Proving Triangle Congruence By Sss
Exploration 1
Work with a partner.Use dynamic geometry software.
a. Construct circles with radii of 2 units and 3 units centered at the origin. Label the origin A. Then draw \ of length 4 units.Answer:
b. Move \ so that B is on the smaller circle and C is on the larger circle. Then draw ABC.Answer:
c. Explain why the side lengths of ABC are 2, 3, and 4 units.Answer:
d. Find mA, mB, and mC.Answer:
e. Repeat parts and several times, moving \ to different locations. Keep track of our results by copying and completing the table below. What can you conclude?USING TOOLS STRATEGICALLYTo be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.Answer:
QT TR, PQ SR, PT STSo, QPT RST by SSS Congruence Theorem
Determine whether the figure is stable. Explain your reasoning.
Question 4.Answer:The figure is not stable. Because it doesnt have a triangle. By the SSS Congruence Theorem, those triangles cannot change shape.
Question 5.
Answer:The figure is stable as the diagonal forms the triangle. By the SSS Congruence Theorem, these triangles cannot change the shape, so the figure is stable.
Question 6.
Answer:The figure is stable as the diagonal forms the triangle. By the SSS Congruence Theorem, these triangles cannot change the shape, so the figure is stable.
Use the diagram.
Redraw ABC and DCB side by side with corresponding parts in the same position.
Answer:
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Basics Of Geometry Cumulative Assessment
Question 1.Use the diagram to determine which segments, if any, are congruent. List all congruent segments.Answer:The Congruency means having the same size and the same lengthBut, from the above coordinate plane,We can observe that there are no congruent linesHence, from the above,We can conclude that there are no congruent line segments in the coordinate plane
Question 2.Order the terms so that each consecutive term builds off the previous term.plane segment line point rayAnswer:a. Plane b. Segment c. Line d. Point e. RayHence,The order of the given terms are:Point, Line, Ray, Segment, and Plane
Question 3.The endpoints of a line segment are and . Which choice shows the correct midpoint and distance between these two points? \\) 18.8 units \\) 18.8 units \\) 9.4 units \\) 9.4 unitsAnswer:The given endpoints of a line segment are: and We know that,The coordinates of the midpoint = , \)= , \)= , \)= , 9)The distance between the coordinates = \= \= 18.78 18.8We can conclude that option shows the correct midpoint and distance
Question 4.Find the perimeter and area of the figure shownAnswer:Q , R , S , and T Now,\ = \= \= 6\ = \= \= 1\ = \= \= 6\ = \= \= 1The perimeter of the given polygon = QR + RS + ST + TQ= 6 + 1 + 6 + 1The area of the given polygon = Length × WidthSo,The area of the polygon = 6 = 6The perimeter of the given polygon is: 14The area of the given polygon is: 6
b. Identify all linear pairs.Answer:
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Lesson 13 Using Midpoint And Distance Formulas
Monitoring Progress
Identify the segment, bisector of \. Then find PQ.
Question 1.The given line segment is:We know that,A segment has a starting point and an ending pointSo,The bisector of \ is: MNNow,From the above figure,The bisector divided the given line segment into 2 equal parts and given the length of 1 partSo,\ = 1\Hence,The length of \ = \ + \= 2 × 1\= 2 × \= \ × \= \We can conclude that the length of \ is: \
Question 2.The given line segment is:We know that,A segment has a starting point and an ending pointSo,The bisector of \ is: MNow,From the above figure,The bisector divided the given line segment into 2 equal parts and given the length of 1 partSo,\ = 2\Hence,The length of \ = \ + \= 2 × 2\= 2 × \= \ × \= \We can conclude that the length of \ is: \
Question 3.Identify the segment bisector of \. Then find MQ.Answer:The given figure is:We know that,A Bisector is a line that divides a line segment into 2 congruent equal partsSo,The bisector of \ is: line lWe know that,We can conclude that the value of MQ is:7
Question 4.Identify the segment bisector or \. Then find RS.Answer:
Question 5.The endpoints of \ are A and B. Find the coordinates of the midpoint M.Answer:The given endpoints of \ are:A , B We know that,M = , \)So,M = , \)M = , \)M = Hence, from the above,We can conclude that the coordinates of the midpoint of \ are:
Using Midpoint And Distance Formulas
EssentiaI QuestionHow can you find the midpoint and length of a line segment in a coordinate plane?Answer:Let the line segment is formed by the points A , B So,The coordinates of the midpoint of the line segment are given as:M = , \)The length of the line segment in a coordinate plane is given as:D = \
Exploration 1
Finding the Midpoint of a Line Segment
Work with a partner.
a. Graph \, where the points A and B are as shown.Answer:We can conclude that the length of \ is: 10cmThe representation of \ is:
b. Explain how to bisect \, that is, to divide AB into two congruent line segments. Thenbisects \ and use the result to find the midpoint M of \.Answer:The steps to find the bisector of \ is:a. Place the compass at one end of the line segmentb. Adjust the compass to slightly longer than half of the line segment lengthc. Draw arcs above and below the lined. Keeping the same compass width, draw arcs from the other end of linee. Place ruler where the arcs cross and draw the line segmentNow,The representation of \ is:Now,The representation of the perpendicular bisector and the midpoint M of \ is:Hence, from the above figure,The midpoint of \ is: 5 cm
c. What are the coordinates of the midpoint M?Answer:The coordinates of the midpoint of the line segment are given as:M = , \)So,M = ,\)M = ,\)M = Hence,The coordinates of the midpoint M is:
Exploration 2
d. Use the Pythagorean Theorem and point M from Exploration 1 to find the lengths of \ and \. What can you conclude?Answer:
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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:
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 HIG
EG 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 WVZ
XY 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.
Question 19.Describe and correct the error in identifying congruent triangles.Answer:
Question 20.ERROR ANALYSISDescribe and correct the error in determining the value of x that makes the triangles congruent.
Answer:
Question 25.A, B, C, D, E, FAnswer:
A, B, C, D, E, F
Answer:
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Exercise 16 Describing Pairs Of Angles
Vocabulary and Core Concept Check
Question 1.Explain what is different between adjacent angles and vertical angles.Answer:
WHICH ONE DID DOESNT BELONG?Which one does hot belong with the other three? Explain your reasoning.Answer:The angle that is the same in more than 1 angle-pair and has a common side is called Adjacent AngleSo,From the above figure,We can observe that only the first figure has an adjacent angle whereas all the three figures dont have any adjacent anglesHence, from the above,We can conclude that the first figure does not belong with the other three
Monitoring Progress and Modeling with Mathematics
In Exercises 3-6, use the figure.
Question 3.Name a pair of adjacent complementary angles.Answer:
Name a pair of adjacent supplementary angles.Answer:The angles that have the sum of the angle measures 180° are called Supplementary anglesHence, from the figure,A pair of adjacent supplementary angles are: LJN + LJK
Question 5.Name a pair of nonadjacent complementary angles.Answer:
Name a pair of nonadjacent supplementary angles.Answer:The angles that have the sum of the angle measures 180° are called Supplementary anglesHence, from the figure,A pair of adjacent supplementary angles are: LJN + LJK and NGP + HGF
In Exercises 7 10. find the angle measure.
Question 7.1 is a complement of 2, and m1 = 23°. Find, m2.Answer:
CAB = 58° and CAD = 32°
Question 13.UVW and XYZ arc complementary angles, mUVW = °. and mXYZ = °.Answer:
c. CAD EAF.
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:
Write a plan to prove that PTU UQP.
Answer:
Describe a situation in which you might choose to use indirect measurement withcongruent triangles to find a measure rather than measuring directly.
Answer:Indirect measurements are calculations based on known lengthsWhen two triangles are similar using AAS, SSS, ASA or SAS indirect measurement can be used to find missing measurements and angles.If you were given a triangle with two angles marked and a similar triangle with two sides marked you would be able to find measurements of the angles indirectly.
Monitoring Progress and Modeling With Mathematics
In Exercise 3-8, explain how to prove that the statement is true.
Question 3.
Answer:QP PT, RP SP, QR STAll pairs of sides are congruent by SSS congruence theorem. QPR STP. Because corresponding parts of congruent triangles are congruent, Q T.
Question 5.\ \Answer:
\ \
Answer:\ \,\ \ by reflexive property of congruenceC BACD BDC by SAS congruence theorem.
Question 7.\ \Answer:
\ \
Answer:VW RT, Q S, W TQVW VRT by the AAS congruence theoremSo, \ \
In Exercises 9-12, write a plan to prove that 1 2.
Question 9.
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Measuring And Constructing Segments
Essential QuestionHow can you measure and construct a line segment?Answer:The steps used to measure a line segment are:a. Pick up a scale to measure the length of a line segment.b. Identify the line segment you want to measurec. Place the tip of the ruler at the starting of the line segment
The steps used to construct a line segment are:a. Place the compass at one end of the lineb. Adjust the compass to slightly longer than half of the lines lengthc. Draw arcs above and below the lined. Keeping the same compass width, draw arcs from the other end of the linee. Place ruler where the arcs cross and draw the line segment
Exploration 1
Measuring Line Segments Using Nonstandard Units
Work with a partner.
a. Draw a line segment that has a length of 6 inches.Answer:We will use a ruler to draw a line segment and the ruler we use generally is the Centimeter rulerBut,It is given that we have to draw a line segment that has a length of 6 inchesBut, it is not possibleSo,6 inches = 15.24 cmHence,The representation of the line segment that has the length of 6 inches in terms of cm is:
c. Write conversion factors from paper clips to inches and vice versa.Answer:We can conclude that the conversion of paper clips into inches and vice-versa is:1 paperclip = 1.377 inch1 inch = 2.54 paperclip
Exploration 2
Measuring Line Segments Using Nonstandard Units
Work with a partner.
Exploration 3
Measuring Heights Using Nonstandard Units
Work with a partner.
Communicate Your Answer:
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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 VYT
Answer:So, the given information is not enough to prove that RST VYT
Lesson 15 Measuring And Constructing Angles
Monitoring Progress
Write three names for the angle.
Question 1.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is QHence,The three names for the given angle are: Q, PQR, and RQP
Question 2.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is YHence,The three names for the given angle are: 1, XYZ, and ZYX
Question 3.An Angle is formed when two rays have the same endpoint or vertex.So,Here, the angle is EHence,The three names for the given angle are: 2, FED, and DEF
Use the diagram in Example 2 to find the angle measure. Then classify the angle.
Question 4.
EFH and HFG are 60° and 30° respectively
Question 10.Angle MNP is a straight angle, and \Q bisects MNP. Draw MNP and \Q. Use arcs to mark the congruent angles in your diagram. Find the angle measures of these congruent angles.Answer:It is given that angle MNP is a straight angle, and \Q bisects MNP. Draw MNP and \Q.So,The representation of the above statement is:Hence, from the above,We can conclude that the angle measure of each congruent angle is: 55.3°
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Lesson 16 Describing Pairs Of Angles
Monitoring Progress
In Exercises 1 and 2. use the figure.
Question 1.Name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.Answer:We know that,The pair of angles that the sum of the angle measures is 90° is called Complementary anglesThe pair of angles that the sum of the angle measures is 180° is called Supplementary anglesThe angle that is the same in more than 1 angle-pair is called Adjacent AngleSo,The pair of complementary angles is: FGK and LKGThe pair of supplementary angles is: HGK and GKLThe pair of adjacent angles is: FGK and HGK
Question 2.Are KGH and LKG adjacent angles? Are FGK and FGH adjacent angles? Explain.Answer:The angle that is the same in more than 1 angle-pair is called Adjacent AngleBut,In KGH and LKG, the angle is not the sameSo,KGH and LKG are not the adjacent anglesIn FGK and FGH, the angle is the sameSo,FGK and FGH are adjacent angles
Question 3.1 is a complement of 2. arid m2 = 5°. Find m1.Answer:We know that,The pair of angles that the sum of the angle measures is 90° is called Complementary anglesIt is given that1 is a complement of 2So,It is also given that2 = 5°
LMN = 26° and PQR = 64°
Measuring And Constructing Angles
Essential Question
How can you measure and classify an angle?Answer:An angle is formed when two rays have the same endpoint or vertex.An Angle will be measured by using a protractorClassification of angles:An angle can be classified according to the size of theopening between its rays.An acute angle measures greater than 0° and lessthan 90°.A right-angle forms a square corner and measures 90°.An obtuse angle measures greater than 90° and less than 180°
Exploration 1
Measuring and Classifying Angles
Work with a partner: Find the degree measure of each of the following angles. Classify each angle as acute, right, or obtuse.a. AOB
a. Use a ruler and protractor to draw the triangular pattern shown at the right.Answer:The representation of the triangular pattern by using a ruler and a protractor is:
b. Cut out the pattern and use it to draw three regular hexagons
C. The sum of the angle measures of a polygon with n sides is equal to 180°. Do the angle measures at your hexagons agree with this rule? Explain.ATTENDING TO PRECISIONTo be proficient in math, you need to calculate and measure accurately and efficiently.Answer:It is given that the sum of the angle measures of a polygon with n sides is equal to 180 °This rule will be applicable for all the angle measures of all polygonsHence,We can conclude that the above rule will be applicable for the angle measures of the hexagon
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