Lesson 72 Properties Of Parallelograms
Monitoring progress
Find FG and mG.Answer:
BCD = 70°
Question 4.Using the figure and the given statement in Example 3, prove that C and F are supplementary angles.Answer:
Find the coordinates of the intersection of the diagonals of STUV with vertices S, T, U, and V.Answer:The given coordinates of the parallelogram STUV are:S , T , U , and V Compare the given points with , We know that,The opposite vertices form a diagonalSo,SU and TV are the diagonalsSo,We know that,The intersection of the diagonals means the midpoint of the vertices of the diagonals because diagonals bisect each otherSo,The midpoint of SU = , \)= , \)= , \)= Hence, from the above,We can conclude that the coordinates of the intersection of the diagonals of parallelogram STUV is:
Question 6.ABCD are A, B, and C. Find the coordinates of vertex D.Answer:The given vertices of parallelogram ABCD are:A , B , and C Let the fourth vertex of the parallelogram ABCD be: We know that,The diagonals bisect each other i.e., the angle between the diagonals is 90°So,We can say that the diagonals are the perpendicular linesSo,AC and BD are the diagonalsNow,Slope of AC = \= \= \Slope of BD = \= \We know that,AC and BD are the perpendicular linesSO,The product of the slopes of the perpendicular lines is equal to -1So, × = -1-5 × \ = -1\ = \Equate the numerator and denominator of both expreesionsWe get,y 2 = 1 x 5 = 5y = 1 + 2 x = 5 + 5y = 3 x = 10We can conclude that the coordinates of the vertex D are:
Question 49.
Lesson 71 Angles Of Polygons
Monitoring Progress
Question 1.The Coin shown is in the shape of an 11-gon. Find the sum of the measures of the interior angles.Answer:It is given that the coin shown is in the shape of an 11-gonSo,The number of sides in a coin = 11We know that,The sum pf the measures of the interior angles = 180° = 180° = 1620°Hence, from the above,We can conclude that the sum of the measures of the interior angles in a coin is: 1620°
Question 2.The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides.Answer:It is given that the sum of the measures of the interior angles of a convex polygon is 1440°We know that,The sum of the measures of the interior angles of a polygon = 180° Wheren is the number of sidesSo,n 2 = \n 2 = 8
Sketch a pentagon that is equilateral but not equiangular.Answer:The Equilateral means all the sides are congruentThe Equiangular means all the angles are congruentHence,A pentagon that is equilateral but not equiangular is:
Quadrilaterals And Other Polygons Cummulative Assessment
Question 1.Copy and complete the flowchart proof of the Parallelogram Opposite Angles Theorem .Given ABCD is a parallelogram.Prove A C, B DAnswer:
Question 2.Use the steps in the construction to explain how you know that the circle is inscribed within ABC.Answer:
Question 3.Your friend claims that he can prove the Parallelogram Opposite Sides Theorem using the SSS Congruence Theorem and the Parallelogram Opposite Sides Theorem . Is your friend correct? Explain your reasoning.Answer:
Question 4.Find the perimeter of polygon QRSTUV Is the polygon equilateral? equiangular? regular? Explain your reasoni ng.Answer:
Question 5.Choose the correct symbols to complete the proof of the Converse of the Hinge Theorem .Given Prove m B > m EStep 1 Assume temporarily that m B m E. Then it follows that either mB____ mE or mB ______ m E.
Step 2 If m B ______ mE. then AC _____ DF by the Hinge Theorem . If, mB _______ m E. then B _____ E. So. ABC ______ DEF by the SAS Congruence Theorem and AC _______ DF.
Step 3 Both conclusions contradict the given statement that AC _______ DF. So, the temporary assumption that m B > m E Cannot be true. This proves that m B ______ m E.> < = Answer:
Question 6.Use the Isosceles Trapctoid Base Angles Conersc to prove that ABCD is an isosceles trapezoid.Given \ || \. EBC ¿ECB, ABE DCEProve ABCD is an isoscelcs trapezoid.Answer:
| Statements |
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Proving That A Quadrilateral Is A Parallelogram
Exploration 1
Proving That a Quadrilateral is a Parallelogram
Work with a partner: Use dynamic geometry software.a. Construct a quadrilateral ABCD whose opposite sides are congruent.Answer:
b. Is the quadrilateral a parallelogram? Justify your answer.Answer:We know that,If a quadrilateral which has opposite sides congruent and each angle measure not equal to 90° and the diagonals bisected each other, then that quadrilateral is called the ParallelogramHence,The representation of the quadrilateral ABCD as a parallelogram is:
c. Repeat parts and for several other quadrilaterals. Then write a conjecture based on your results.Answer:From the above quadrilateral,The conjecture about the quadrilaterals is:If the opposite sides of a quadrilateral are equal, then the opposite angles of a quadrilateral are equal
d. Write the converse of your conjecture. Is the Converse true? Explain.
REASONING ABSTRACTLYTo be proficient in math, you need to know and flexibly use different properties of objects.Answer:From part ,The conjecture about quadrilaterals is:If the opposite sides of a quadrilateral are equal, then the opposite angles of a quadrilateral are equalThe converse of your conjecture is:If the opposite angles of a quadrilateral are equal, then the opposite sides of a quadrilateral are equalNow,When B and D are 90°,AD = BC = 2.9We can conclude that the converse of the conjecture from part is true
Exploration 2
Proving That a Quadrilateral Is a Parallelogram
Communicate Your Answer
Balbharati Solutions For Mathematics 2 Geometry 9th Standard Maharashtra State Board Chapter 7 Co
On a graph paper plot the points A , B, C. Join A, B and B, C. What is the figure formed?
Write the equation of the line parallel to the Y-axis at a distance of 7 units from it to its left.
Write the equation of the line parallel to the X-axis at a distance of 5 units from it and below the X-axis.
The point Q lies on a line parallel to the Y-axis. Write the equation of the line and draw its graph.
X-axis and line x = – 4 are parallel lines. What is the distance between them?
Which of the equation given below have graph parallel to the X-axis, and which ones have graph parallel to the Y-axis ?
x = 3
Which of the equation given below have graph parallel to the X-axis, and which ones have graph parallel to the Y-axis ?
y – 2 = 0
Which of the equation given below have graph parallel to the X-axis, and which ones have graph parallel to the Y-axis ?
x + 6 = 0
Which of the equation given below have graph parallel to the X-axis, and which ones have graph parallel to the Y-axis ?
y = – 5
On a graph paper, plot the points A, B and C. If those points are collinear then draw the line which includes them. Write the co-ordinates of the points at which the line intersects the X-axis and the Y-axis.
Draw the graph of the equation given below
x + y = 2
Draw the graph of the equation given below
3x – y = 0
Draw the graph of the equation given below
2x + y = 1
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Exercise 72 Page No: 167
1. Find the coordinates of the point which divides the join of and in the ratio 2:3.
Solution:
Let P be the required point. Using the section formula, we get
x = )/ = /5 = 1
y = / = /5 = 3
Therefore, the point is .
2. Find the coordinates of the points of trisection of the line segment joining and .
Solution:
Let P and Q are the points of trisection of the line segment joining the given points i.e., AP = PQ = QB
Therefore, point P divides AB internally in the ratio 1:2.
x1 = + 2×4)/3 = /3 = 6/3 = 2
y1 = + 2×)/ = /3 = -5/3
Therefore: P = P
Point Q divides AB internally in the ratio 2:1.
x2 = + 1×4)/ = /3 = 0
y2 = + 1×)/ = /3 = -7/3
The coordinates of the point Q is
3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
Solution:
From the given instruction, we observed that Niharika posted the green flag at 1/4th of the distance AD i.e., m = 25 m from the starting point of 2nd line. Therefore, the coordinates of this point are .
Hence, P =
Solution:
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Big Ideas Math Book Geometry Answer Key Chapter 7 Quadrilaterals And Other Polygons
Learning the concepts of Ch 7 Quadrilaterals and Other Polygons with the help of our provided BIM Geometry Answers is the best way to answer all the questions asked in the annual exams. The fact that you can observe in this Big Ideas Math Book Geometry Solution Key is covering all exercise questions of Chapter 7 Quadrilaterals and Other Polygons. So, just click on the quick links mentioned below and score better marks in any final exams or competitive examinations.
The coordinates of line a are: , The coordinates of line b are: , The coordinates of line c are: , The coordinates of line d are: , Compare the given coordinates with , Now,The slope of the line = \So,The slope of line a = \= \The slope of line b = \= \The slope of line c = \= \The slope of line d = \= \
The coordinates of line a are: , The coordinates of line b are: , The coordinates of line c are: , The coordinates of line d are: , Compare the given coordinates with , Now,The slope of the line = \So,The slope of line a = \= \The slope of line b = \= \The slope of line c = \= \The slope of line d = \= \
The coordinates of line a are: , The coordinates of line b are: , The coordinates of line c are: , The coordinates of line d are: , Compare the given coordinates with , Now,The slope of the line = \So,The slope of line a = \= \The slope of line b = \= \The slope of line c = \= \The slope of line d = \= \line b and line c are parallel linesline b and line d and line c and line d are perpendicular lines
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Practice Set 72 Geometry 9th Std Maths Part 2 Answers Chapter 7 Co
Question 1.On a graph paper plot the points A, B, C. Join A, B and B, C. What is the figure formed?Soiution:d = 3 cm, d = 3 cm, d = 3 cm, d = 3 cm and each angle of OABC is 90° OABC is a square.
Question 2.Write the equation of the line parallel to the Y-axis at a distance of 7 units from it to its left.Solution:The equation of a line parallel to the Y-axis is x = a.Since, the line is at a distance of 7 units to the left of Y-axis, a = -7 x = -1 is the equation of the required line.
Question 3.Write the equation of the line parallel to the X-axis at a distance of 5 units from it and below the X-axis.Solution:The equation of a line parallel to the X-axis is y = b.Since, the line is at a distance of 5 units below the X-axis. b = -5 y = -5 is the equation of the required line.
Question 4.The point Q lies on a line parallel to the Y-axis. Write the equation of the line and draw its graph.Solution:The equation of a line parallel to the Y-axis is x = a.Here, a = -3 x = -3 is the equation of the required line.
Question 5.Y-axis and line x = 4 are parallel lines. What is the distance between them?Solution:Equation of Y-axis is x = 0.Equation of the line parallel to the Y-axis is x = 4. Distance between the Y-axis and the line x = 4 is 0 = 0 + 4 = 4 units The distance between the Y-axis and the line x = 4 is 4 units.
ii. y 2 = 0The equation of a line parallel to the X-axis is y = b. The line y 2 = 0 is parallel to the X-axis.
= 1 + 2= 3
Lesson 73 Proving That A Quadrilateral Is A Parallelogram
Monitoring Progress
Question 1.In quadrilateral WXYZ, mW = 42°, mX = 138°, and mY = 42°. Find mZ. Is WXYZ a parallelogram? Explain your reasoning.Answer:
The quadrilateral JKLM is a parallelogram according to the Opposite sides parallel and congruent Theorem
Question 8.Refer to the Concept Summary. Explain two other methods you can use to show that quadrilateral ABCD in Example 5 is a parallelogram.Concept SummaryWays to Prove a Quadrilateral is a ParallelogramAnswer:The other ways to prove a quadrilateral parallelogram are:a. Prove that the sum of the consecutive angles is supplementaryb. Prove that an angle is supplementary to both the consecutive angles
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Lesson 75 Properties Of Trapezoids And Kites
Monitoring progress
Question 1.The points A, B C, and D form the vertices of a quadrilateral. Show that ABCD is a trapezoid. Then decide whether it is isosceles.Answer:
In Exercises 2 and 3, use trapezoid EFGH.
Question 2.If EG = FH, is trapezoid EFGH isosceles? Explain.Answer:
If mHEF = 70° and ,mFGH = 110°, is trapezoid EFGH isosceles? Explain.Answer:
Question 4.In trapezoid JKLM, J and M are right angles, and JK = 9 centimeters. The length of midsegment \ of trapezoid JKLW is 12 centimeters. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning.Answer:
Question 5.Explain another method you can use to find the length of \ in Example 4.Answer:
Question 6.In a kite. the measures of the angles are 3x° 75°, 90°, and 120°. Find the value of x. What are the measures of the angles that are congruent?Answer:
Question 7.Quadrilateral DEFG has at least one pair of opposite sides congruent. What types of quadrilaterals meet this condition?Answer:
Give the most specific name for the quadrilateral. Explain your reasoning.
Question 8.
Which is different? Find both answers.Is there enough information to prove that trapezoid ABCD is isosceles?Answer:
Is there enough information to prove that \ \?Answer:
Is there enough information to prove that the non-parallel sides of trapezoid ABCD are congruent?Answer:
Is there enough information to prove that the legs of trapezoid ABCD are congruent?Answer:
Monitoring Progress and Modeling with Mathematics
Question 7.
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