the tet easier to understand. Eample onsider three points,,, and on a coordinate grid. he y-coordinates of and are the same. he -coordinate of is greater than the -coordinate of. oth coordinates of are greater than the corresponding coordinates of. Is triangle acute, right, or obtuse? o answer this question, first draw a sample triangle that fits the description. ide must be a horizontal segment because the y-coordinates are the same. Point must be located to the right and up from point. From the diagram you can see that triangle must be obtuse. y Q nswer each question. raw a simple triangle on the grid above to help you.. onsider three points,,, and 2. onsider three noncollinear points, on a coordinate grid. he J, K, and L on a coordinate grid. he -coordinates of and are the y-coordinates of J and K are the same. he y-coordinate of is same. he -coordinates of K and L between the y-coordinates of are the same. Is triangle JKL acute, and. he -coordinate of is less right, or obtuse? than the -coordinate of. Is angle of triangle acute, right, or obtuse? 3. onsider three noncollinear points, 4. onsider three points, G, H, and I, E, and F on a coordinate grid. on a coordinate grid. Points G and he -coordinates of and E are H are on the positive y-ais, and opposites. he y-coordinates of and the y-coordinate of G is twice the E are the same. he -coordinate of y-coordinate of H. Point I is on the F is 0. What kind of triangle must positive -ais, and the -coordinate EF be: scalene, isosceles, or of I is greater than the y-coordinate equilateral? of G. Is triangle GHI scalene, isosceles, or equilateral? Glencoe/McGraw-Hill 88 Glencoe Geometry
4-2 NME E PEI tudy Guide and Intervention ngles of riangles ngle um heorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found. ngle um he sum of the measures of the angles of a triangle is 80. heorem In the figure at the right, m m m 80. Eample Eample 2 25 35 Find m. m m m 80 ngle um heorem 25 35 m 80 ubstitution 60 m 80 dd. m 20 ubtract 60 Eercises from each side. Find the missing angle measures. 58 90 2 3 08 m m m 80 m 58 90 80 m 48 80 m 32 m 2 32 m 3 m 2 m E 80 m 3 32 08 80 m 3 40 80 m 3 40 E ngle um heorem ubstitution dd. ubtract 48 from each side. Vertical angles are congruent. ngle um heorem ubstitution dd. ubtract 40 from each side. Lesson 4-2 Find the measure of each numbered angle.. M 2. 62 P 90 N Q 30 3. V 4. 60 W 30 2 U M 66 58 P Q 2 N 3 50 5. 6. 20 2 52 60 30 W G Glencoe/McGraw-Hill 89 Glencoe Geometry
4-2 NME E PEI tudy Guide and Intervention (continued) ngles of riangles Eterior ngle heorem t each verte of a triangle, the angle formed by one side and an etension of the other side is called an eterior angle of the triangle. For each eterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that eterior angle. In the diagram below, and are the remote interior angles for eterior. Eterior ngle heorem he measure of an eterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m m m Eample Eample 2 60 80 m m m 60 80 40 Find m. Eterior ngle heorem ubstitution dd. P 78 Q 55 Find. m PQ m m Eterior ngle heorem 78 55 ubstitution 23 ubtract 55 from each side. Eercises Find the measure of each numbered angle.. X 2. 50 35 65 Y Z W 25 2 3. N M 4. 60 60 3 2 Q P U 80 2 35 V 3 36 Find. 5. 6. E 95 2 45 58 H G F Glencoe/McGraw-Hill 90 Glencoe Geometry
4-2 NME E PEI kills Practice ngles of riangles Find the missing angle measures.. 2. 80 IGE 73 46 Find the measure of each angle. 3. m 4. m 2 40 85 2 55 3 5. m 3 Find the measure of each angle. 6. m 7. m 2 2 55 70 3 50 Lesson 4-2 8. m 3 Find the measure of each angle. 9. m 0. m 2. m 3 2. m 4 60 80 40 4 5 2 05 3 3. m 5 Find the measure of each angle. 4. m 5. m 2 2 63 Glencoe/McGraw-Hill 9 Glencoe Geometry
4-2 NME E PEI Practice ngles of riangles Find the missing angle measures.. 2. 72? 40 55 Find the measure of each angle. 3. m 58 2 3 4. m 2 5. m 3 39 35 Find the measure of each angle. 6. m 7. m 4 68 36 2 70 3 4 65 82 5 8 6 8. m 3 9. m 2 0. m 5. m 6 Find the measure of each angle if and are right angles and m 84. 64 2. m 3. m 2 2 4. NUIN he diagram shows an eample of the Pratt russ used in bridge construction. Use the diagram to find m. 45 Glencoe/McGraw-Hill 92 Glencoe Geometry
4-2 NME E PEI eading to Learn Mathematics ngles of riangles Pre-ctivity How are the angles of triangles used to make kites? ead the introduction to Lesson 4-2 at the top of page 85 in your tetbook. he frame of the simplest kind of kite divides the kite into four triangles. escribe these four triangles and how they are related to each other. eading the Lesson. efer to the figure. a. Name the three interior angles of the triangle. (Use three letters to name each angle.) b. Name three eterior angles of the triangle. (Use three letters to name each angle.) c. Name the remote interior angles of E. d. Find the measure of each angle without using a protractor. i. ii. iii. F iv. E 2. Indicate whether each statement is true or false. If the statement is false, replace the underlined word or number with a word or number that will make the statement true. a. he acute angles of a right triangle are supplementary. b. he sum of the measures of the angles of any triangle is 00. c. triangle can have at most one right angle or acute angle. d. If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. e. he measure of an eterior angle of a triangle is equal to the difference of the measures of the two remote interior angles. f. If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35. g. n eterior angle of a triangle forms a linear pair with an interior angle of the triangle. E 39 23 F Lesson 4-2 Helping You emember 3. Many students remember mathematical ideas and facts more easily if they see them demonstrated visually rather than having them stated in words. escribe a visual way to demonstrate the ngle um heorem. Glencoe/McGraw-Hill 93 Glencoe Geometry
4-2 NME E PEI Enrichment Finding ngle Measures in riangles You can use algebra to solve problems involving triangles. Eample In triangle, m, is twice m, and m is 8 more than m. What is the measure of each angle? Write and solve an equation. Let m. m m m 80 2 ( 8) 80 4 8 80 4 72 43 o, m 2(43) or 86, m 43, and m 43 8 or 5. olve each problem.. In triangle EF, m E is three times m, and m F is 9 less than m E. 2. In triangle, m is 5 more than m, and m is 0 less than m. What is the measure of each angle? What is the measure of each angle? 3. In triangle JKL, m K is four times m J, and m L is five times m J. 4. In triangle XYZ, m Z is 2 more than twice m X, and m Y is 7 less than twice m X. What is the measure of each angle? What is the measure of each angle? 5. In triangle GHI, m H is 20 more than 6. In triangle MN, m M is equal to m N, m G, and m G is 8 more than m I. and m is 5 more than three times What is the measure of each angle? m N.What is the measure of each angle? 7. In triangle U, m U is half m, and m is 30 more than m.what 8. In triangle PQ, m P is equal to m Q, and m is 24 less than m P. is the measure of each angle? What is the measure of each angle? 9. Write your own problems about measures of triangles. Glencoe/McGraw-Hill 94 Glencoe Geometry
4-3 NME E PEI tudy Guide and Intervention ongruent riangles orresponding Parts of ongruent riangles riangles that have the same size and same shape are congruent triangles. wo triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure,. Eample If XYZ, name the pairs of congruent angles and congruent sides. X, Y, Z X Y, X Z, Y Z X Z Y Eercises Identify the congruent triangles in each figure.. K 2. 3. J L K J L M Name the corresponding congruent angles and sides for the congruent triangles. 4. F G L K 5. 6. U E J Lesson 4-3 Glencoe/McGraw-Hill 95 Glencoe Geometry
4-3 NME E PEI tudy Guide and Intervention (continued) ongruent riangles Identify ongruence ransformations If two triangles are congruent, you can slide, flip, or turn one of the triangles and they will still be congruent. hese are called congruence transformations because they do not change the size or shape of the figure. It is common to use prime symbols to distinguish between an original and a transformed. Eample Name the congruence transformation that produces from. he congruence transformation is a slide. ; ; ; ; ; y Eercises escribe the congruence transformation between the two triangles as a slide, a flip, or a turn. hen name the congruent triangles.. y 2. N y M N P M P 3. y P 4. y Q Q P 5. y 6. N y P M P N Glencoe/McGraw-Hill 96 Glencoe Geometry
4-3 NME E PEI kills Practice ongruent riangles Identify the congruent triangles in each figure.. P V 2. J X L Y W 3. Q 4. E P F G Name the congruent angles and sides for each pair of congruent triangles. 5. FGH 6. PQ U Verify that each of the following transformations preserves congruence, and name the congruence transformation. Lesson 4-3 7. 8. EF E F y E y E F F Glencoe/McGraw-Hill 97 Glencoe Geometry
4-3 NME E PEI Practice ongruent riangles Identify the congruent triangles in each figure.. 2. M N P L Q Name the congruent angles and sides for each pair of congruent triangles. 3. GKP LMN 4. N V Verify that each of the following transformations preserves congruence, and name the congruence transformation. 5. P P 6. LMN L M N y M y L N P P L N M QUILING For Eercises 7 and 8, refer to the quilt design. 7. Indicate the triangles that appear to be congruent. E 8. Name the congruent angles and congruent sides of a pair of congruent triangles. I H G F Glencoe/McGraw-Hill 98 Glencoe Geometry
4-3 NME E PEI eading to Learn Mathematics ongruent riangles Pre-ctivity Why are triangles used in bridges? ead the introduction to Lesson 4-3 at the top of page 92 in your tetbook. In the bridge shown in the photograph in your tetbook, diagonal braces were used to divide squares into two isosceles right triangles. Why do you think these braces are used on the bridge? eading the Lesson. If UWV, complete each pair of congruent parts. W U W W V 2. Identify the congruent triangles in each diagram. a. b. Q P c. M Q d. V N P 3. etermine whether each statement says that congruence of triangles is refleive, symmetric, or transitive. a. If the first of two triangles is congruent to the second triangle, then the second triangle is congruent to the first. b. If there are three triangles for which the first is congruent to the second and the second is congruent to the third, then the first triangle is congruent to the third. c. Every triangle is congruent to itself. U Lesson 4-3 Helping You emember 4. good way to remember something is to eplain it to someone else. Your classmate en is having trouble writing congruence statements for triangles because he thinks he has to match up three pairs of sides and three pairs of angles. How can you help him understand how to write correct congruence statements more easily? Glencoe/McGraw-Hill 99 Glencoe Geometry
4-3 NME E PEI Enrichment ransformations in he oordinate Plane he following statement tells one way to map preimage points to image points in the coordinate plane. (, y) ( 6, y 3) (, y) ( 6, y 3) y his can be read, he point with coordinates (, y) is mapped to the point with coordinates ( 6, y 3). With this transformation, for eample, (3, 5) is mapped to (3 6, 5 3) or (9, 2). he figure shows how the triangle is mapped to triangle XYZ. X Y Z. oes the transformation above appear to be a congruence transformation? Eplain your answer. raw the transformation image for each figure. hen tell whether the transformation is or is not a congruence transformation. 2. (, y) ( 4, y) 3. (, y) ( 8, y 7) y y 4. (, y) (, y) 5. (, y) 2, y y y Glencoe/McGraw-Hill 200 Glencoe Geometry
4-4 NME E PEI tudy Guide and Intervention Proving ongruence, Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. he ide-ide-ide () Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle. Postulate If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Eample Write a two-column proof. Given: and is the midpoint of. Prove: tatements easons.. Given 2. is the midpoint of. 2. Given 3. 3. efinition of midpoint 4. 4. efleive Property of 5. 5. Postulate Eercises Write a two-column proof.. Y 2. U Z X Given: X Y, X Z, Y Z Prove: XYZ tatements easons. X Y.Given Given: U, U Prove: U tatements easons. U. Given Lesson 4-4 2. XYZ 2. Post. 2. 3. U 2. efl. Prop. 3. Post. Glencoe/McGraw-Hill 20 Glencoe Geometry
4-4 NME E PEI tudy Guide and Intervention (continued) Proving ongruence, Postulate nother way to show that two triangles are congruent is to use the ide-ngle-ide () Postulate. Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Eample For each diagram, determine which pairs of triangles can be proved congruent by the Postulate. a. X b. G H c. In, the angle is not he right angles are he included angles, included by the sides congruent and they are the and 2, are congruent and. o the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the Postulate. EF JGH by the for two parallel lines. Postulate. P QP by the Postulate. Eercises Y Z F For each figure, determine which pairs of triangles can be proved congruent by the Postulate. E J P 2 Q. P 2. X 3. Q Y N M U W Z N P M L 4. V W 5. 6. F G K M J H Glencoe/McGraw-Hill 202 Glencoe Geometry
NME E PEI 4-4 kills Practice Proving ongruence, etermine whether KLM given the coordinates of the vertices. Eplain.. ( 3, 3), (, 3), ( 3, ), K(, 4), L(3, 4), M(, 6) 2. ( 4, 2), ( 4, ), (, ), K(0, 2), L(0, ), M(4, ) 3. Write a flow proof. Given: P E, P F E, F Prove: P EF P F E P E Given P F Given P EF E Given F Given P hird ngle heorem Lesson 4-4 etermine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 4. 5. 6. Glencoe/McGraw-Hill 203 Glencoe Geometry
NME E PEI 4-4 Practice Proving ongruence, etermine whether EF PQ given the coordinates of the vertices. Eplain.. ( 6, ), E(, 2), F(, 4), P(0, 5), Q(7, 6), (5, 0) 2. ( 7, 3), E( 4, ), F( 2, 5), P(2, 2), Q(5, 4), (0, 5) 3. Write a flow proof. Given: V is the midpoint of. Prove: V V V Given V V efleive Property V is the midpoint of. Given V V V V efinition of midpoint etermine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 4. 5. 6. 7. INIE MEUEMEN o measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths and are equal? Glencoe/McGraw-Hill 204 Glencoe Geometry
4-4 NME E PEI eading to Learn Mathematics Proving ongruence, Pre-ctivity How do land surveyors use congruent triangles? ead the introduction to Lesson 4-4 at the top of page 200 in your tetbook. Why do you think that land surveyors would use congruent right triangles rather than other congruent triangles to establish property boundaries? eading the Lesson. efer to the figure. a. Name the sides of LMN for which L is the included angle. N b. Name the sides of LMN for which N is the included angle. L M c. Name the sides of LMN for which M is the included angle. 2. etermine whether you 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 that you would use. If not, write not possible. a. b. E F G c. E H and G bisect each other. d. G E F H U Lesson 4-4 Helping You emember 3. Find three words that eplain what it means to say that two triangles are congruent and that can help you recall the meaning of the Postulate. Glencoe/McGraw-Hill 205 Glencoe Geometry
4-4 NME E PEI Enrichment ongruent Parts of egular Polygonal egions ongruent figures are figures that have eactly the same size and shape. here are many ways to divide regular polygonal regions into congruent parts. hree ways to divide an equilateral triangular region are shown. You can verify that the parts are congruent by tracing one part, then rotating, sliding, or reflecting that part on top of the other parts.. ivide each square into four congruent parts. Use three different ways. 2. ivide each pentagon into five congruent parts. Use three different ways. 3. ivide each heagon into si congruent parts. Use three different ways. 4. What hints might you give another student who is trying to divide figures like those into congruent parts? Glencoe/McGraw-Hill 206 Glencoe Geometry