Lines & Angles
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ACT Math › Lines & Angles
In the diagram, lines $l$ and $m$ are parallel and are intersected by a transversal line $t$. If the measure of angle 1 is $70°$, what is the measure of angle 7?
$110°$
$70°$
$160°$
$20°$
Explanation
This is a parallel lines and transversals question testing supplementary angle relationships. Choice C (110°) is correct — using standard transversal labeling (angles 1–4 at the upper intersection, 5–8 at the lower), angles 1, 3, 5, and 8 are all congruent, and angles 2, 4, 6, and 8 are all supplementary to 1, 3, 5, and 8. So if angle 1 is 70, that mens that angle 8 is 70, and that means that angle 7 is 110.
Two lines intersect. One of the angles formed is $120^\circ$. Angle $x$ is adjacent to the $120^\circ$ angle, sharing a side with it, and the other sides form a straight line (a linear pair). What is the measure of angle $x$?
$60^\circ$
$240^\circ$
$30^\circ$
$120^\circ$
Explanation
The angles form a linear pair, being adjacent and forming a straight line. Linear pairs are supplementary, so their measures add to $180^\circ$. Subtract the given $120^\circ$ from $180^\circ$: x = $180^\circ$ - $120^\circ$ = $60^\circ$. This applies the straight-angle property. Choice B of $120^\circ$ might result from assuming equality instead of supplement.
Lines $m$ and $n$ are parallel and cut by a transversal. An interior angle on the left side of the transversal at the top intersection is $70^\circ$. The alternate interior angle at the bottom intersection is labeled $x$.
m ⇒ ⇒ ⇒
70°\
\
\
n ⇒ ⇒ ⇒
/ x
If lines $m$ and $n$ are parallel, what is the measure of angle $x$?
$110^\circ$
$70^\circ$
$20^\circ$
$90^\circ$
Explanation
The 70° angle and angle x are alternate interior angles created by a transversal intersecting parallel lines m and n. Alternate interior angles are equal in measure when the lines are parallel, as they lie on opposite sides of the transversal between the parallels. Therefore, angle x measures 70°. This property helps prove lines are parallel or find unknown angles in such configurations. Choice A of 110° might stem from incorrectly treating them as supplementary instead of alternate interior.
In a pair of parallel lines cut by a transversal, angle $3$ is $85^\text{o}$. What is the measure of angle $6$, the alternate interior angle?
95°
85°
90°
75°
Explanation
Angles 3 and 6 are alternate interior angles formed by parallel lines cut by a transversal. When parallel lines are cut by a transversal, alternate interior angles are equal. Since angle 3 is 85°, angle 6 must also be 85°. Choice A incorrectly uses 95°, which has no geometric relationship to the given angle.
At point $O$, two lines intersect. The angle labeled $48^\circ$ and the angle labeled $x$ are vertical angles.
\ 48° /
\ /
-----O-----
/ x \
What is the measure of angle $x$?
$48^\circ$
$180^\circ$
$132^\circ$
$90^\circ$
Explanation
The angles 48° and x are vertical angles formed when two lines intersect at point O. Vertical angles are always equal because they are opposite each other when two lines cross. Therefore, x = 48°. Choice A (132°) incorrectly treats these as supplementary angles, which would be the relationship between adjacent angles on a straight line rather than vertical angles.
Lines $p$ and $q$ are parallel, and line $r$ is a transversal. If angle $2$ is $110^\circ$, what is the measure of the corresponding angle $4$?
80°
70°
110°
100°
Explanation
Angles $2$ and $4$ are corresponding angles formed by parallel lines p and q cut by transversal r. When parallel lines are cut by a transversal, corresponding angles are equal. Since angle $2$ is $110^\circ$, angle $4$ must also be $110^\circ$. Choice A incorrectly uses $70^\circ$, which would be the supplementary angle.
Lines $m$ and $n$ are parallel (⇒). A transversal $t$ intersects them. The angle labeled $x$ is an exterior angle at line $n$ on the right side of the transversal. The angle labeled $95^\circ$ is the corresponding exterior angle at line $m$ on the right side of the transversal.
m ⇒ ───────────────
/ 95°
/ t
/
/
/ x
n ⇒ ───────────────
If lines $m$ and $n$ are parallel, what is the measure of angle $x$?
$90^\circ$
$95^\circ$
$180^\circ$
$85^\circ$
Explanation
The angles x and 95° are corresponding angles formed by parallel lines m and n with transversal t. When two parallel lines are cut by a transversal, corresponding angles are always equal. Therefore, x = 95°. Choice A (85°) might result from incorrectly treating these as supplementary angles, which would be the case for same-side interior angles but not corresponding angles.
Two lines intersect at point $O$. The angle labeled $x$ is adjacent to an angle labeled $80^\circ$ and together they form a straight line.
\ 80° /
\ /
-----O-----
/ x \
What is the measure of angle $x$?
$160^\circ$
$10^\circ$
$100^\circ$
$80^\circ$
Explanation
The angles 80° and x are adjacent angles that form a linear pair on a straight line. Adjacent angles on a straight line are supplementary, meaning they add up to 180°. Therefore, 80° + x = 180°, which gives us x = 180° - 80° = 100°. Choice A (80°) incorrectly assumes the angles are vertical angles (equal) rather than supplementary adjacent angles.
If angle $C$ and angle $D$ are supplementary and angle $C$ measures $110^\circ$, what is the measure of angle $D$?
$70^\circ$
$110^\circ$
$90^\circ$
$120^\circ$
Explanation
Supplementary angles are two angles whose measures sum to 180°. If angle C and angle D are supplementary, then C + D = 180°. Since angle C measures 110°, we have 110° + D = 180°, so D = 180° - 110° = 70°. Choice B (110°) incorrectly assumes the angles are equal.
If angle $A$ and angle $B$ are vertical angles and angle $A$ is $50^\text{o}$, what is the measure of angle $B$?
100°
80°
130°
50°
Explanation
Angles A and B are vertical angles formed when two lines intersect. Vertical angles are always equal in measure. Since angle A is 50°, angle B must also be 50°. Choice A incorrectly uses 130°, which would be the supplementary (adjacent) angle.