ssrs barcode font Parallel Lines, Distances, and Angle Sums in Objective-C

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CHAPTER 4 Parallel Lines, Distances, and Angle Sums
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(a) See Fig. 4-59(a). ^ I > ^ II by hy-leg. (b) See Fig. 4-59(b). ^ I > ^ II by SAA.
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4.20 Proving a congruency problem
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Given: Quadrilateral ABCD DF ' AC, BE ' AC AE > FC, BC > AD To Prove: BE > FD Plan: Prove ^ I > ^ II
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Statements BC > AD DF ' AC, BE ' AC j1 > j2 AE > FC EF > EF AF > EC
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7. ^ I > ^ II 8. BE > FD
Reasons Given Given Perpendiculars form rt. , and all rt. are congruent. Given Identity If equals are added to equals, the sums are equal. Definition of congruent segments. 7. Hy-leg s 8. Corresponding parts of congruent n are congruent.
4.21 Proving a congruency problem stated in words Prove that in an isosceles triangle, altitudes to the congruent sides are congruent.
Given:
Isosceles ^ ABC (AB > BC) AD is altitude to BC CE is altitude to AB
To Prove:
AD > CE
Plan: Prove ^ ACE > ^ CAD or ^ I > ^ II
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
PROOF:
Statements 1. AB > BC 2. jA > jC 3. AD is altitude to BC, CE is altitude to AB. 4. j1 > j2 5. AC > AC 6. ^ I > ^ II 7. AD > CE
Reasons 1. Given 2. In a ^, angles opposite equal sides are equal. 3. Given 4. 5. 6. 7. Altitudes form rt. and rt. are congruent. Identity SAA s Corresponding parts of congruent n are congruent.
SUPPLEMENTARY PROBLEMS
4.1. In each part of Fig. 4-60, find x and y. (4.1)
Fig. 4-60
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
In each part of Fig. 4-61, find x and y. (4.3)
Fig. 4-61
If two parallel lines are cut by a transversal, find (a) Two alternate interior angles represented by 3x and 5x 70 (b) Two corresponding angles represented by 2x 10 and 4x 50
(4.3)
(c) Two interior angles on the same side of the transversal represented by 2x and 3x 4.4. Provide the proofs requested in Fig. 4-62 (4.4)
Fig. 4-62
Provide the proofs requested in Fig. 4-63
(4.4)
Fig. 4-63
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
4.6. Prove each of the following: (a) If the opposite sides of a quadrilateral are parallel, then they are also congruent. (b) If AB and CD bisect each other at E, then AC i BD.
(4.5)
(c) In quadrilateral ABCD, let BC i AD. If the diagonals AC and BD intersect at E and AE i DE, then BE > CE. (d) AB and CD are parallel lines cut by a transversal at E and F. If EG and FH bisect a pair of corresponding S S angles, then EG i FH. (e) If a line through vertex B of ^ABC is parallel to AC and bisects the angle formed by extending AB through B, then ^ ABC is isosceles. 4.7. In Fig. 4-64, find the distance from (a) A to B; (b) E to AC; (c) A to BC; (d) ED to BC. (4.6)
4 4 S S
Fig. 4-64
Fig. 4-65
In Fig. 4-65, find the distance (a) from P to the outer circle; (b) from P to the inner circle; (c) between the concentric circles; (d) from P to O. (4.6) In Fig. 4-66 (4.6)
(a) Locate P, a point on AD, equidistant from B and C. Then locate Q, a point on AD, equidistant from AB and BC. (b) Locate R, a point equidistant from A, B, and C. Then locate S, a point equidistant from B, C, and D. (c) Locate T, a point equidistant from BC, CD, and AD. Then locate U, a point equidistant from AB, BC, and CD. 4.10. In each part of Fig. 4-67, describe P, Q, and R as equidistant points and locate them on a bisector.
Fig. 4-66
Fig. 4-67
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
In each part of Fig. 4-68, describe P, Q, and R as equidistant points. (4.9)
Fig. 4-68
Find x and y in each part of Fig. 4-69.
(4.11)
Fig. 4-69
Find x and y in each part of Fig. 4-70.
(4.12)
Fig. 4-70
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
4.14. Find the measure of each angle (a) Of a triangle if its angle measures are in the ratio 1:3:6 (b) Of a right triangle if its acute angle measures are in the ratio 4:5 (c) Of an isosceles triangle if the ratio of the measures of its base angle to a vertex angle is 1:3 (d) Of a quadrilateral if its angle measures are in the ratio 1:2:3:4
(4.13)
(e) Of a triangle, one of whose angles measures 55 and whose other two angle measures are in the ratio 2:3 ( f ) Of a triangle if the ratio of the measures of its exterior angles is 2:3:4 4.15. Prove each of the following: (a) In quadrilateral ABCD if jA > jD and jB > jC, then BC i AD. (b) Two parallel lines are cut by a transversal. Prove that the bisectors of two interior angles on the same side of the transversal are perpendicular to each other. 4.16. Show that a triangle is (a) Equilateral if its angles are represented by x (b) Isosceles if its angles are represented by x 15, 3x 75, and 2x 30 15, 3x 35, and 4x (4.14) (4.14)
(c) A right triangle if its angle measures are in the ratio 2:3:5 (d) An obtuse triangle if one angle measures 64 and the larger of the other two measures 10 less than five times the measure of the smaller 4.17. (a) Find the sum of the measures of the interior angles (in straight angles) of a polygon of 9 sides; of 32 sides. (4.15) (b) Find the sum of the measures of the interior angles (in degrees) of a polygon of 11 sides; of 32 sides; of 1002 sides. (c) Find the number of sides a polygon has if the sum of the measures of the interior angles is 28 straight angles; 20 right angles; 4500 ; 36,000 . 4.18. (a) Find the measure of each exterior angle of a regular polygon having 18 sides; 20 sides; 40 sides. (b) Find the measure of each interior angle of a regular polygon having 18 sides; 20 sides; 40 sides. (c) Find the number of sides a regular polygon has if each exterior angle measures 120 ; 40 ; 18 ; 2 . (d) Find the number of sides a regular polygon has if each interior angle measures 60 , 150 ; 170 , 175 ; 179 . 4.19. (a) Find each interior angle of a quadrilateral if its interior angles are represented by x 5, x 20, 2x 45, and 2x 30. (4.17) (b) Find the measure of each interior angle of a quadrilateral if the measures of its exterior angles are in the ratio of 1:2:3:3. 4.20. 4.21. In each part of Fig. 4-71, select congruent triangles and state the reason for the congruency. (4.18) (4.16)
In each part of Fig. 4-72, two triangles can be proved congruent. Make a diagram showing the congruent parts of both triangles, and state the reason for the congruency. (4.19)
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