Lk Is Parallel To Z Z
11. Find YK. 12. Find JK.
13. Find XK. 14. Find JL.
15. Find YL. 16. Find KL.
17. Draw a triangle and label it ABC. Draw all the midpoints and label them.
Identify pairs of parallel sides and congruent angles in your triangle.
Use the fi gure at the right for Exercises 14.
1. What is the relationship between LN and MO?
2. What is the value of x?
3. Find LM. 4. Find LO.
Use the fi gure at the right for Exercises 58.
5. From the information given in the fi gure, how is TV related to SU?
6. Find TS. 7. Find UV. 8. Find SU.
9. At the right is a layout for the lobby of a building
placed on a coordinate grid.
a. At which of the labeled points would a receptionist chair
be equidistant from both entrances?
b. Is the statue equidistant from the entrances? How
do you know?
10. In baseball, the baseline is a segment connecting the bases. A
shortstop is told to play back 3 yd from the baseline and exactly
the same distance from second base and third base. Describe how
the shortstop could estimate the correct spot. Th ere are 30 yd
between bases. Assume that the shortstop has a stride of 36 in.
Use the fi gure at the right for Exercises 1115.
11. According to the fi gure, how far is A from CD? From CB?
12. How is CA
bisects l DCB Converse of l Bis. Thm.
58 58
32 32
Yes the statue is at a point that lieson the perpendicular bisector of a segment joining the entrances.
TVis the perpendicular bisector ofSU.
Use the fi gure at the right for Exercises 1619.
29. 30.
P 10
Lesson 51 Angles Of Triangles
Monitoring Progress
Draw an obtuse isosceles triangle and an acute scalene triangle.Answer:The figures of an obtuse isosceles triangle and an acute triangle are as follows:
Question 2.ABC has vertices A, B, and C, Classify the triangle by its sides. Then determine whether it is a right triangle.Answer:A , B , and C and the triangle is ABCWe know that,To find whether the given triangle is a right-angled triangle or not,We have to prove,AC² = AB² + BC²Where,AC is the distance between A and C pointsAB is the distance between A and B pointsBC is the distance between B and C pointsWe know that,The distance between 2 points = ² + ²Now,Let the given points be considered as A, B, and CSo,AB = ² + ² = 3² + 3²= 9 + 9 = 18BC = ² + ²= ² + 0²AC = ² + ²= ² + 3²
The measure of each acute angle is 90°, 64°, and 26°
Exercise 58 Coordinate Proofs
Vocabulary and Core Concept Check
Question 1.How is a coordinate proof different from other types of proofs you have studied?How is it the same?Answer:
Question 2.WRITINGExplain why it is convenient to place a right triangle on the grid as shown when writing a coordinate proof.Answer: Because the right triangle has the base and another leg on the same line in the coordinate plane.
Maintaining Progress and Modeling with Mathematics
In Exercises 3-6, place (he figure in a coordinate plane in a convenient way. Assign coordinates to each vertex. Explain the advantages of your placement.
Question 3.a right triangle with leg lengths of 3 units and 2 unitsAnswer:
a square with a side length of 3 unitsAnswer:Place the sides on the x-axis, y-axisIt is easy to find the lengths of horizontal and vertical segments and distances from the origin.
Question 5.an isosceles right triangle with leg length pAnswer:
a scalene triangle with one side length of 2mAnswer:
In Exercises 7 and 8, write a plan for the proof.
Question 7.Given Coordinates of vertices of OPM and ONM Prove OPM and ONM are isosceles triangles.Answer:
Given G is the midpoint of \.Prove GHJ GFOThe coordinates of G are OG = ² + ² = 9 + 4 = 13OF = ² + ² = 5GF = ² + ² = 2² + 2² = 8GH = ² + ² = 2² + 2² = 8HJ = ² + ² = 5² = 5GJ = ² + ² = 9 + 4 = 13OG GJ, OF HJ, GF GHAll the sides are congruent. So, GHJ GFO by SSS congruence theorem.
The length of one of the legs = 58.31
Question 13.A, B, CAnswer:
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Exercise 51 Angles Of Triangles
Vocabulary and Core Concept Check
Question 1.Can a right triangle also be obtuse? Explain our reasoning.Answer:
Question 2.COMPLETE THE SENTENCEThe measure of an exterior angle of a triangle is equal to the sum of the measures of the two ____________ interior angles.Answer:The given statement is:The measure of an exterior angle of a triangle is equal to the sum of the measures of the two ____________ interior angles.Hence,The completed form of the given statement is:The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
Monitoring Progress and Modeling with Mathematics
In Exercises 3-6, classify the triangle by its sides and by measuring its angles.
Question 3.
The given figure is:We know that,If any side is not equal to each other in the triangle, then the triangle is called a Scalene triangleThe angle greater than 90° is called as Obtuse angleAn angle less than 90° is called an Acute angleHence, from the above,We can conclude that ABC is an Acute scalene triangle
In Exercises 7-10, classify ABC by its sides. Then determine whether it is a right triangle.
Question 7.A, B, Answer:
Question 9.A, B, CAnswer:
We can conclude that the given angles forms a triangle
Question 40.THOUGHT-PROVOKINGFind and draw an object that can be modeled by a triangle and an exterior angle. Describe the relationship between the interior angles of the triangle and the exterior angle in terms of the object.Answer:
Lesson 53 Proving Triangle Congruence By Sas
Monitoring Progress
In the diagram, ABCD is a square with four congruent sides and four rightangles. R, S, T, and U are the midpoints of the sides of ABCD. Also, \ \ and \ \.Question 1.
Answer:Given that,ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides of ABCD.So, SV = VU, RS = RU, S = UAll the corresponding sides and angles are congruent. So, SVR UVR.
Question 2.
B = D, S = U, R = T and BS = UD, BR = TD, SR = TUSo, BSR DUT
Question 3.You are designing the window shown in the photo. You want to make DRA congruent to DRG. You design the window so that \ \ and ADR GDR. Use the SAS Congruence Theorem to prove DRA DRG.
Answer:When you rotate DRG 90° towards left, then DRG coincides DRA.so, according to the Side Angle Side Congruence theorem, DRA DRG
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Congruent Triangles Cumulative Assessment
Question 1.Your friend claims that the Exterior Angle Theorem can be used to prove the Triangle Sum Theorem . Is your friend correct? Explain your reasoning.Answer: Yes
Question 2.Use the steps in the construction to explain how you know that the line through point P is parallel to line m.Answer: From step 4 the red line is parallel to m and passes through the pint P. So, point P is parallel to line m.
Question 3.The coordinate plane shows JKL and XYZa. Write a composition of transformations that maps JKL to XYZAnswer:The coordinates of J, L K, X, Y, ZJL = ² + ² = 3XZ = ² + ² = 3JK = ² + ² = 5XY = ² + ² = 5KL = ² + ² = 8YZ = ² + ²= 8JK XY, JL XZ, KL YZ
b. Is the composition a congruence transformation? If so, identify all congruent corresponding parts.Answer: JKL XYZ using the SSS Congruence theorem.
Question 4.The directed line segment RS is shown. Point Q is located along \ so that the ratio of RQ to QS is 2 to 3. What are the coordinates of point Q? Q
Lesson 54 Equilateral And Isosceles Triangles
Monitoring Progress
Copy and complete the statement.
Question 1.If \ \, then _______ _______ .
Answer:G = K by using the base angles theorem.
Question 2.
What is the relationship between the base angles of an isosceles triangle? Explain.Answer:The base angle of an isosceles triangle is congruent.
Explanation:An isosceles triangle consists of two equal legs and one base. There are two equal angles on the base. These are called base angles. So, the base angles are opposite to the congruent sides. Using the base angle theorem, we can conclude that the base angles of an isosceles triangle are congruent.
Monitoring Progress and Modeling with Mathematics
In Exercises 3-6. copy and complete the statement. State which theorem you used.
Question 3.If \ \ then _____ _____ .Answer:
Question 4.If \ \ then _____ _____ .Answer: If \ \ then A E by the Base Angles Theorem.
Question 5.
1.25 inchesAnswer:Construct the base leg with length 1.25 inches and draw an arc with center at one of the endpoints and a radius 1.25 in. Draw another arc that intersects the first arc with the same radius and from another endpoint of the base leg. Connect intersection of arcs to two endpoints of the base to form an equilateral triangle.
Question 19.Describe and correct the error in finding the length of \.Answer:
a. Explain why ABC is isosceles.Answer:AB and BC have the same length.So, ABC is isosceles.
d. Find the measure of BAE.Answer: BAE BCE.
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Big Ideas Math Book Geometry Answer Key Chapter 5 Congruent Triangles
Prepare using the Big Ideas Math Book Geometry chapter 5 Congruent Triangles Answer Key and get a good hold of the entire concepts. Clarify all your doubts taking help of the BIM Book Geometry Ch 5 Congruent Triangles Solutions provided. Simply tap on the BIM Geometry Chapter 5 Congruent Triangles Answers and prepare the corresponding topic in no time. Big Ideas Math Geometry Congruent Triangles Solution Key covers questions from Lessons 5.1 to 5.8, Practice Tests, Assessment Tests, Chapter Tests, etc. Attempt the exam with confidence and score better grades in exams.
Lesson 57 Using Congruent Triangles
Monitoring Progress
Explain how you can prove that A C.Answer:If you can prove that ABD CBD, then A C.AB BC, AD CDBD BD by reflexive property of congruenceSo, ABD CBD by SSS congruence theorem.
Question 2.In Example 2, does it mailer how far from point N you place a stake at point K? Explain.Answer:Change the placement of stake at point K away from the original point.The midpoint M changes but the congruency remain the same.Similarly, move point K close to N.The midpoint changes but the congruency is not affected.
Question 3.Write a plan to prove that PTU UQP.Answer:PU PU by reflexive property of congruencePTU UQP by SAS congruence theorem.
Question 4.Use the construction of an angle bisector on page 42. What segments can you assume are congruent?Answer:Refer to angle bisector on page 42.Distance from A to B is the same as A to C.Thus AB and AC line segments have the same length.Since the line segments, AB and AC have the same length.AB ACAlso, CAG = GAB CAG GAB
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Lesson 56 Proving Triangle Congruence By Asa And Aas
Monitoring Progress
Question 1.Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use.Answer:WX YZ, XY WZ, and 1 3So, WXY WYZ by the AAS congruence theorem.
Question 2.In the diagram, \ \, \ \, and \ \ . Prove ABC DEF.Answer:\ \, A DSo, the given information is not enough to prove that ABC DEF.
Question 3.In the diagram, S U and \\ . Prove that RST VYTAnswer:So, the given information is not enough to prove that RST VYT
Exercise 53 Proving Triangle Congruence By Sas
vocabulary and core concept check
Question 1.What is an included angle?Answer:
Question 2.COMPLETE THE SENTENCEIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then __________ .Answer:If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then two triangles are congruent according to the SAS Congruence Theorem.
Monitoring progress and Modeling with Mathematics
In Exercises 3-8, name the included an1e between the pair of sides given.
Question 3.\ and \Answer:
\ and \Answer: KLP is the included angle between \ and \
Question 5.\ and \Answer:
\ and \Answer: KJL is the included angle between \ and \
Question 7.\ and \Answer:
\ and \Answer: KPL is the included angle between \ and \
In Exercises 9-14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem . Explain.
Question 9.
Given \ \, \ \Prove PQT RST\ \\ \In a parallelogram, diagonals bisect at 90 degrees.PTQ = STR PQT RST
In Exercises 19-22, use the given information to name two triangles that are congruent. Explain your reasoning.
Question 19.SRT URT, and R is the center of the circle.Answer:
RSTUV is a regular pentagon.Answer:
Question 22.\ \, \ \, and M and L are centers of circles.Answer:NK = NK by the reflexive property of congruenceSo, MKN LKN
Question 23.
Question 33.
Question 35.
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Proving Triangle Congruence By Asa And Aas
Exploration 1
Determining Whether SSA Is Sufficient
Work with a partner.a. Use dynamic geometry software to construct ABC. Construct the triangle so that vertex B is at the origin. \ has a length of 3 units. and \ has a length of 2 units.Answer:
b. Construct a circle with a radius of 2 units centered at the origin. Locate point D where the circle intersects \. Draw \.Answer:
c. ABC and ABD have two congruent sides and a non included congruent angle.Name them.
d. Is ABC ABD? Explain your reasoning.Answer:
e. Is SSA sufficient to determine whether two triangles are congruent? Explain your reasoning.Answer:
Exploration 2
Determining Valid Congruence Theorems
Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. For those that are not valid. write a counter example. Explain your reasoning.CONSTRUCTING VIABLE ARGUMENTSTo be proficient in math, you need to recognize and use counterexamples.
| Possible Congruence Theorem |
What information is sufficient to determine whether two triangles are congruent?Answer:
Question 4.Is it possible to show that two triangles are congruent using more than one congruence theorem? If so, give an example.Answer:1. Proving ABC BAD using ASAIn ABC and BADHence using the ASA Congruence theorem, ABC and BAD are congruent.2. Proving ABC and BAD using SSSABC and BADAC = BD AB = AB BC = AD Hence using the SSS congruence theorem, ABC and BAD are congruent.
Congruent Triangles Mathematical Practices
Classify each statement as a definition, a postulate, or a theorem. Explain your reasoning.
Question 1.In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is 1.Answer:The given statement is:In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is 1.We know that,According to the parallel and perpendicular lines theorem, two non-vertical lines are perpendicular if and only if the product of their slopes is -1Hence, from the above,We can conclude that the given statement is a Theorem
Question 2.If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.Answer:The given statement is:If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.We know that,According to the Linear pair perpendicular theorem,When two straight lines intersect at a point and form a linear pair of congruent angles, then the lines are perpendicularHence, from the above,We can conclude that the given statement is a Theorem
Question 3.If two lines intersect to form a right angle. then the lines are perpendicular.Answer:If two lines intersect to form a right angle. then the lines are perpendicular.We know that,According to the Perpendicular lines theorem,When two lines intersect to form a right angle, the lines are perpendicularHence, from the above,We can conclude that the given statement is a Theorem
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Exercise 55 Proving Triangle Congruence By Sss
Vocabulary and Core Concept Check
Question 1.The side opposite the right angle is called the __________of the right triangle.Answer:
JL JL by reflexive property of congruenceSo, JKL LJM by SSS Congruence Theorem.
In Exercises 11 and 12, determine whether the figure is stable. Explain your reasoning.
Question 11.
Question 12.Answer: No, the figure has no diagonals. So the figure is not stable.
In Exercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. Then write a proof.
Question 13.Given \ \\ \\ \Prove BAD CDAAnswer:
Question 14.Given G is the midpoint of \, \ \, E and H are right angles.Prove EFG HIGEG GH as G is the midpoint of \\ \, E HEFG HIG by SAS congruence theorem.
In Exercises 15 and 16. write a proof.
Question 15.Given \ \, \ \Prove LMJ JKLAnswer:
Question 16.Given \ \, \ \, \ \Prove VWX WVZXY YZ, WY VY, WX VZVWX WVZ by SSS Congruence theorem
CONSTRUCTIONIn Exercises 17 and 18, construct a triangle that is congruent to QRS using the SSS Congruence Theorem Theorem 5.8).
Question 17.
Question 18.Answer:At first, construct a side that is congruent to QS. Draw an arc with the compass with Q as center and radius as QR. Draw another arc that intersects the first arc with S as center and radius as SR. Join the point to Q and S to form a circle that is congruent to QRS.
In Exercises 25-28. use the given coordinates to determine whether ABC DEF.