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regular decagon. What is the measure of ∠CBD? � � � � ∠CBD is an exterior angle of a regular decagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angle measures is 360°. m∠CBD = 360° _ = 36° 10 A regular decagon has 10 ≅ ext. , so divide the sum by 10. 5. What if…? Suppose the shutter were form... |
T are right angles, and ∠Q ≅ ∠S. What are m∠Q and m∠S? � � � � PRACTICE AND PROBLEM SOLVING Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 16. 17. 18. Independent Practice For See Exercises Example 16–18 19–21 22–24 25–26 27–28 1 2 3 4 5 Tell whether each polygon is regul... |
), B (-4, -1), C (-1, 2), D (4, 0), and E (3, -5). Estimate the measure of each interior angle. Make a conjecture about whether the polygon is equiangular. Now measure each interior angle with a protractor. Was your conjecture correct? 45. This problem will prepare you for the Multi-Step TAKS Prep on page 406. In this ... |
each interior angle. 57. Polygon PQRST is a regular pentagon. Find the values of x, y, and z. 58. Multi-Step Polygon ABCDEFGHJK is a regular decagon. ̶̶ DE are extended so that they meet at point L Sides in the exterior of the polygon. Find m∠BLD. ̶̶ AB and 59. Critical Thinking Does the Polygon Angle Sum Theorem work... |
. From the graph, you can see that only one y-value exists for each x-value, so the relation is a function. Try This TAKS Grades 9–11 Obj. 2 Give the domain and range of each relation. Tell whether the relation is a function. 1. y = (x - 2) 180 2. y = 360 4. y = 360_ x 7. x = -2 5. x = 3y - 10 8. y = x 2 + 4 3. y = (x ... |
so that ̶̶ BC? 4 Lay ABCD over QRST so that ∠A overlays ∠S. What do you notice about their measures? What does this tell you about ∠A and ∠C? Now move ABCD so that ∠B overlays ∠T. What do you notice about their measures? What does this tell you about ∠B and ∠D? 5 Arrange the pieces of patty paper so that ̶̶ QR and ̶̶ ... |
ram ABCD ABCD ̶̶ AB ǁ ̶̶ CD, ̶̶ BC ǁ ̶̶ DA Theorem 6-2-1 Properties of Parallelograms THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is a parallelogram, then its opposite sides are congruent. ( → opp. sides ≅) ̶̶ AB ≅ ̶̶ BC ≅ ̶̶ CD ̶̶ DA PROOF PROOF Theorem 6-2-1 Given: JKLM is a parallelogram. Prove: ̶̶ KL ≅ ̶̶ JK... |
180° m∠D + m∠A = 180° ̶̶ AZ ≅ ̶̶ BZ ≅ ̶̶ CZ ̶̶ DZ You will prove Theorems 6-2-3 and 6-2-4 in Exercises 45 and 44. E X A M P L E 1 Racing Application Racing The diagram shows the parallelogram-shaped linkage that joins the frame of a race car to one wheel of the car. In PQRS, QR = 48 cm, RT = 30 cm, and m∠QPS = 73°. F... |
by 2. B m∠B m∠A + m∠B = 180° → cons. supp. (10y - 1) + (6y + 5) = 180 16y + 4 = 180 16y = 176 y = 11 ⎤ ⎦ ⎡ ⎣ m∠B = (6y + 5) ° = 6 (11) + 5 ° = 71° Substitute the given values. Combine like terms. Subtract 4 from both sides. Divide both sides by 16. EFGH is a parallelogram. Find each measure. 2a. JG 2b. FH E X A M ... |
ABCD is a parallelogram. Prove: ∠BAD ≅ ∠DCB, ∠ABC ≅ ∠CDA Proof: Statements Reasons 1. ABCD is a parallelogram. ̶̶̶ DA ≅ ̶̶ BC 2. ̶̶ AB ≅ ̶̶ BD ≅ ̶̶ CD, ̶̶ BD 3. 4. △BAD ≅ △DCB 5. ∠BAD ≅ ∠DCB ̶̶ AC ≅ ̶̶ AC 6. 7. △ABC ≅ △CDA 8. ∠ABC ≅ ∠CDA 1. Given 2. → opp. sides ≅ 3. Reflex. Prop. of ≅ 4. SSS Steps 2, 3 5. CPCTC 6. ... |
���������������������������������������������������������������������� 6-2 Exercises Exercises KEYWORD: MG7 6-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Explain why the figure at right is NOT a parallelogram. 2. Draw PQRS. Name the opposite sides ... |
-1). Find the coordinates of vertex T. 26. Write a two-column proof. Given: ABCD and AFGH are parallelograms. Prove: ∠C ≅ ∠G 6- 2 Properties of Parallelograms 395 395 ���������������������������������������������������������������� Algebra The perimeter of PQRS is 84. Find the length of each side of PQRS under the g... |
3 ≅ c. by e. bisect each other at E by the definition of g.?. By the Alternate Interior Angles Theorem, ∠1 ≅ b. ̶̶̶̶?. ̶̶̶̶?, and ̶̶̶̶ ̶̶ CD because d.?. This means that △ABE ≅ △CDE ̶̶̶̶ ̶̶ ̶̶ DE. Therefore CE, and?, ̶̶̶̶ ̶̶ AB ≅?. So by f. ̶̶̶̶ ̶̶ AC and ̶̶ BE ≅ ̶̶ AE ≅ ̶̶ BD?. ̶̶̶̶ 45. Write a two-column proof of The... |
2 JL = KM KM JZ JL = 2JZ 53. Gridded Response In ABCD, BC = 8.2, and CD = 5. What is the perimeter of ABCD? CHALLENGE AND EXTEND The coordinates of three vertices of a parallelogram are given. Give the coordinates for all possible locations of the fourth vertex. 54. (0, 5), (4, 0), (8, 5) 55. (-2, 1), (3, -1), (-1, ... |
ram or the conditions below. Theorems Conditions for Parallelograms THEOREM EXAMPLE In the converse of a theorem, the hypothesis and conclusion are exchanged. 6-3-1 6-3-2 6-3-3 If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (quad. with pair of opp... |
�� → ) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (quad. with diags. bisecting each other → ) You will prove Theorems 6-3-4 and 6-3-5 in Exercises 27 and 30. E X A M P L E 1 Verifying Figures are Parallelograms A Show that ABCD is a parallelogram for x = 7 and y ... |
mine if each quadrilateral must be a parallelogram. Justify your answer. A B No. One pair of opposite sides are parallel. A different pair of opposite sides are congruent. The conditions for a parallelogram are not met. Yes. The diagonals bisect each other. By Theorem 6-3-5, the quadrilateral is a parallelogram. Determ... |
�� ̶̶̶ GH and Since GH = JF, FGHJ is a parallelogram. ̶̶ JF have the same slope, so ̶̶ JF. So by Theorem 6-3-1, ̶̶̶ GH ≅ ̶̶̶ GH ǁ ̶̶ JF. 400 400 Chapter 6 Polygons and Quadrilaterals ������������������������������� 3. Use the definition of a parallelogram to show that the quadrilateral with vertices K (-3, 0), L (-... |
parallelogram. � � � � � � 4. The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same? THINK AND DISCUSS 1. What do all the theorems in this lesson have in common? 2. How are the theorems ... |
GH is a parallelogram 10. Show that TUVW is a parallelogram for for x = 3.2 and y = 7. for a = 19.5 and b = 22. Determine if each quadrilateral must be a parallelogram. Justify your answer. Skills Practice p. S14 Application Practice p. S33 11. 12. 13. Show that the quadrilateral with the given vertices is a parallelog... |
̶̶ DA 3. △DAB ≅ b.? ̶̶̶̶̶ 4. ∠1 ≅ d. ̶̶ CD, ̶̶ AB ǁ?, ∠4 ≅ e. ̶̶̶̶̶ ̶̶ ̶̶̶ DA BC ǁ 5.? ̶̶̶̶̶ 6. ABCD is a parallelogram. 1. Given 2. a. 3. c.? ̶̶̶̶̶? ̶̶̶̶̶ 4. CPCTC 5. f. 6. g.? ̶̶̶̶̶? ̶̶̶̶̶ 6- 3 Conditions for Parallelograms 403 403 ��������������������������������������������������������������������������������������... |
∠E + m∠F = 180° and m∠E + m∠H = 180°. ̶̶ FG ǁ Then you can conclude that ̶̶̶ GH and ̶̶ EF ǁ ̶̶ HE. � � � � � � 30. Theorem 6-3-5 ̶̶ JL and Given: Prove: JKLM is a parallelogram. ̶̶̶ KM bisect each other. Plan: Show that △JNK ≅ △LNM and △KNL ≅ △MNJ. Then use the fact that the corresponding angles are congruent to show ̶... |
), B (1, 3), and C (6, 1)? D (8, 5) D (4, -3) D (13, 3) D (3, 7) 37. Short Response The vertices of quadrilateral RSTV are R (-5, 0), S (-1, 3), T (5, 1), and V (2, -2). Is RSTV a parallelogram? Justify your answer. CHALLENGE AND EXTEND 38. Write About It As the upper platform of the movable staircase is raised and low... |
of polygons that the faces form. 1. What type of polygon is ABCDE in the fluorite crystal? Given that m∠B = 120°, m∠E = 65°, and ∠C ≅ ∠D, find m∠A. ̶̶ AE ǁ ̶̶ CD, � � � � � � � � � 2. The pink crystals are called rhodochrosite. The face FGHJ is a parallelogram. Given that m∠F = (9x - 13) ° and m∠J = (7x + 1) °, find m... |
KLM WXYZ is a parallelogram. Find each measure. 16. WX 18. m∠X 17. YZ 19. m∠W � � � � � 6-3 Conditions for Parallelograms 20. Show that RSTV is a parallelogram 21. Show that GHJK is a parallelogram for x = 6 and y = 4.5. for m = 12 and n = 9.5. Determine if each quadrilateral must be a parallelogram. Justify your answe... |
lograms that you learned in Lesson 6-2. E X A M P L E 1 Craft Application An artist connects stained glass pieces with lead strips. In this rectangular window, the strips are cut so that FG = 14 in. and FH = 20 in. Find JG. ̶̶ ̶̶ EG ≅ FH EG = FH = 20 JG = 1 _ 2 JG = 1 _ (20) = 10 in. 2 EG Rect. → diags. ≅ Def. of ≅ seg... |
̶̶ JK ≅ ̶̶ JM, and ̶̶ KL ≅ ̶̶̶ ML by the definition of a rhombus. By the Reflexive Property of Congruence, Thus △JKL ≅ △JML by SSS. Then ∠1 ≅ ∠2, and ∠3 ≅ ∠4 by CPCTC. ̶̶ JL bisects ∠KJM and ∠KLM by the definition of an angle bisector. So ̶̶̶ KM bisects ∠JKL and ∠JML. By similar reasoning, ̶̶ JL ≅ ̶̶ JL. Like a rectang... |
CD = (b + 3) ° and m∠CDF = (6b - 40) ° Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the pr... |
ectors of each other. 410 410 Chapter 6 Polygons and Quadrilaterals �������������������������������������������������������������������������������������������� Special Parallelograms To remember the properties of rectangles, rhombuses, and squares, I start with a square, which has all the properties of the others. To ... |
What special properties does a rhombus have? 3. GET ORGANIZED Copy and complete the graphic organizer. Write the missing terms in the three unlabeled sections. Then write a definition of each term. 6- 4 Properties of Special Parallelograms 411 411 �������������������������������������� 6-4 Exercises Exercises KEYWORD:... |
-8). Show that the diagonals of square PQRS are congruent perpendicular bisectors of each other. 17. Given: RHMB is a rhombus with diagonal ̶̶ HB. Prove: ∠HMX ≅ ∠HRX Find the measures of the numbered angles in each rectangle. 18. 19. 20. 412 412 Chapter 6 Polygons and Quadrilaterals �����������������������������������... |
states that if a quadrilateral is a parallelogram, then its opposite sides are congruent. So JKLM is a ̶̶̶ LM, and ̶̶ KL ≅ parallelogram by Theorem 6-2-1. 35. Complete the two-column proof of Theorem 6-4-2 by filling in the blanks. Given: EFGH is a rectangle. ̶̶ GE Prove: ̶̶ FH ≅ Proof: Statements Reasons 1. EFGH is a... |
ZX ≅ △YZX. Then use CPCTC to show that ∠WZX and ∠YZX are right angles. 38. Write a paragraph proof of Theorem 6-4-1. Given: ABCD is a rectangle. Prove: ABCD is a parallelogram. 39. Write a two-column proof. Given: ABCD is a rhombus. E, F, G, and H are the midpoints of the sides. Prove: EFGH is a parallelogram. Multi-St... |
gment of a rectangle divides the rectangle into two congruent rectangles. 51. The figure is formed by joining eleven congruent squares. How many rectangles are in the figure? SPIRAL REVIEW 52. The cost c of a taxi ride is given by c = 2 + 1.8 (m - 1), where m is the length of the trip in miles. Mr. Hatch takes a 6-mile... |
̶̶ BC two lines. Hide the lines and construct ̶̶ CD to complete the parallelogram. and 3 Measure the four sides and angles of the parallelogram. 4 Move A so that m∠ABC = 90°. What type of special parallelogram results? 5 Move A so that m∠ABC ≠ 90°. 6 Construct ̶̶ AC and ̶̶ BD and measure their lengths. Move A so that A... |
to determine whether the parallelogram is a rectangle. Theorems Conditions for Rectangles THEOREM EXAMPLE 6-5-1 6-5-2 If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. ( with one rt. ∠ → rect.) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectan... |
ram bisects a pair of opposite angles, then the parallelogram is a rhombus. ( with diag. bisecting opp. → rhombus) You will prove Theorems 6-5-3 and 6-5-4 in Exercises 32 and 30. PROOF PROOF Theorem 6-5-5 Given: JKLM is a parallelogram. ̶̶ JL bisects ∠KJM and ∠KLM. Prove: JKLM is a rhombus. Proof: Statements Reasons... |
right angle. Def. of ⊥ ABCD is a rectangle. with one rt. ∠ → rect. Step 3 Determine if ABCD is a rhombus. ̶̶ AC ⊥ ̶̶ BD ABCD is a rhombus. Given with diags. ⊥ → rhombus Step 4 Determine if ABCD is a square. Since ABCD is a rectangle and a rhombus, it has four right angles and four congruent sides. So ABCD is a squ... |
� 5, ABCD is not a rectangle. Thus ABCD is not a square. 420 420 Chapter 6 Polygons and Quadrilaterals ���������������������������������������������� Step 3 Determine if ABCD is a rhombus. slope of ̶̶ AC = 6 - 2 _ ) (-2) = -1, B E (-4, -1), F (-3, 2), G (3, 0), H (2, -3) Since ( 1 _ = 1 _ 2 ̶̶ AC ⊥ 8 - 0 2 slope of ̶̶ ... |
3a. K (-5, -1), L (-2, 4), M (3, 1), N (0, -4) 3b. P (-4, 6), Q (2, 5), R (3, -1), S (-3, 0) THINK AND DISCUSS 1. What special parallelogram is formed when the diagonals of a parallelogram are congruent? when the diagonals are perpendicular? when the diagonals are both congruent and perpendicular? 2. Draw a figure tha... |
6, -1), S (-3, -4) 5. W (-6, 0), X (1, 4), Y (2, -4), Z (-5, -8) Independent Practice For See Exercises Example 6 7–8 9–10 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S15 Application Practice p. S33 PRACTICE AND PROBLEM SOLVING 6. Crafts A framer uses a clamp to hold together the pieces of a picture frame. ̶̶ RS and T... |
Give one characteristic of the diagonals of each figure that would make the conclusion valid. 18. Conclusion: JKLM is a rhombus. 19. Conclusion: PQRS is a square. The coordinates of three vertices of ABCD are given. Find the coordinates of D so that the given type of figure is formed. 20. A (4, -2), B (-5, -2), C (4,... |
������������������ 29. This problem will prepare you for the Multi-Step TAKS Prep on page 436. A state fair takes place on a plot of land given by the coordinates A (-2, 3), B (1, 2), C (2, -1), and D (-1, 0). a. Show that the opposite sides of quadrilateral ABCD are parallel. b. A straight path connects A and C, and a... |
lines are represented by the equations below. m: y = -x + 7 ℓ: y = -x + 1 a. Graph the four lines in the coordinate plane. b. Classify the quadrilateral formed by the lines. c. What if…? Suppose the slopes of lines n and p change to 1. n: y = 2x + 1 p: y = 2x + 7 Reclassify the quadrilateral. 34. Write a two-column pr... |
̶̶ BC ǁ ̶̶ AC ≅ ̶̶ BE ⊥ ̶̶ DE, ̶̶ EF ̶̶ AB ⊥ ̶̶ DE ⊥ ̶̶ EF, Prove: EBCF is a rectangle. 43. Critical Thinking Consider the following statement: If a quadrilateral is a rectangle and a rhombus, then it is a square. a. Explain why the statement is true. b. If a quadrilateral is a rectangle, is it necessary to show that ... |
on ̶̶ AB. Construct a parallel line ℓ through C. 2 Draw point D on line ℓ. Construct ̶̶ AC and ̶̶ BD. 3 Measure AC, BD, ∠CAB, ∠ABD, ∠ACD, and ∠CDB. 4 Move D until AC = BD. What do you notice about m∠CAB and m∠ABD? What do you notice about m∠ACD and m∠CDB? 5 Move D so that AC ≠ BD. Now move D so that m∠CAB = m∠ABD. Wha... |
oid base of a trapezoid leg of a trapezoid base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid Theorems Properties of Kites THEOREM HYPOTHESIS CONCLUSION 6-6-1 6-6-2 If a quadrilateral is a kite, then its diagonals are perpendicular. (kite → diags. ⊥) If a quadrilateral is a kite, then exactly one p... |
a cloth binding. There are 2 yards of binding in one package. What is the total amount of binding needed to cover the edges of the kite? How many packages of binding must Alicia buy? � � � ������ ������ � ������ ������ � Understand the Problem The answer has two parts. • the total length of binding Alicia needs • the ... |
185 ≈ 27. The perimeter of the kite is approximately 1. What if...? Daryl is going to make a kite by doubling all the measures in the kite above. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy? 428 428 Chapter 6 Polygons and Quadrilaterals 1234 E ... |
pezoids THEOREM DIAGRAM EXAMPLE 6-6-3 6-6-4 If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. (isosc. trap. → base ≅) If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. (trap. with pair base ≅ → isosc. trap.) 6-6-5 A trapezoid is isosceles... |
. 3b. JN = 10.6, and NL = 14.8. Find KM. E X A M P L E 4 Applying Conditions for Isosceles Trapezoids A Find the value of y so that EFGH is isosceles. ∠E ≅ ∠H Trap. with pair base ≅ m∠E = m∠H 2y 2 - 25 = y 2 + 24 → isosc. trap. Def. of ≅ Substitute 2 y 2 - 25 for m∠E and y 2 + 24 for m∠H. y 2 = 49 Subtract y 2 from... |
RU) 2 31 = 1 _ (ST + 38 ) 2 62 = ST + 38 24 = ST Trap. Midsegment Thm. Substitute the given values. Multiply both sides by 2. Subtract 38 from both sides. 5. Find EH. THINK AND DISCUSS 1. Is it possible for the legs of a trapezoid to be parallel? Explain. 2. How is the midsegment of a trapezoid similar to a midsegment... |
. Find the value of y so that LMPQ is isosceles 11. Find QR. 12. Find AZ. p. 431 432 432 Chapter 6 Polygons and Quadrilaterals ��������������������������������������������������������������������������� PRACTICE AND PROBLEM SOLVING 13. Design Each square section in the iron railing contains four small kites. The figure... |
l06005aAB�������������������������������������������������������������� 33. This problem will prepare you for the Multi-Step TAKS Prep on page 436. The boundary of a fairground is a quadrilateral with vertices at E (-1, 3), F (3, 4), G (2, 0), and H (-3, -2). a. Use the Distance Formula to show that EFGH is a kite. b. ... |
is made from eight pieces of wood shaped like congruent isosceles trapezoids. What are m∠A, m∠B, m∠C, and m∠D? 45. Write About It Compare an isosceles trapezoid to a trapezoid that is not isosceles. What properties do the figures have in common? What properties does one have that the other does not? � � � � 46. Use co... |
AB, BC, and CD. SPIRAL REVIEW 52. An empty pool is being filled with water. After 10 hours, 20% of the pool is full. If the pool is filled at a constant rate, what fraction of the pool will be full after 25 hours? (Previous course) Write and solve an inequality for x. (Lesson 3-4) 53. 54. Tell whether a parallelogram ... |
̶̶ RS, and ̶̶ PR and 3. Use the paths ̶̶ SQ to tell whether PQRS is a rhombus, rectangle, or square. 4. One section of the fair will contain all the rides and games. The organizers will fence off this area within the fairground by using the existing fences along ̶̶ CE, where E has coordinates (-1, 0). What type of al... |
, -4) ̶̶ ZX are midsegments of △TWY. 11. M (-4, 5), N (1, 7), P (3, 2), Q (-2, 0) ̶̶ VX and 12. Given: ̶̶̶ TW ≅ ̶̶ TY Prove: TVXZ is a rhombus. 6-6 Properties of Kites and Trapezoids In kite EFGH, m∠FHG = 68°, and m∠FEH = 62°. Find each measure. 13. m∠FEJ 15. m∠FGJ 14. m∠EHJ 16. m∠EHG 17. Find m∠R. 18. YZ = 34.2, and V... |
................ 429 diagonal................... 382 rectangle................... 408 vertex of a polygon.......... 382 isosceles trapezoid.......... 429 regular polygon............. 382 Complete the sentences below with vocabulary words from the list above. 1. The common endpoint of two sides of a polygon is a(n)?. ̶̶... |
. If it is a polygon, name it by the number of its sides. 5. 6. 7. Tell whether each polygon is regular or irregular. Tell whether it is concave or convex. 8. 10. 9. Find each measure. 11. the sum of the interior angle measures of a convex dodecagon 12. the measure of each interior angle of a regular 20-gon 13. the mea... |
. m∠Y 24. m∠X 26. m∠Z 27. Three vertices of RSTV are R (-8, 1), S (2, 3), and V (-4, -7). Find the coordinates of vertex T. 28. Write a two-column proof. Given: GHLM is a parallelogram. ∠L ≅ ∠JMG Prove: △GJM is isosceles. 6-3 Conditions for Parallelograms (pp. 398–405) E X A M P L E S ■ Show that MNPQ is a parallelogr... |
A M P L E S In rectangle JKLM, KM = 52.8, and JM = 45.6. Find each length. ■ KL JKLM is a . KL = JM = 45.6 Rect. → → opp. sides ≅ ■ NL JL = KM = 52.8 NL = 1_ 2 JL = 26.4 Rect. → diags. ≅ → diags. bisect each other ■ PQRS is a rhombus. Find m∠QPR, given that m∠QTR = (6y + 6) ° and m∠SPR = 3y°. m∠QTR = 90° 6y + 6... |
�TZV = (8n + 18) °, and m∠SRV = (9n + 1) °. Find each measure. 42. m∠TRS 43. m∠RSV 44. m∠STV 45. m∠TVR Find the measures of the numbered angles in each figure. 46. rectangle MNPQ 47. rhombus CDGH Show that the diagonals of the square with the given vertices are congruent perpendicular bisectors of each other. 48. R (-5... |
diagonals to tell whether a parallelogram with vertices P (-5, 3), Q (0, 1), R (2, -4), and S (-3, -2) is a rectangle, rhombus, or square. Give all the names that apply. PR = √ 98 = 7 √ 2 QS = √ 18 = 3 √ 2 Distance Formula Distance Formula Use the diagonals to tell whether a parallelogram with the given vertic... |
Side Int. Thm. Substitute 51 for m∠C. Subtract. 51 + m∠D = 180 m∠D = 129° ■ In trapezoid HJLN, In kite WXYZ, m∠VXY = 58°, and m∠ZWX = 50°. Find each measure. 56. m∠XYZ 57. m∠ZWV 58. m∠VZW 59. m∠WZY Find each measure. 60. m∠R and m∠S 61. BZ if ZH = 70 and EK = 121.6 62. MN 63. EQ JP = 32.5, and HL = 50. Find PN. ̶̶ ̶̶... |
of the interior angle measures of a convex nonagon. 5. Find the measure of each exterior angle of a regular 15-gon. � ��� ���� � � ���� ���� � 6. In EFGH, EH = 28, HZ = 9, and m∠EHG = 145°. Find FH and m∠FEH. 7. JKLM is a parallelogram. Find KL and m∠L. 8. Three vertices of PQRS are P (-2, -3), R (7, 5), and S (6, 1... |
∠JFR = 43°, and m∠JNB = 68°. Find m∠FBN. 20. Find HR. � � � 442 442 Chapter 6 Polygons and Quadrilaterals ��������� � ������ ������ 19. PV = 61.1, and YS = 24.7. Find MY. � � � ��������������������������������������������������������������������������������������� FOCUS ON SAT The scores for each SAT section range from... |
Choices For some multiple-choice test items, you can eliminate one or more of the answer choices without having to do many calculations. Use estimation or logic to help you decide which answer choices can be eliminated. What is the value of x in the figure? 3° 63° 83° 153° The sum of the exterior angle measures of a c... |
̶̶ than, or equal to the measure of AC? What answer choices can you eliminate and why? ̶̶ GB be more than, less 4.7 square meters 5.4 meters 9.4 square meters 12.8 meters 1. Are there any answer choices you can eliminate immediately? If so, which choices and why? 2. Describe how to use estimation to eliminate at least ... |
is congruent to ∠CAB? ∠QRP ∠XZY ∠YXZ ∠XYZ 4. Which line coincides with the line 2y + 3x = 4? x + 2 3y + 2x = 4 y = 2 _ 3 a line through (-1, 1) and (2, 3) a line through (0, 2) and (4, -4) 5. What is the value of x in polygon ABCDEF? 12 18 24 36 446 446 Chapter 6 Polygons and Quadrilaterals 6. If ̶̶ JK ǁ ̶̶̶ ML, what ... |
Short Response 17. In △ABC, AE = 9x - 11.25, and AF = x + 4. 11. Quadrilateral RSTU is a kite. What is the length ̶̶ RV? of 4 inches 5 inches 6 inches 13 inches 12. What is the measure of each interior angle in a regular dodecagon? 30° 144° 150° 162° 13. The coordinates of the vertices of quadrilateral RSTU are R (1, ... |
obtuse. Justify your reasoning and state any theorems or postulates used. b. If AB = 3, BC = 5, AC = 5, and m∠B > m∠Y, ̶̶ XZ so that △XYZ is a right find the length of triangle. Justify your reasoning and state any theorems or postulates used. c. If AB = 8 and BC = 4, find the range of possible ̶̶ AC. Justify your ans... |
and Quadrilaterals ����������� Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Titan When it opened in April 2001 at Six Flags Over Texas in Arlington, Titan became th... |
Duro Canyon or “the Grand Canyon of Texas.” 450 450 Chapter 7 Vocabulary Match each term on the left with a definition on the right. 1. side of a polygon A. two nonadjacent angles formed by two intersecting lines 2. denominator 3. numerator 4. vertex of a polygon 5. vertical angles B. the top number of a fraction, whi... |
. Bike riders often talk about gear ratios. Give examples of situations where the word ratio is used. What do these examples have in common? Geometry TEKS G.1.B Geometric structure* recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes G.2.A Geometric stru... |
orem is used in photography. Photography Application The aperture of the camera shown is formed by ten blades. The blades overlap to form a regular decagon. What is the measure of ∠CBD? � � � � Step Understand the Problem Procedure Result • List the important information. • The answer will be the measure of ∠CBD. ∠CBD ... |
values. Simplify. 1. Given that two points on m are C (-2, 3) and D (6, 5), write a ratio expressing the slope of m. A ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3 : 7 : 3 : 7. E X A M P L E 2 Using Ratios The ratio of the side lengths of a quadrilateral... |
Subtract 2 from both sides. Solve each proportion. = x _ 3a. 3 _ 8 56 3c. d _ = 6 _ 2 3 3b. 3d. 2y = 8 _ _ 4y The following table shows equivalent forms of the Cross Products Property. Properties of Proportions ALGEBRA _ _ c = d a The proportion b the following: is equivalent to NUMBERS = 2 _ 3 6 The proportion 1 _ th... |
of 9.2 m and a width of 6 m. What is the height of the actual tower? 456 456 Chapter 7 Similarity 1234 THINK AND DISCUSS 1. Is the ratio 6 : 7 the same ratio as 7 : 6? Why or why not? 2. Susan wants to know if the fractions 3 __ 7 and 12 __ 28 are equivalent. Explain how she can use the properties of proportions to fi... |
18. m 19. n 17–19 20–21 22–27 28–29 30 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 20. The ratio of the side lengths of an isosceles triangle is 4 : 4 : 7, and its perimeter is 52.5 cm. What is the length of the base of the triangle? 21. The ratio of the angle measures in a paralle... |
, a skyscraper is 1.25 in. wide and 15 in. tall. The actual skyscraper is 800 ft tall. a. Write a proportion that you can use to find the width of the actual skyscraper. b. Solve the proportion from part a. What is the width of the actual skyscraper? 458 458 Chapter 7 Similarity ��������������� 40. Critical Thinking Th... |
. What is the probability that the two ratios will form a proportion? ___? 51. Express the ratio x 2 + 9x + 18 _________ x 2 - 36 in simplest form. SPIRAL REVIEW Complete each ordered pair so that it is a solution to y - 6x = -3. (Previous course) 52. (0, 54. (-4, 53. (, 3) ) ) Find each angle measure. (Lesson 3-2) 55.... |
segments to complete the square. 3 Find the midpoint of ̶̶ AB and label it M. Create a segment from M to C. Construct a ̶̶̶ MC. circle with its center at M and radius of Construct a ray with endpoint A through B. Where the circle and the ray intersect, label the point E. Create a line through E that is AB. Show the... |
2 Technology Lab 461 461 ���������������������� 7-2 Ratios in Similar Polygons TEKS G.5.B Geometric patterns: use... geometric patterns to make generalizations about ratios in similar figures... Objectives Identify similar polygons. Apply properties of similar polygons to solve problems. Vocabulary similar similar pol... |
gruent angles. ∠P ≅ ∠T, ∠Q ≅ ∠U, ∠R ≅ ∠V, and ∠S ≅ ∠W All of a rect. are rt. and are ≅. Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. Step 2 Compare corresponding sides., PS _ = 3 _ 4 TW = 2 _ = 4 _ 3 6 = 12 _ 16 PQ _ TU Since corresp... |
7x = (50) (2) 7x = 100 x ≈ 14.3 Cross Products Prop. Simplify. Divide both sides by 7. The width of the model is approximately 14.3 ft. 3. A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch. THINK AND DISCUSS 1. If you combine the symbol for simi... |
RSQ and △UXZ 10. rectangles ABCD and JKLM 7- 2 Ratios in Similar Polygons 465 465 ��������������������������������������������������������������������������������������������������������������������������������������������������������� 11. Hobbies The ratio of the model car’s dimensions to the actual car’s dimensions i... |
It Two similar polygons have a similarity ratio of 1 : 1. What can you say about the two polygons? Explain. 26. This problem will prepare you for the Multi-Step TAKS Prep on page 478. A stage set consists of a painted backdrop with some wooden flats in front of it. One of the flats shows a tree that has a similarity r... |
proportion for ℓ. (Hint : Use the Quadratic Formula.) d. The value of ℓ is known as the golden ratio. Use a calculator to find ℓ to the nearest tenth. SPIRAL REVIEW 34. There are four runners in a 200-meter race. Assuming there are no ties, in how many different orders can the runners finish the race? (Previous course... |
the triangles are also congruent? 2. Will the ratios of corresponding sides found in Step 3 always be equal? Drag ̶̶ DE to investigate this question. State a a vertex of △ABC or an endpoint of conjecture based on your results. Activity 2 1 Construct a new △ABC. Create P in the interior of the triangle. Create △DEF by ... |
as the Pyramid Building in San Diego, California. (See Example 5.) There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent. Postulate 7-3-1 An... |
�. JK = 3 _ = 1 _ _ 3 9 JM Therefore △JKL ∼ △JMN by SAS ∼. JL = 1 _ = 2 _ _, 3 6 JN 2. Verify that △TXU ∼ △VXW. E X A M P L E 3 Finding Lengths in Similar Triangles Explain why △ABC ∼ △DBE and then find BE. Step 1 Prove triangles are similar. ̶̶ AC ǁ ̶̶ ED, ∠A ≅ ∠D, and ∠C ≅ ∠E As shown by the Alternate Interior Angles... |
= 8. Trans. Prop. of = 9. Reflex. Prop. of ≅ 10. SAS ∼ Steps 8, 9 ̶̶ JK. 4. Given: M is the midpoint of ̶̶ KL, N is the midpoint of and P is the midpoint of ̶̶ JL. Prove: △JKL ∼ △NPM (Hint : Use the Triangle Midsegment Theorem and SSS ∼.) E X A M P L E 5 Engineering Application The photo shows a gable roof. △ABC ∼ △FB... |
Similarity Postulate? 2. What additional information, if any, would you need in order to show that △ABC ∼ △DEF by the SAS Similarity Theorem? 3. Do corresponding sides of similar triangles need to be proportional and congruent? Explain. 4. GET ORGANIZED Copy and complete the graphic organizer. If possible, write a con... |
VW and △XYZ Independent Practice For See Exercises Example 11–12 13–14 15–16 17–18 19 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S16 Application Practice p. S34 Multi-Step Explain why the triangles are similar and then find each length. 15. AB 16. PS 17. Given: CD = 3AC, CE = 3BC 18. Given: PR _ MR = QR _ NR Prov... |
and MN? 28. Prove the Transitive Property of Similarity. Given: △ABC ∼ △DEF, △DEF ∼ △XYZ Prove: △ABC ∼ △XYZ 29. 29. Draw and label △PQR and △STU such that PQ ___ ST = QR ___ TU Meteorology but △PQR is NOT similar to △STU. 30. 30. Given: △KNJ is isosceles with ∠N as the vertex angle. ∠H ≅ ∠L Prove: △GHJ ∼ △MLK This sat... |
Which similarity postulate or theorem lets you conclude that △BCD ∼ △FGH? AA SSS SAS None of these 37. Gridded Response If 6, 8, and 12 and 15, 20, and x are the lengths of the corresponding sides of two similar triangles, what is the value of x? CHALLENGE AND EXTEND 38. Prove the SSS Similarity Theorem. = BC _ EF = A... |
45. 46. b - 5 _ 28 = 7 _ b - 5 7- 3 Triangle Similarity: AA, SSS, and SAS 477 477 �������������������������������������������� SECTION 7A Similarity Relationships Lights! Camera! Action! Lorenzo, Maria, Sam, and Tia are working on a video project for their history class. They decide to film a scene where the character... |
two polygons are similar. If so, write the similarity ratio and a similarity statement. 10. rectangles ABCD and WXYZ 11. △JMR and △KNP 12. Leonardo da Vinci’s famous portrait the Mona Lisa is 30 in. long and 21 in. wide. Janelle has a refrigerator magnet of the painting that is 3.5 cm wide. What is the length of the m... |
bisectors of ̶̶ ∠E and ∠F. Find the point of intersection of EF and the bisector of ∠D. Label the intersection G. 2 Find DI, DG, and the perimeter of △DEF. ̶̶ 3 Divide the length of DI by the length of DG. ̶̶ ̶̶ Add the lengths of DF. Then divide DE and this sum by the perimeter of △DEF. Compare the two quotients. Dra... |
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