Geometry · Unit 3
Two parallel lines and a bent path between them. Draw a parallel through the bend, then add the two pieces. Some diagrams need a system for x and y.
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A crook
Arrow marks show the parallel lines. A crook is the bent path between them. x is the angle at the bend.
One straight transversal is not enough. The path turns once.
Draw a parallel through the bend
Draw a line through the bend, parallel to the given lines. It splits x into an upper piece and a lower piece.
Each piece equals the alternate interior angle on that side: the angle between that parallel and the segment, on the bend side.
Add the two pieces
x = upper piece + lower piece
x = 36° + 53°
x = 89°
A straight-line angle
If the marked angle sits next to the piece on a straight line, subtract from 180° first.
Lower piece = 180° − 152°
Lower piece = 28°
x = 39° + 28°
x = 67°
Two variables
A linear pair adds to 180°. Alternate interior angles are equal when the lines are parallel.
The bend is the sum of the two pieces.
You need two equations. Solve for x and y.
The helper parallel is already drawn.
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 32°.
Alternate interior angles are equal, so the lower piece is 47°.
x = upper piece + lower piece
x = 32° + 47°
x = 79°
Answer: x = 79°
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 36°.
Alternate interior angles are equal, so the lower piece is 41°.
x = upper piece + lower piece
x = 36° + 41°
x = 77°
Answer: x = 77°
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 44°.
Alternate interior angles are equal, so the lower piece is 38°.
x = upper piece + lower piece
x = 44° + 38°
x = 82°
Answer: x = 82°
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 49°.
Alternate interior angles are equal, so the lower piece is 34°.
x = upper piece + lower piece
x = 49° + 34°
x = 83°
Answer: x = 83°
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 42°.
Alternate interior angles are equal, so the lower piece is 39°.
x = upper piece + lower piece
x = 42° + 39°
x = 81°
Answer: x = 81°
The dashed line splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 50°.
Alternate interior angles are equal, so the lower piece is 34°.
x = upper piece + lower piece
x = 50° + 34°
x = 84°
Answer: x = 84°
Draw the helper parallel yourself. Some marked angles are next to the piece on a straight line.
Draw a line through the bend, parallel to the given lines. It splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 46°.
Alternate interior angles are equal, so the lower piece is 32°.
x = upper piece + lower piece
x = 46° + 32°
x = 78°
Answer: x = 78°
Draw a line through the bend, parallel to the given lines. It splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 43°.
Alternate interior angles are equal, so the lower piece is 32°.
x = upper piece + lower piece
x = 43° + 32°
x = 75°
Answer: x = 75°
Draw a line through the bend, parallel to the given lines. It splits x into two pieces.
Alternate interior angles are equal, so the upper piece is 44°.
The 144° angle and the lower piece form a straight line.
Lower piece = 180° − 144°
Lower piece = 36°
x = upper piece + lower piece
x = 44° + 36°
x = 80°
Answer: x = 80°
The middle line passes through the bend and splits x into two pieces.
The 52° angle is vertical to the alternate interior angle, so the upper piece is 52°.
Alternate interior angles are equal, so the lower piece is 35°.
x = upper piece + lower piece
x = 52° + 35°
x = 87°
Answer: x = 87°
Draw a line through the bend, parallel to the given lines. It splits x into two pieces.
The 134° angle and the upper piece form a straight line.
Upper piece = 180° − 134°
Upper piece = 46°
The 138° angle and the lower piece form a straight line.
Lower piece = 180° − 138°
Lower piece = 42°
x = upper piece + lower piece
x = 46° + 42°
x = 88°
Answer: x = 88°
Draw a line through the bend, parallel to the given lines. It splits x into two pieces.
The 137° angle and the upper piece form a straight line.
Upper piece = 180° − 137°
Upper piece = 43°
The 141° angle and the lower piece form a straight line.
Lower piece = 180° − 141°
Lower piece = 39°
x = upper piece + lower piece
x = 43° + 39°
x = 82°
Answer: x = 82°
An expression is on the diagram. The crook is still the sum of the two pieces.
Draw a line through the bend, parallel to the given lines. It splits the 84° angle into two pieces.
The upper piece is (4x)°.
The lower piece is (3x)°.
84° = (4x)° + (3x)°
84 = 7x
x = 12
Upper piece = 4(12) = 48°
Lower piece = 3(12) = 36°
48 + 36 = 84. Check.
Answer: x = 12
Draw a line through the bend, parallel to the given lines. It splits the crook into two pieces.
Alternate interior angles are equal, so the upper piece is 44°.
Alternate interior angles are equal, so the lower piece is 34°.
(6x)° = 44° + 34°
6x = 78
x = 13
Crook = 6(13) = 78°
Answer: x = 13
Draw a line through the bend, parallel to the given lines. It splits the 70° angle into two pieces.
The upper piece is (2x)°.
Alternate interior angles are equal, so the lower piece is 34°.
70° = (2x)° + 34°
70 = 2x + 34
36 = 2x
x = 18
Upper piece = 2(18) = 36°
36 + 34 = 70. Check.
Answer: x = 18
Draw a line through the bend, parallel to the given lines. It splits the 78° angle into two pieces.
The upper piece is (5x)°.
The 142° angle and the lower piece form a straight line.
Lower piece = 180° − 142°
Lower piece = 38°
78° = (5x)° + 38°
78 = 5x + 38
40 = 5x
x = 8
Upper piece = 5(8) = 40°
40 + 38 = 78. Check.
Answer: x = 8
Draw a line through the bend, parallel to the given lines. It splits the 74° angle into two pieces.
The (4x)° angle and the upper piece form a straight line.
Upper piece = 180° − (4x)°
The 146° angle and the lower piece form a straight line.
Lower piece = 180° − 146°
Lower piece = 34°
74° = upper piece + lower piece
74 = (180 − 4x) + 34
74 = 214 − 4x
4x = 140
x = 35
Upper piece = 180 − 140 = 40°
40 + 34 = 74. Check.
Answer: x = 35
Draw a line through the bend, parallel to the given lines. It splits the 80° angle into two pieces.
The upper piece is (5x)°.
The lower piece is (3x)°.
80° = (5x)° + (3x)°
80 = 8x
x = 10
Upper piece = 5(10) = 50°
Lower piece = 3(10) = 30°
50 + 30 = 80. Check.
Answer: x = 10
Three parallel lines. Write two equations, then find x and y.
Not drawn to scale.
(5x − y)° + (5x + y)° = 180°
10x = 180
x = 18
(5x − y)° = 48°
5(18) − y = 48
90 − y = 48
y = 90 − 48
y = 42
Check: 132° + 48° = 180°.
Answer: x = 18, y = 42
Not drawn to scale.
(3x − y)° + (3x + y)° = 180°
6x = 180
x = 30
(3x − y)° = 70°
3(30) − y = 70
90 − y = 70
y = 90 − 70
y = 20
Check: 110° + 70° = 180°.
Answer: x = 30, y = 20
Not drawn to scale.
(2x + y)° + (4x − y)° = 180°
6x = 180
x = 30
(2x + y)° = 85°
2(30) + y = 85
60 + y = 85
y = 85 − 60
y = 25
Check: 95° + 85° = 180°.
Answer: x = 30, y = 25
Not drawn to scale.
(2x + y)° = 64°
Bend = upper piece + lower piece
100° = (2x + y)° + (3x − y)°
100 = 5x
x = 20
2(20) + y = 64
40 + y = 64
y = 24
Lower piece = 3(20) − 24 = 36°
64° + 36° = 100°. Check.
Answer: x = 20, y = 24
Not drawn to scale.
(4x − y)° + (2x + y)° = 180°
6x = 180
x = 30
(4x − y)° = 100°
4(30) − y = 100
120 − y = 100
y = 120 − 100
y = 20
Check: 80° + 100° = 180°.
Answer: x = 30, y = 20
Not drawn to scale.
(2x + 3y)° + (4x − 3y)° = 180°
6x = 180
x = 30
(2x + 3y)° = 84°
2(30) + 3y = 84
60 + 3y = 84
3y = 24
y = 8
Check: 4(30) − 3(8) = 96°, and 84° + 96° = 180°.
Answer: x = 30, y = 8
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