is a transversa 6-3 Proving Triangles Congruent-SSS, SAS are parallel

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1 c Sample answer: ; is a transversa CAB ACD are alternate interior angles Sinc CAB ACD are congruent corresponding angl are parallel 6-3 Proving Triangles Congruent-SSS SAS 1 OPTICAL ILLUSION The figure below is a pattern formed using four large congruent squares four small congruent squares 2 EXTENDED RESPONSE Triangle ABC has vertices A( 3 5) B( 1 1) C( 1 5) Triangle XYZ has vertices X(5 5) Y(3 1) Z(3 5) a Graph both triangles on the same coordinate plane b Use your graph to make a conjecture as to whether the triangles are congruent Explain your reasoning c Write a logical argument using coordinate geometry to support your conjecture a a How many different-sized triangles are used to cre b Use the Side-Side-Side Congruence Postulate to p c What is the relationship between reasoning? a There are two differently sized triangles in the figu smaller triangle in the interior of b ABCD is a square 1 ABCD is a square 2 (Def of a square) 3 4 (Reflex Prop (SSS) c Observe the graph the triangles are right triangles In triangle ABC AC = 2 BC = 4 Use the Pythagorean Theorem to find AB ) c Sample answer: ; is a transversa CAB ACD are alternate interior angles Sinc CAB ACD are congruent corresponding angl are parallel 2 EXTENDED RESPONSE Triangle ABC has vertices A( 3 5) B( 1 1) C( 1 5) Triangle XYZ has vertices X(5 5) Y(3 1) Z(3 5) a Graph both triangles on the same coordinate plane b Use your graph to make a conjecture as to whether the triangles are congruent Explain your reasoning c Write a logical argument using coordinate esolutions Manualto - Powered Cognero geometry supportbyyour conjecture a b The triangles look the same size shape so we can conjecture that they are congruent Similarly In triangle XYZ ZX = 2 ZY = 4 Use the Pythagorean Theorem to find XY The corresponding sides have the same measure are congruent So by SSS 3 EXERCISE In the exercise diagram if Page 1 is LPM NOM equilateral write a paragraph proof to show that

2 1 The corresponding have the samesas measure 6-3 Proving Triangles sides Congruent-SSS are congruent So by SSS 3 EXERCISE In the exercise diagram if is LPM NOM equilateral write a paragraph proof to show that (Reflex Prop (SAS) ) PROOF Write the specified type of proof 5 paragraph proof Sample answer: We are given that Since is equilateral by the definition of an equilateral triangle Therefore is congruent to by the Side-Angle-Side Congruent Postulate We know that by the Reflexive Property Since by SSS 6 two-column proof Write a two-column proof (Reflex Prop (SAS) ) PROOF Write the specified type of proof 5 paragraph proof 1 2 C is the midpoint of (Def of Segment Bisectors) 3 (Midpoint Thm) 4 (SSS) 7 BRIDGES The Sunshine Skyway Bridge in Florida is the world s longest cable-stayed bridge spanning 41 miles of Tampa Bay It is supported using steel cables suspended from two concrete supports If the supports are the same height above the roadway perpendicular to the roadway the topmost cables meet at a point midway between the supports prove that the two triangles shown in the photo are congruent We know that the Reflexive esolutions Manual - Powered byby Cognero Since by SSS Property Page 2

3 2 C is the midpoint of (Def of Segment Bisectors) 3 (Midpoint Thm) 6-3 Proving Triangles Congruent-SSS SAS 4 (SSS) has end points N(5 2) O(1 1) 7 BRIDGES The Sunshine Skyway Bridge in Florida is the world s longest cable-stayed bridge spanning 41 miles of Tampa Bay It is supported using steel cables suspended from two concrete supports If the supports are the same height above the roadway perpendicular to the roadway the topmost cables meet at a point midway between the supports prove that the two triangles shown in the photo are congruent has end points O(1 1) M (2 5) ABC EDC are right angles C is the midpoint of 1 are right angles C is the midpoint of 2 (All rt s ) 3 (Midpoint Thm) 4 (SAS) Similarly find the lengths of has end points Q( 4 4) R( 7 1) CCSS SENSE-MAKING Determine whether Explain 8 M (2 5) N(5 2) O(1 1) Q( 4 4) R( 7 1) S( 3 0) Use the Distance Formula to find the lengths of has end points R( 7 1) S( 3 0) has end points M (2 5) N(5 2) has end points S( 3 0) Q( 4 4) has end points N(5 2) O(1 1) esolutions Manual - Powered by Cognero Page 3

4 has end points S( 3 0) Q( 4 4) 6-3 Proving Triangles Congruent-SSS SAS Similarly find the lengths of has end points Q(3 3) R(4 4) So Each pair of corresponding sides has the same measure so they are congruent SSS by 9 M (0 1) N( 1 4) O( 4 3) Q(3 3) R(4 4) S (3 3) Use the Distance Formula to find the lengths of has end points R(4 4) S(3 3) has end points M (0 1) N( 1 4) has end points S(3 3) Q(3 3) has end points N( 1 4) O( 4 3) The corresponding sides are not congruent so the triangles are not congruent 10 M (0 3) N(1 4) O(3 1) Q(4 1) R(6 1) S(9 1) Use the Distance Formula to find the lengths of has end points O( 4 3) M (0 1) has end points M (0 3) N(1 4) Similarly find the lengths of esolutions Manual - Powered by Cognero has end points Q(3 3) R(4 4) Page 4

5 has end points M (0 3) N(1 4) has end points R(6 1) S(9 1) 6-3 Proving Triangles Congruent-SSS SAS has end points N(1 4) O(3 1) has end points S(9 1) Q(4 1) The corresponding sides are not congruent so the triangles are not congruent has end points O(3 1) M (0 3) 11 M (4 7) N(5 4) O(2 3) Q(2 5) R(3 2) S(0 1) Use the Distance Formula to find the lengths of has end points M (4 7) N(5 4) Similarly find the lengths of has end points Q(4 1) R(6 1) has end points N(5 4) O(2 3) has end points R(6 1) S(9 1) has end points O(2 3) M (47) esolutions Manual - Powered by Cognero Page 5

6 6-3 Proving Triangles Congruent-SSS SAS So Each pair of corresponding sides has the same measure so they are congruent SSS has end points O(2 3) M (47) by PROOF Write the specified type of proof 12 two-column proof bisects Similarly find the lengths of has end points Q(2 5) R(3 2) 1 bisects 2 are right angles (Def of ) 3 (all right angles are ) 4 (Def of bisects) 5 (Ref Prop) 6 (SAS) has end points R(3 2) S(0 1) 13 paragraph proof R is the midpoint of Since R is the midpoint of has end points S(0 1) Q(2 5) midpoint Theorem So by definition of a by the Vertical Angles by SAS PROOF Write the specified type of proof 14 flow proof L is the midpoint of So Each pair of corresponding sides has the same measure so they are congruent esolutions Manual - Powered by Cognero SSS PROOF Write the specified type of proof by Page 6

7 Property Since is equilateral it is a special type of isosceles triangle so By the SideAngle-Side Congruence Postulate by definition of a by the Vertical Angles midpoint TheoremTriangles Congruent-SSS SAS 6-3 Proving So by SAS CCSS ARGUMENTS Determine which postulate can be used to prove that the triangles are congruent If it is not possible to prove congruence write not possible PROOF Write the specified type of proof 14 flow proof L is the midpoint of 16 The corresponding sides are congruent; SSS 17 The triangles have two corresponding sides congruent but no information is given about the included angle or the third pair of sides; not possible paragraph proof is equilateral Information is given about two pairs of corresponding sides one pair of corresponding angles but the congruent angles are not in the interior of the congruent sides; not possible bisects Y 19 We know that bisects so Also by the Reflexive Property Since is equilateral it is a special type of isosceles triangle so By the SideAngle-Side Congruence Postulate CCSS ARGUMENTS Determine which postulate can be used to prove that the esolutions Manual are - Powered by Cognero triangles congruent If it is not possible to prove congruence write not possible The triangles have two pairs of corresponding sides congruent the interior angles are congruent; SAS 20 SIGNS Refer to the diagram a Identify the three-dimensional figure represented by the wet floor sign b If prove that c Why do the triangles not look congruent in the diagram? Page 7

8 The triangles have two pairs of corresponding sides congruenttriangles the interior angles are congruent; 6-3 Proving Congruent-SSS SAS SAS 20 SIGNS Refer to the diagram a Identify the three-dimensional figure represented by the wet floor sign b If prove that 3 (SSS) c Sample answer: The object is three-dimensional so when it is viewed in two dimensions the perspective makes it look like the triangles are differently shaped PROOF Write a flow proof K is the midpoint of 21 c Why do the triangles not look congruent in the diagram? a triangular pyramid b 1 2 (Refl Prop) 3 (SSS) c Sample answer: The object is three-dimensional so when it is viewed in two dimensions the perspective makes it look like the triangles are differently shaped PROOF Write a flow proof K is the midpoint of esolutions Manual - Powered by Cognero SOFTBALL Use the diagram of a fast-pitch softball diamond shown Let F = first base S = second base T = third base P = pitcher s point R = home plate a Write a two-column proof to prove that the distance from first base to third base is the same as the distance from home plate to second base b Write a two-column proof to prove that the angle formed between second base home plate third base is the same as the angle formed between Page 8 second base home plate first base

9 R = home plate a Write a two-column proof to prove that the distance from first base to third base is the same as the distance from home plate to secondsas base 6-3 Proving Triangles Congruent-SSS b Write a two-column proof to prove that the angle formed between second base home plate third base is the same as the angle formed between second base home plate first base 1 2 angles 3 (All rt s are 4 (SAS) 5 are right ) PROOF Write a two-column proof 24 a are right angles 1 2 (Alt Int s) 3 (Reflex Prop) 4 (SAS) angles 3 (All rt s are 4 (SAS) 5 b esolutions Manual - Powered by Cognero 1 are right ) are right angles (Reflex Prop) AB = CB DB = EB (Def segments) 7 AB + DB = CB + EB (Add Prop =) 8 AD = AB + DB CE = CB + EB (Segment addition) 9 AD = CE (Subst Prop =) 10 (Def segments) 11 (SSS) Page 9 26 CCSS ARGUMENTS Write a paragraph proof

10 congruence ML = LM By the Segment Addition Postulate PL = PM + ML KM = KL + LM By substitution PL = KL + LM PL = KM By the definition of congruence so by SSS 1 2 (Alt Int s) 3 (Reflex Prop) 6-3 Proving Triangles (SAS) Congruent-SSS SAS 4 ALGEBRA Find the value of the variable that yields congruent triangles Explain (Reflex Prop) AB = CB DB = EB (Def segments) 7 AB + DB = CB + EB (Add Prop =) 8 AD = AB + DB CE = CB + EB (Segment addition) 9 AD = CE (Subst Prop =) 10 (Def segments) 11 (SSS) Given that by CPCTC The value of y can be found using the sides or the angles 26 CCSS ARGUMENTS Write a paragraph proof 28 so by the definition of congruence GH = JH HL = HM By the Segment Addition Postulate GL = GH + HL JM = JH + HM By substitution GL = JH + HM GL = JM By the definition of congruence so by the definition of congruence PM = KL by the Reflexive Property of Congruence so by the definition of congruence ML = LM By the Segment Addition Postulate PL = PM + ML KM = KL + LM By substitution PL = KL + LM PL = KM By the definition of congruence so by SSS ALGEBRA Find the value of the variable that Explain yields congruent esolutions Manual - Poweredtriangles by Cognero 27 By the definition of congruence By the definition of congruence Page 10

11 6-3 Proving Triangles Congruent-SSS SAS 29 CHALLENGE Refer to the graph shown a Describe two methods you could use to prove that is congruent to You may not use a ruler or a protractor Which method do you think is more efficient? Explain b Are congruent? Explain your reasoning 28 By the definition of congruence By the definition of congruence 29 CHALLENGE Refer to the graph shown a Describe two methods you could use to prove that is congruent to You may not use a ruler or a protractor Which method do you think is more efficient? Explain b Are congruent? Explain your reasoning a Sample answer: Method 1: You could use the Distance Formula to find the length of each of the sides then use the Side-Side-Side Congruence Postulate to prove the triangles congruent Method 2 : You could find the slopes of to prove that they are perpendicular that are both right angles You can use the Distance Formula to prove that is congruent to Since the triangles share the leg you can use the Side-Angle-Side Congruence Postulate; Sample answer: I think that method 2 is more efficient because you only have two steps instead of three b Sample answer: Yes; the slope of is 1 the slope of is are opposite reciprocals so Since is perpendicular to they are perpendicular are both 90 Using the Distance Formula the length of is the length of Since to is congruent to Angle-Side Congruent Postulate a Sample answer: Method 1: You could use the Distance Formula to find the length of each of the sides then use the Side-Side-Side Congruence Postulate to prove the triangles congruent esolutions Manual - Powered by Cognero Method 2 : You could find the slopes of is is congruent by the Side- 30 REASONING Determine whether the following statement is true or false If true explain your reasoning If false provide a counterexample If the congruent sides in one isosceles triangle have the same measure as the congruent sides in another isosceles triangle then the trianglespage are 11 congruent

12 the length of Since is is congruent to is congruent to by the Side6-3 Proving Triangles Congruent-SSS SAS Angle-Side Congruent Postulate 30 REASONING Determine whether the following statement is true or false If true explain your reasoning If false provide a counterexample If the congruent sides in one isosceles triangle have the same measure as the congruent sides in another isosceles triangle then the triangles are congruent False the triangles shown below are isosceles triangles with two pairs of sides congruent the third pair of sides not congruent 31 ERROR ANALYSIS Bonnie says that by SAS Shada disagrees She says that there is not enough information to prove that the two triangles are congruent Is either of them correct? Explain Sample answer: Using a ruler I measured all of the sides they are congruent so the triangles are congruent by SSS 33 WRITING IN MATH Two pairs of corresponding sides of two right triangles are congruent Are the triangles congruent? Explain your reasoning Case 1: The two pairs of congruent angles includes the right angle You know the hypotenuses are congruent one of the legs are congruent Then the Pythagorean Theorem says that the other legs are congruent so the triangles are congruent by SSS Case 2: The two pairs of congruent angles do not include the right angle You know the legs are congruent the right angles are the included angles all right angles are congruent then the triangles are congruent by SAS 34 ALGEBRA The Ross Family drove 300 miles to visit their grparents Mrs Ross drove 70 miles per hour for 65% of the trip 35 miles per hour or less for 20% of the trip that was left Assuming that Mrs Ross never went over 70 miles per hour how many miles did she travel at a speed between miles per hour? A 195 B 84 C 21 D 18 For the 65% of the distance she drove 70 miles per Shada; for SAS the angle must be the included angle here it is not included 32 OPEN ENDED Use a straightedge to draw obtuse triangle ABC Then construct so that it is congruent to using either SSS or SAS Justify your construction mathematically verify it using measurement Sample answer: Using a ruler I measured all of the sides they are congruent so the triangles are congruent by SSS or 195 miles hour So she covered The remaining distance is = 105 miles For the 20% of the remaining distance she drove less than 35 miles per hour So she covered or 21 miles So totally she covered or 216 miles at the speed 70 miles per hour less than 30 miles per hour Bu the trip has 300 miles so the remaining distance is or 84 miles She has traveled 84 miles at a speed between miles per hour So the correct option is B 35 In the figure 33 WRITING IN MATH Two pairs of corresponding sides of two right triangles are congruent Are the triangles congruent? Explain your reasoning Case 1: The two pairs of congruent angles includes the right angle You know the hypotenuses are congruent one of the legs are congruent Then esolutions Manual - Powered by Cognero the Pythagorean Theorem says that the other legs are congruent so the triangles are congruent by SSS Case 2: The two pairs of congruent angles do not What additional information could be used to prove that F Page 12 G H

13 21 or 216 miles at the speed 70 miles per hour less than 30 miles per hour Bu the trip has 300 miles so the remaining distance is or 84 miles She has traveled 84 Congruent-SSS miles at a speed between 6-3 Proving Triangles SAS miles per hour So the correct option is B 35 In the figure What additional information could be used to prove that F G H J We can prove by SAS if we knew that So the needed information is The correct option is F 36 EXTENDED RESPONSE The graph below shows the eye colors of all of the students in a class What is the probability that a student chosen at rom from this class will have blue eyes? Explain your reasoning the probability of romly choosing a student with blue eyes is the number of students with blue eyes divided by 20 Since there are 3 students with blue eyes the probability is 37 SAT/ACT If 4a + 6b = 6 2a + b = 7 what is the value of a? A 2 B 1 C2 D3 E4 Solve the equation 2a + b = 7 for b Substitute b = 2a 7 in 4a + 6b = 6 So the correct option is D In the diagram 38 Find x By the definition of congruence PL = TQ First you have to find how many students there are in the class There are or 20 Then the probability of romly choosing a student with blue eyes is the number of students with blue eyes divided by 20 Since there are 3 students with blue eyes the probability is 37 SAT/ACT If 4a + 6b = 6 2a + b = 7 what is the value of a? A 2 B 1 C2 D 3 Manual - Powered by Cognero esolutions E4 39 Find y By the definition of congruence Page 13

14 6-3 Proving Triangles Congruent-SSS SAS 39 Find y By the definition of congruence 40 ASTRONOMY The Big Dipper is a part of the larger constellation Ursa Major Three of the brighter stars in the constellation form If m R = 41 m S = 109 find m A The sum of the measures of the angles of a triangle is 180 So esolutions Manual - Powered by Cognero Page 14

8-2 Parallelograms. Refer to Page 489. a. If. b. If c. If MQ = 4, what is NP? SOLUTION:

8-2 Parallelograms. Refer to Page 489. a. If. b. If c. If MQ = 4, what is NP? SOLUTION: 1 NAVIGATION To chart a course, sailors use a parallel ruler One edge of the ruler is placed along the line representing the direction of the course to be taken Then the other ruler is moved until its