Geometry · Unit 3
One transversal cuts four lines. Use the angle marks, not the picture, to decide which lines are parallel.
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Same corner
If every mark sits in the same corner, those angles are corresponding.
Equal corresponding angles mean the lines are parallel.
Example: both upper-left angles are 62°. Those two lines are parallel.
Move the mark first
A mark in another corner has to move to the upper-left before you compare.
Upper-left and lower-right are vertical angles, so they are equal.
Upper-left and upper-right are a linear pair. So are upper-left and lower-left.
A linear pair adds to 180°.
Same number, different corner
The same number in two different corners is not automatically a match.
Move both marks to the upper-left, then compare.
Two corresponding angles that add to 180° do not make the lines parallel.
Check every line
You might get one pair, two separate pairs, three parallel lines, or no parallel lines.
The drawing is not to scale. Use the angle measures.
The four angle marks are in the same corner.
Transversal t cuts lines a, b, c, and d.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the upper-left corner, so they are corresponding angles.
a is 64°, b is 78°, c is 51°, d is 64°.
a and d are both 64°.
a ∥ d by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: B. a ∥ d
Transversal g cuts lines m, n, p, and q.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the lower-right corner, so they are corresponding angles.
m is 110°, n is 70°, p is 110°, q is 85°.
110° and 70° add to 180°, but corresponding angles have to be congruent.
70° and 110° add to 180°, but corresponding angles have to be congruent.
m and p are both 110°.
m ∥ p by the corresponding angles converse.
No other pair of lower-right angles is equal.
Answer: B. m ∥ p
Transversal v cuts lines w, x, y, and z.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the upper-right corner, so they are corresponding angles.
w is 47°, x is 47°, y is 63°, z is 81°.
w and x are both 47°.
w ∥ x by the corresponding angles converse.
No other pair of upper-right angles is equal.
Answer: A. w ∥ x
Transversal k cuts lines r, s, t, and u.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the lower-left corner, so they are corresponding angles.
r is 125°, s is 98°, t is 55°, u is 98°.
125° and 55° add to 180°, but corresponding angles have to be congruent.
s and u are both 98°.
s ∥ u by the corresponding angles converse.
No other pair of lower-left angles is equal.
Answer: C. s ∥ u
Transversal p cuts lines j, k, h, and n.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the upper-left corner, so they are corresponding angles.
j is 36°, k is 72°, h is 108°, n is 36°.
72° and 108° add to 180°, but corresponding angles have to be congruent.
j and n are both 36°.
j ∥ n by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: C. j ∥ n
Transversal v cuts lines a, b, c, and d.
Not drawn to scale.
Which two lines are parallel?
All four marks are in the lower-right corner, so they are corresponding angles.
a is 88°, b is 53°, c is 88°, d is 127°.
53° and 127° add to 180°, but corresponding angles have to be congruent.
a and c are both 88°.
a ∥ c by the corresponding angles converse.
No other pair of lower-right angles is equal.
Answer: B. a ∥ c
The marks sit in different corners. Move each one to the upper-left, then compare.
Transversal t cuts lines a, b, c, and d.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
a: the upper-right is 115°.
Upper-left = 180° − 115°
Upper-left = 65°.
b: the upper-left is 65°.
c: the lower-right is 102°.
The upper-left is 102°.
d: the lower-left is 40°.
Upper-left = 180° − 40°
Upper-left = 140°.
a and b are both 65°.
a ∥ b by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: A. a ∥ b
Transversal g cuts lines a, b, c, and d.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
a: the upper-left is 58°.
b: the upper-right is 75°.
Upper-left = 180° − 75°
Upper-left = 105°.
c: the lower-left is 58°.
Upper-left = 180° − 58°
Upper-left = 122°.
d: the lower-right is 122°.
The upper-left is 122°.
a and c are both marked 58°.
Those two marks are in different corners.
The upper-left angles are not all equal.
c and d are both 122°.
c ∥ d by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: D. c ∥ d
Transversal n cuts lines p, q, r, and s.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
p: the lower-right is 47°.
The upper-left is 47°.
q: the upper-left is 133°.
r: the upper-right is 133°.
Upper-left = 180° − 133°
Upper-left = 47°.
s: the lower-left is 70°.
Upper-left = 180° − 70°
Upper-left = 110°.
q and r are both marked 133°.
Those two marks are in different corners.
The upper-left angles are not all equal.
p and r are both 47°.
p ∥ r by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: B. p ∥ r
Transversal t cuts lines e, f, g, and h.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
e: the upper-left is 96°.
f: the lower-right is 84°.
The upper-left is 84°.
g: the upper-right is 84°.
Upper-left = 180° − 84°
Upper-left = 96°.
h: the lower-left is 110°.
Upper-left = 180° − 110°
Upper-left = 70°.
f and g are both marked 84°.
Those two marks are in different corners.
The upper-left angles are not all equal.
e and g are both 96°.
e ∥ g by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: B. e ∥ g
Transversal v cuts lines j, k, h, and n.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
j: the lower-left is 142°.
Upper-left = 180° − 142°
Upper-left = 38°.
k: the upper-left is 38°.
h: the upper-right is 38°.
Upper-left = 180° − 38°
Upper-left = 142°.
n: the lower-right is 100°.
The upper-left is 100°.
k and h are both marked 38°.
Those two marks are in different corners.
The upper-left angles are not all equal.
j and k are both 38°.
j ∥ k by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: A. j ∥ k
Transversal g cuts lines w, x, y, and z.
Not drawn to scale.
Which two lines are parallel?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
w: the upper-right is 124°.
Upper-left = 180° − 124°
Upper-left = 56°.
x: the lower-left is 124°.
Upper-left = 180° − 124°
Upper-left = 56°.
y: the lower-right is 71°.
The upper-left is 71°.
z: the upper-left is 109°.
w and x are both 56°.
w ∥ x by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: A. w ∥ x
Two pairs, a supplement that is not a match, or a grid of two parallel pairs.
Transversal v cuts lines w, x, y, and z.
Not drawn to scale.
Which is a true statement?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
w: the upper-left is 68°.
x: the lower-right is 105°.
The upper-left is 105°.
y: the upper-right is 112°.
Upper-left = 180° − 112°
Upper-left = 68°.
z: the lower-left is 75°.
Upper-left = 180° − 75°
Upper-left = 105°.
w and y are both 68°.
w ∥ y by the corresponding angles converse.
x and z are both 105°.
x ∥ z by the corresponding angles converse.
A line from one pair is not parallel to a line from the other pair.
Answer: A. w ∥ y and x ∥ z
Transversal t cuts lines a, b, c, and d.
Not drawn to scale.
Which is a true statement?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
a: the upper-left is 80°.
b: the upper-left is 100°.
c: the lower-right is 100°.
The upper-left is 100°.
d: the upper-right is 100°.
Upper-left = 180° − 100°
Upper-left = 80°.
b, c, and d are all marked 100°.
Those marks are not all in the same corner.
The upper-left angles are not all equal.
a and d are both 80°.
a ∥ d by the corresponding angles converse.
b and c are both 100°.
b ∥ c by the corresponding angles converse.
A line from one pair is not parallel to a line from the other pair.
Answer: B. a ∥ d and b ∥ c
Transversal n cuts lines p, q, r, and s.
Not drawn to scale.
Which is a true statement?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
p: the lower-left is 63°.
Upper-left = 180° − 63°
Upper-left = 117°.
q: the upper-right is 63°.
Upper-left = 180° − 63°
Upper-left = 117°.
r: the upper-left is 63°.
s: the lower-right is 48°.
The upper-left is 48°.
p, q, and r are all marked 63°.
Those marks are not all in the same corner.
The upper-left angles are not all equal.
p and q are both 117°.
p ∥ q by the corresponding angles converse.
No other pair of upper-left angles is equal.
Answer: A. only p ∥ q
In this drawing, j ∥ k and m ∥ n.
Which angle is not necessarily congruent to ∠4?
j ∥ k with transversal m: ∠4 and ∠2 are corresponding angles.
Corresponding angles are congruent.
∠4 ≅ ∠2.
∠4 and ∠7 are vertical angles.
Vertical angles are congruent.
∠4 ≅ ∠7.
m ∥ n with transversal j: ∠2 and ∠10 are corresponding angles.
∠2 ≅ ∠10.
So ∠4 ≅ ∠10.
∠3 and ∠4 are a linear pair.
∠3 = 180° − ∠4.
m ∥ n with transversal k: ∠11 and ∠3 are corresponding angles.
∠11 ≅ ∠3.
∠11 = 180° − ∠4.
∠11 is not necessarily congruent to ∠4.
Answer: D. ∠11
Transversal t cuts lines a, b, c, and d.
Not drawn to scale.
Which is a true statement?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
a: the upper-left is 72°.
b: the lower-right is 108°.
The upper-left is 108°.
c: the upper-right is 65°.
Upper-left = 180° − 65°
Upper-left = 115°.
d: the lower-left is 40°.
Upper-left = 180° − 40°
Upper-left = 140°.
No two upper-left angles are equal.
No two of these lines are parallel.
Answer: D. no two of these lines are parallel
Transversal g cuts lines a, b, c, and d.
Not drawn to scale.
Which is a true statement?
Move every mark to the upper-left corner. Those angles are corresponding.
Upper-left and lower-right are vertical angles, so they are equal.
A linear pair adds to 180°.
a: the upper-left is 54°.
b: the lower-right is 54°.
The upper-left is 54°.
c: the upper-right is 126°.
Upper-left = 180° − 126°
Upper-left = 54°.
d: the lower-left is 80°.
Upper-left = 180° − 80°
Upper-left = 100°.
a, b, and c are all 54°.
a ∥ b ∥ c by the corresponding angles converse.
d does not match, so it is not parallel to these lines.
Answer: A. a ∥ b and b ∥ c
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