Parallel Lines and Transversals
1, 4, 8
Which angles have the same measure as angle 5?
parallel
|| means ___.
intersect
Parallel planes are planes that do not ___.
skew
Are the lines parallel, perpendicular, or skew?
x = 8, m∠8 = 115°
If m∠3 = (9x - 7)° and m∠1 = (x + 57)°, What is m∠8?
x = 33, m∠4 = 63°
If m∠6 = (3x - 36)° and m∠5 = (6x - 81)°, What is m∠4?
3, 6, 7
Which angles have the same measure as angle 2?
Parallel
Are lines CG and BF parallel or perpendicular?
Perpendicular
Are lines GH and FH parallel or perpendicular?
x = 20, m∠5 = 155°
If m∠7 = (9x - 25)° and m∠1 = (4x + 75)°, What is m∠5?
x = 10, m∠2 = 150°
If m∠7 = (x + 20)° and m∠1 = (5x - 20)°, What is m∠2?
x = 47, m∠3 = 56°
If m∠7 = (x + 9)° and m∠1 = (3x - 85)°, What is m∠3?
same side interior angles
Angles between 2 lines and on the same side of a transversal.
x = 45 and y = 5
Find x an y.
x = 20, m∠7 = 41°
If m∠2 = (5x + 39)° and m∠4 = (2x + 99)°, What is m∠7?
x = 14, m∠7 = 69°
If m∠2 = (8x - 1)° and m∠4 = (6x + 27)°, What is m∠7?
Converse of Alternate Interior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
x = 59, m∠7 = 124°
If m∠6 = (x - 3)° and m∠1 = (3x - 53)°, What is m∠7?
x = 18, m∠3 = 55°
If m∠8 = (12x - 91)° and m∠4 = (2x + 89)°, What is m∠3?
x = 21, m∠2 = 78°
If m∠4 = (2x + 36)° and m∠5 = (5x - 3)°, What is m∠2?
Same Side Interior Angles Theorem
If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.
AC
AF < AB and AF < _____
EFH or HFE
Angle ABF and angle ___ are corresponding angles.
alternate interior angles
Angles between 2 lines and on opposite sides of a transversal.
alternate exterior angles
Angles that lie on the same side of the transversal and in corresponding positions.
corresponding angles
Angles that lie on the same side of the transversal and in corresponding positions.
Perpendicular
Are planes EFG and BDH parallel or perpendicular?
x = 22, m∠4 = 60°
If m∠3 = (4x + 32)° and m∠2 = (x + 38)°, What is m∠4?
x = 18, m∠1 = 120°
If m∠3 = (4x + 48)° and m∠5 = (6x + 12)°, What is m∠1?
x = 9, m∠5 = 18°
If m∠3 = (5x - 27)° and m∠7 = (x + 9)°, What is m∠5?
x = 10, m∠7 = 82°
If m∠3 = (7x + 12)° and m∠8 = (11x - 12)°, What is m∠7?
x = 15, m∠5 = 93°
If m∠3 = (7x - 12)° and m∠2 = (8x - 33)°, What is m∠5?
x = 59, m∠8 = 107°
If m∠3 = (x + 14)° and m∠2 = (3x - 70)°, What is m∠8?
x = 5, m∠2 = 71°
If m∠4 = (4x + 51)° and m∠7 = (2x + 99)°, What is m∠2?
x = 18, m∠2 = 101°
If m∠4 = (7x - 25)° and m∠1 = (9x - 83)°, What is m∠2?
x = 14, m∠2 = 85°
If m∠4 = (8x - 27)° and m∠8 = (2x + 57)°, What is m∠2?
x = 19, m∠2 = 102°
If m∠4 = (8x - 50)° and m∠8 = (4x + 26)°, What is m∠2?
x = 2, m∠2 = 84°
If m∠5 = (11x + 74)° and m∠6 = (15x + 54)°, What is m∠2?
x = 69, m∠4 = 120°
If m∠6 = (3x - 87)° and m∠5 = (x - 9)°, What is m∠4?
x = 13, m∠5 = 83°
If m∠6 = (4x + 45)° and m∠7 = (6x + 5)°, What is m∠5?
x = 2, m∠5 = 99°
If m∠6 = (4x + 73)° and m∠7 = (x + 97)°, What is m∠5?
x = 35, m∠2 = 102°
If m∠6 = (4x - 38)° and m∠5 = (2x + 8)°, What is m∠2?
x = 5, m∠5 = 87°
If m∠6 = (9x + 48)° and m∠3 = (10x + 37)°, What is m∠5?
Converse of Alternate Exterior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.
congruent
If two intersecting lines form a linear pair of ___ angles, then the lines are perpendicular.
Corresponding Angles Postulate
If two lines parallel, then corresponding angles are congruent.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
perpendicular
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is ___ to the other line.
Line n
Which line is the transversal?
None (151 + 28 = 179)
Which lines, if any, can be proved parallel given the diagram?
Skew
___ lines are not coplanar. They are not parallel and they do not intersect.
perpendicular
⊥ means ___.
x = 77, y = 63
Solve for x and y.
n and p
If angles 1 and 3 are congruent, which lines must be parallel?
x = 33, m∠8 = 150°
If m∠1 = (2x - 36)° and m∠3 = (3x - 69)°, What is m∠8?
x = 8, m∠8 = 132°
If m∠1 = (3x + 24)° and m∠3 = (4x + 16)°, What is m∠8?
x = 21, m∠6 = 72°
If m∠5 = (8x - 60)° and m∠2 = (4x - 12)°, What is m∠6?
supplementary
When two lines are parallel, the acute angles they form with a transversal are ___ to the obtuse angles.
Congruent
Congruent or Supplementary?
16
Find x if f || g.
perpendicular
The shortest segment from a point to a line is ___ to the line.
x = 12, m∠5 = 94°
If m∠8 = (4x + 38)° and m∠1 = (5x + 34)°, What is m∠5?
x = 3, m∠7 = 170°
If m∠8 = (6x - 8)° and m∠6 = (14x - 32)°, What is m∠7?
x = 3, m∠1 = 85°
If m∠3 = (9x + 58)° and m∠6 = (14x + 53)°, What is m∠1?
x = 11, m∠4 = 87°
If m∠8 = (13x - 56)° and m∠7 = (16x - 83)°, What is m∠4?
x = 14, m∠4 = 34°
If m∠8 = (2x + 6)° and m∠2 = (6x - 50)°, What is m∠4?
x = 39, m∠4 = 85°
If m∠8 = (2x + 7)° and m∠3 = (3x - 22)°, What is m∠4?
95 degrees
Find x.
l and m
If angles 2 and 5 are congruent, what two lines must be parallel?
x = 16, m∠5 = 80°
If m∠7 = (7x - 32)° and m∠3 = (3x + 32)°, What is m∠5?
Vertical Angles
What type of angle pair are 6 and 7?
Parallel
___ lines are coplanar and do not intersect.
Parallel
Are planes ABF and DHG parallel or perpendicular?
x = 16, m∠1 = 57°
If m∠2 = (12x - 69)° and m∠3 = (x + 41)°, What is m∠1?
x = 46, m∠2 = 95°
If m∠8 = (2x + 3)° and m∠3 = (4x - 99)°, What is m∠2?
True
TRUE OR FALSE. You can use the measures of the angles formed by two lines and a transversal to determine whether the two lines are parallel.
Corresponding Angles
Type of Angles?
Corresponding Angles
What type of angle pair are 1 and 5?
Alternate Exterior Angles
What type of angle pair are 2 and 7?
Alternate Interior Angles
What type of angle pair are 3 and 6?
Perpendicular
Lines p and q together contain a linear pair in which the angles are congruent. What is the relationship between lines p and q?
Alternate Exterior Angles
Type of Angles?
Converse of Corresponding Angles Postulate
If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.
Converse of Same Side Interior Angles Theorem
If two coplanar lines are cut by a transversal so that a pair of same side interior angles are supplementary, then the two lines are parallel.
parallel
If two coplanar lines are perpendicular to the same line, then the two lines are ___ to each other.
perpendicular; midpoint
The perpendicular bisector of a segment is a line ___ to a segment at the segment's ___.
Alternate Interior Angles
Type of Angles?
Same Side Interior Angles
Type of Angles?
Same Side Interior Angles
What type of angle pair are 4 and 6?
Linear Pair
What type of angle pair are 5 and 6?
1, 5, 7
Which angles are on the same side as angle 3?
2, 3, 6, 7
Which angles are supplementary to angle 4?
1, 4, 5, 8
Which angles are supplementary to angle 6?
x = 8, m∠2 = 125°
If m∠7 = (17x - 81)° and m∠8 = (7x + 69)°, What is m∠2?
x = 10, m∠3 = 83°
If m∠7 = (18x - 97)° and m∠4 = (4x + 57)°, What is m∠3?
x = 22, m∠4 = 100°
If m∠1 = (4x - 8)° and m∠3 = (x + 58)°, What is m∠4?
x = 89, m∠1 = 4°
If m∠2 = (x + 87)° and m∠8 = (3x - 91)°, What is m∠1?
x = 87, m∠4 = 86°
If m∠2 = (x - 1)° and m∠1 = (2x - 80)°, What is m∠4?
x = 18, m∠6 = 27°
If m∠2 = (4x - 45)° and m∠8 = (2x - 9)°, What is m∠6?
x = 11, m∠6 = 96°
If m∠2 = (5x + 41)° and m∠7 = (11x - 37)°, What is m∠6?
x = 29, m∠8 = 118°
If m∠2 = (6x - 56)° and m∠4 = (x + 89)°, What is m∠8?
x = 17, m∠1 = 160°
If m∠2 = (7x - 99)° and m∠4 = (3x - 31)°, What is m∠1?
x = 11, m∠1 = 110°
If m∠2 = (8x - 18)° and m∠3 = (16x - 66)°, What is m∠1?
x = 10, m∠8 = 66°
If m∠7 = (13x - 16)° and m∠1 = (11x + 4)°, What is m∠8?
x = 9, m∠4 = 113°
If m∠7 = (14x - 59)° and m∠3 = (9x - 14)°, What is m∠4?
x = 44, m∠8 = 85°
If m∠7 = (2x + 7)° and m∠6 = (x + 41)°, What is m∠8?
x = 6, m∠3 = 32°
If m∠7 = (3x + 14)° and m∠1 = (x + 26)°, What is m∠3?
x = 11, m∠1 = 108°
If m∠7 = (4x + 64)° and m∠5 = (14x - 46)°, What is m∠1?
x = 39, m∠5 = 72°
If m∠7 = (4x - 84)° and m∠2 = (x + 69)°, What is m∠5?
x = 22, m∠6 = 48°
If m∠7 = (5x + 22)° and m∠5 = (2x + 88)°, What is m∠6?
x = 7, m∠4 = 63°
If m∠7 = (5x + 82)° and m∠1 = (4x + 89)°, What is m∠4?
x = 4, m∠6 = 89°
If m∠8 = (12x + 41)° and m∠7 = (10x + 51)°, What is m∠6?
x = 12, m∠2 = 105°
If m∠8 = (14x - 63)° and m∠6 = (16x - 87)°, What is m∠2?
x = 85, m∠1 = 27°
If m∠8 = (2x - 17)° and m∠4 = (x + 68)°, What is m∠1?
x = 56, m∠1 = 141°
If m∠8 = (2x - 73)° and m∠6 = (x - 17)°, What is m∠1?
x = 18, m∠1 = 81°
If m∠8 = (4x + 27)° and m∠4 = (2x + 63)°, What is m∠1?
x = 3, m∠6 = 86°
If m∠8 = (7x + 65)° and m∠3 = (11x + 61)°, What is m∠6?
x = 6, m∠1 = 80°
If m∠8 = (9x + 46)° and m∠5 = (13x + 2)°, What is m∠1?
x = 31, m∠8 = 68°
If m∠1 = (6x - 74)° and m∠4 = (2x + 6)°, What is m∠8?
x = 61, m∠3 = 120°
If m∠1 = (x + 59)° and m∠5 = (2x - 2)°, What is m∠3?
x = 10, m∠5 = 108°
If m∠1 = (x + 98)° and m∠8 = (2x + 52)°, What is m∠5?
x = 18, m∠4 = 138°
If m∠2 = (11x - 60)° and m∠6 = (13x - 96)°, What is m∠4?
x = 15, m∠1 = 38°
If m∠2 = (15x - 83)° and m∠7 = (2x + 8)°, What is m∠1?
x = 21, m∠1 = 55°
If m∠2 = (2x + 83)° and m∠8 = (8x - 43)°, What is m∠1?
x = 7, m∠5 = 96°
If m∠2 = (3x + 63)° and m∠7 = (6x + 54)°, What is m∠5?
x = 25, m∠1 = 97°
If m∠2 = (4x - 17)° and m∠7 = (3x + 22)°, What is m∠1?
x = 25, m∠6 = 112°
If m∠2 = (x + 87)° and m∠5 = (2x + 18)°, What is m∠6?
x = 27, m∠1 = 96°
If m∠3 = (3x + 15)° and m∠8 = (5x - 51)°, What is m∠1?
x = 35, m∠7 = 81°
If m∠3 = (x + 46)° and m∠2 = (3x - 6)°, What is m∠7?
x = 13, m∠1 = 75°
If m∠3 = (x + 62)° and m∠4 = (11x - 38)°, What is m∠1?
x = 4, m∠7 = 139°
If m∠4 = (11x - 3)° and m∠2 = (3x + 29)°, What is m∠7?
x = 12, m∠2 = 146°
If m∠4 = (18x - 70)° and m∠5 = (4x - 14)°, What is m∠2?
x = 3, m∠4 = 160°
If m∠5 = (13x - 19)° and m∠7 = (x + 17)°, What is m∠4?
x = 14, m∠4 = 63°
If m∠5 = (15x - 93)° and m∠3 = (3x + 75)°, What is m∠4?
x = 15, m∠6 = 113°
If m∠5 = (2x + 37)° and m∠4 = (x + 98)°, What is m∠6?
x = 16, m∠6 = 93°
If m∠5 = (3x + 39)° and m∠4 = (9x - 51)°, What is m∠6?
x = 2, m∠4 = 153°
If m∠5 = (4x + 19)° and m∠3 = (14x - 1)°, What is m∠4?
x = 31, m∠8 = 95°
If m∠5 = (4x - 39)° and m∠3 = (x + 54)°, What is m∠8?
x = 25, m∠2 = 85°
If m∠5 = (5x - 30)° and m∠6 = (3x + 10)°, What is m∠2?
x = 3, m∠4 = 102°
If m∠5 = (6x + 60)° and m∠6 = (4x + 90)°, What is m∠4?
x = 18, m∠2 = 102°
If m∠5 = (7x - 48)° and m∠6 = (x + 84)°, What is m∠2?
x = 35, m∠7 = 150°
If m∠5 = (7x - 95)° and m∠6 = (x - 5)°, What is m∠7?
x = 2, m∠3 = 66°
If m∠5 = (x + 64)° and m∠7 = (7x + 52)°, What is m∠3?
x = 13, m∠8 = 87°
If m∠5 = (x + 80)° and m∠4 = (4x + 35)°, What is m∠8?
x = 21, m∠6 = 162°
If m∠5 = (x - 3)° and m∠1 = (2x - 24)°, What is m∠6?
x = 15, m∠4 = 93°
If m∠6 = (10x - 57)° and m∠5 = (7x - 18)°, What is m∠4?
x = 18, m∠8 = 118°
If m∠6 = (12x - 98)° and m∠1 = (4x - 10)°, What is m∠8?
x = 13, m∠2 = 139°
If m∠6 = (18x - 95)° and m∠3 = (4x - 11)°, What is m∠2?
x = 44, m∠7 = 93°
If m∠6 = (2x - 1)° and m∠1 = (3x - 39)°, What is m∠7?
x = 16, m∠3 = 94°
If m∠6 = (5x + 6)° and m∠4 = (3x + 38)°, What is m∠3?
x = 33, m∠1 = 111°
If m∠6 = (5x - 96)° and m∠2 = (x + 36)°, What is m∠1?
x = 4, m∠4 = 5°
If m∠6 = (7x - 23)° and m∠2 = (10x - 35)°, What is m∠4?
x = 5, m∠5 = 102°
If m∠6 = (8x + 38)° and m∠7 = (14x + 32)°, What is m∠5?
x = 36, m∠3 = 94°
If m∠6 = (x + 50)° and m∠8 = (3x - 22)°, What is m∠3?
x = 5, m∠5 = 86°
If m∠7 = (11x + 31)° and m∠8 = (15x + 19)°, What is m∠5?