hapter 4 ongruent Triangles That is water, not cement Section 4-1 lassifying Triangles lassification by ngle cute Triangle - a triangle with 3 acute angles! Equiangular Triangle - a triangle with 3 congruent acute angles! Obtuse Triangle - a triangle with one obtuse angle and two acute angles! Right triangle - a triangle with one right angle and two acute angles cute Equiangular Obtuse Right
Example 1 lassify the triangle by its angles 60 30 60 60 Equiangular 120 Obtuse 30 Example 2 lassify YZ as acute, equiangular, obtuse or right and explain your reasoning. 0 W YZ is a right triangle, because the sum of YW and WYZ is 90 Y 40 0 40 Z lassification by Sides Equilateral Triangle - a triangle with 3 congruent sides! Isosceles Triangle - a triangle with at least 2 congruent sides! Scalene Triangle - a triangle with no congruent sides Equilateral Isosceles Scalene
Example 3 lassify JMN, JKO, and OLN as Equilateral, Isosceles, or Scalene M JMN is Isosceles! L JKO is Scalene! K OLN is Equilateral J O N Example 4 Y is the midpoint of V and WY = 3.0. lassify VWY as equilateral, isosceles or scalene. VW = 4., W =.7 and V = 8.4 Explain your reasoning. V 4. W VWY is Scalene because it is a triangle that has three different side lengths. 3.0 4.2 Y 8.4 4.2.7 Example Find the measure of the sides of the isosceles triangle KLM with a base of KL. L K d + 6 4d 13 M 12 d 4d 13 = 12 d d 13 = 12 d = 2 d = KL = d + 6 KL = ( ) + 6 KL = 11 KM = ML = 12 d ( ) KM = ML = 12 KM = ML = 7
Still water Section 4-2 ngles of Triangles Triangle ngle Sum Theorem Triangle angle sum theorem (4.1) - The sum of the measures of the interior angles of a triangle is 180. 1 2 3 m 1+ m 2 + m 3 = 180 Example 1 Find the measures of each numbered angle. 42 8 1 2 81 3 m 1+ 42 + 8 = 180 m 1 = 3 m 1 = m 2 m 2 = 3 m 3+ 81+ 3 = 180 m 3 = 46
Exterior ngle Theorem Exterior angle - an angle formed by the side of a polygon and the extension of the other intersected side! Remote interior angle - ngles that are not addicted to the exterior angle of a polygon! Exterior angle theorem (4.2) - The measure of an exterior angle of a triangle is the sum of the measures of the two remote People always ask, interior angles. 1 2 3 4 is an exterior angle! 1 and 2 are remote interior angles to 4 m 4 = m 1+ m 2 4 what if we extended side and called that angle 4, would it still work? Yes yes it would. Example 2 Find the measure of FLW W m FLW = m W + m O 2x 48 = x + 32 x 48 = 32 x = 80 m FLW = 2x 48 32 x O m FLW = 2 ( 80 ) 48 ( 2x 48 ) L m FLW = 112 F orollaries orollary - a theorem with a proof that follows as a direct result from another theorem. orollaries may be used as reasons in a proof.! orollary to triangle angle sum theorem (4.1) - The acute angles of a right triangle are complementary! orollary to triangle angle sum theorem (4.2) - There can be at most one right or obtuse angle in a triangle
Example 3 Find the measure of each numbered angle 38 32 4 3 2 1 41 64 29 m 1 = 32 + 38 m 1 = 70 m 2 + 70 = 180 m 2 = 1 m 3 + 64 + 70 = 180 m 3 = 46 m 4 + 46 + 32 = 180 m 4 = 2 m + 2 + 41 = 180 m = 37 Still waters run deep or is it Still water runs deep whatever Section 4-3 ongruent Triangles ongruence by orresponding Parts ongruent polygons - polygons in which all the parts are congruent, every line segment and every angle.! orresponding parts - the matching parts of polygons. These can be corresponding angles or corresponding sides.! Two polygons are congruent if and only if their corresponding parts are congruent. PT E D F orresponding Parts of ongruent Triangles are ongruent
Example 1 Show the polygons are congruent by identifying all of the congruent corresponding parts and writing a statement of congruence for the polygons. D E Z DE VWYZ V Y W V W D Y E Z VW W D Y DE YZ E ZV Example 2 Given that ITP NGO find the values of x and y. O T 7. 7 11 6 40 I 13 P G x 2y ( 6y 14) 40 = 6y 14 7. = x 2y N Example 2 continued Given that ITP NGO find the values of x and y. 40 = 6y 14 7. = x 2y 4 = 6y 9 = y 7. = x 2( 9) 7. = x 18 2. = x
Prove Triangles ongruent Third angles theorem (4.3) - If two angles of one triangle are congruent to two angles of another triangle then the third angles of the triangles are congruent. E F D If D and E then F Example 3 Solve for x and y given: m = 30,m = 2x,m E = 70,m = 2y 60,m F = x + 4 D E F m = m D = 30 m = m E 2x = 70 x = 3 m = m F 2y 60 = x + 4 2y 60 = 3 2y 60 = 80 ( ) + 4 2y = 140 y = 70 Example 4 Given: DE GE, DF GF, D G, DFE GFE Prove: DEF GEF Statements DE GE, DF GF, D G, DFE GFE D FE FE DEF GEF F E DEF GEF Reasons Given Reflexive prop ( ) Third angles thm Def of polygons G
Properties of ongruence again I am not putting these in a table again. You have already seen them and written them down twice before! Reflexive! Symmetric! Transitive I think Stillwater is the name of a band Section 4-4 Proving Triangles ongruent SSS, SS Side Side Side Side Side Side Postulate (4.1) - If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. If Y, YZ and Z Then YZ Z Y You can use PT to say the corresponding angles of the triangles are congruent.
Example 1 Given: QU D,QD U Prove: QUD = DU Q D U QU! D Given UD! UD Ref. prop ( ) QD! U Given QUD DU! SSS Flow proof - a proof method where statements and reasons are organized like a flow chart, not requiring them to be in the standard two column structure. Example 2 Given: D(-, -1), V(-1, -2), W(-7, -4), L(1, -), P(2, -1), M(4, -7) Prove: DVW LPM In order to prove two triangles congruent, you must first prove the corresponding parts congruent. We have no skill set to deal with angles in coordinate geometry, so the only way to do this problem is to use the SSS postulate to prove the corresponding sides are congruent. The only method we have of calculating segment length in coordinate geometry is the distance formula. WD = ML = 29 DV = LP = 17 VW = PM = 40 = 2 Side ngle Side Side ngle Side postulate (4.2) - if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent If Y, Z and Then YZ Z Y You can use PT to say the other t wo corresponding angles and the missing side are also congruent.
Example 3 Given: is the midpoint of Q & DU Statement Prove: QD = U is the midpoint of Q & DU Q, D U Q U QD U Reason Given Midpoint thm Vertical thm D QUD = DU SS Example 4 Given: RQ! TS, RQ TS Prove: Q S Q T R S Statement RQ! TS, RQ TS QRT STR RT RT RQT TSR Q S Reason Given lt. Int. thm Ref. prop ( ) SS PT Maybe I am thinking of R, redence learwater Revival yeah, that s it Section 4- Proving Triangles ongruent - S, S
ngle Side ngle ngle side angle postulate (4.3) - if two angles and the included side of one triangle are congruent to two angles and the included angle of another triangle then the triangles are congruent. If Z, Z and Then YZ Z Y Inculded side - the side of a polygon bet ween t wo consecutive angles. The side shared by t wo consecutive angles of a polygon Oh, and you can use PT to make the congruence statement about the missing sides and angles Example 1 Given: is the midpoint of Q,QD! U Prove: QD = U Q U is the midpoint of Q,QD! U D Statement Reason Given Q Midpoint thm QD U Vertical thm Q lt. int thm QUD = DU S ngle ngle Side ngle angle side postulate (4.4) - If two angles and the non included side of one triangle are congruent to two angles and the non included side of another triangle then the triangles are congruent. If Y, Z and Then YZ Z Y gain you can use PT to make the congruence statements about the other angles and sides.
Example 2 Given: NKL NJM, KL JM Prove: LN MN Statement NKL NJM, KL JM J K KNL JNM KNL JNM L M LN MN Reason Given Ref. prop ( ) S PT N The ncient Secret of Geometry If you try to use ngle Side Side, then that is what you are ngle Side Side is called the mbiguous ase, and is discussed in Precalculus. asically sometimes it works, sometimes it doesn t and sometimes it makes t wo different triangles that work Right Triangle ongruence Leg leg theorem (4.6) - if the legs of one right triangle are congruent to the legs of another right triangle then the right triangles are congruent.! Hypotenuse angle theorem (4.7) - if the hypotenuse and one acute angle of one right triangle are congruent to the hypotenuse and one acute angle of another right triangle then the right triangles are congruent! Leg angle theorem (4.8) - If one leg and one acute angle of a right triangle are congruent to one led and one acute angle of another right triangle then the right triangles are congruent! Hypotenuse leg theorem (4.9) - If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle then the right triangles are congruent
Illustrations H HL L LL Summary of ongruence Methods SSS SS S S LL H L HL Is this Florida??? Section 4-6 Isosceles and Equilateral Triangles
Isosceles Triangles Legs - the two congruent sides of an isosceles triangle. Each leg is opposite a base angle! Vertex angle - the angled formed by the intersection of the two legs. the vertex angle is opposite the base! ase angles - the two congruent angles formed by the intersection of the base and each of the legs. each base angle is opposite one of the legs.! ase - the side of the isosceles triangle that is not either of the legs, opposite the vertex angle. Leg Vertex ngle ase! ngles ase Leg Theorems Isosceles Triangle Theorem (4.) - If two sides of a triangle are congruent then the angles opposite those sides are congruent! onverse of Isosceles triangle theorem (4.11) - If two angles of a triangle are congruent then the sides opposite those angles are congruent If then If then Example 1 Given:, Prove: Statement Reason, Given Let x be the midpoint of Every segment has one midpoint Draw Two points determine a line Midpoint thm Reflexive prop ( ) SSS PT
Equilateral Triangles orollary 4.3 - triangle is equilateral if and only if it is equiangular! orollary 4.4 - Each angle of an equilateral triangle measures 60 iff If then m = m = m = 60 Example 2 Find the m SRT T 0 S R Start with what we know m SRT + m RTS + m TSR = 180 m SRT = m RST m TSR = 0 Plug in and clean up m SRT + m SRT + m TSR = 180 2m SRT + m TSR = 180 2m SRT + ( 0) = 180 2m SRT = 130 m SRT = 6 Example 3 m SRT + m RTS + m SRT = 180 Find the measure of each variable 2m SRT + m RTS = 180 T 2( 6y 2) + ( 2) = 180 2 S R ( 4x + 20) ( 6y 2) m SRT + m RTS + m TSR = 180 m STR = 2 m TSR = 4x + 20 m SRT = 6y 2 m TSR = m SRT 12y 4 + 2 = 180 12y + 48 = 180 12y = 132 y = 11 6y 2 = 4x + 20 6( 11) 2 = 4x + 20 66 2 = 4x + 20 64 = 4x + 20 44 = 4x 11 = x x = 11
Example 4 H E Given: HEGO is regular, ONG is equilateral, O N N is the midpoint of GE,E " OG Prove: EN is equilateral G Given: HEGO is regular, ONG is equilateral, N is the midpoint of GE,E " OG Statement Prove: EN is equilateral HEGO is regular, ONG is equilateral, H E N is the midpoint of GE,E " OG HE E G GO OH Reason Given Def of regular polygon O N GO ON NG, GON ONG NGO Def. of Equilateral G OGN EN GN NE ONG EN N GN, NE NGO, EN ONG E N NE, NE EN NE EN is equilateral lt. Int. thm Midpoint thm SS PT Transitive prop ( ) Def. of Equilateral I think it may be Florida like in the Keys? Section 4-7 ongruence Transformations
Transformations Transformation - an operation that maps the original geometric figure onto a new figure! Preimage - the original geometric figure, your starting object! Image - the new geometric figure, your ending object! ongruence transformation - a transformation in which the position and orientation of the image may differ from the preimage, but the objects are still congruent. lso called a rigid transformation or an isometry. There are 3 congruence transformations we will discuss The ongruence Transformations Reflection - a transformation over a line called the line of reflection. Each point in the image is exactly the same distance away from the line of reflection as its corresponding point in the preimage! Translation - a transformation that moves all the points of the original figure the same distance in the same direction! Rotation - a transformation around a fixed point called the center through a specific angle in a specific direction. Each point in the image and its corresponding points in the preimage are the same distance away from the center Translation 1 1 - - 0 1 - - -1
Reflection 1-1 - - 0 1 1 - - -1 Rotation 1-1 - - 0 - - -1 Example 1-1 Identify the transformations. - - 1 1 1 0 1-1 - - 0 1-1 - - 0 - - - - - - -1-1 Translation Translation! Reflection! Rotation -1 Reflection! Rotation 1
Example 2 What transformation is modeled by these real world events:! The sliding door of a van opening! Translation Seeing the mirror image of a bridge in the water below it! Reflection revolving door! Rotation pendulum swinging! Rotation Opening the cover on my pristine TI-8 I bought in 1993! Translation Two twins staring at each other nose to nose Reflection Verify ongruence Verification of congruence (on triangles) after a transformation can work 2 ways! Use the distance formula to verify the side lengths and use SSS to state the image and preimage are congruent (easiest in the case of translation)! Use the distance formula to calculate the distance from each vertex of the triangle to the line of reflection or center (easiest for reflection and rotation) Example 3 Verify the transformation from YZ 1 1 - - 0 1 Y Z - - O O = 3 2 O = 149 O = 2 29 O = 3 2 OY = 149 OZ = 2 29-1