Geometry and Trigonometry: Lines, Angles, and Triangles

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational SAT)

1. The angles of a triangle measure $x^\circ$, $2x^\circ$, and $60^\circ$. What is the value of $x$? (Grid-in)
2. Lines $l$ and $m$ are parallel and are intersected by transversal line $t$. If one of the acute angles formed is $45^\circ$, what is the measure of one of the obtuse angles formed?
   • A) $45^\circ$
   • B) $90^\circ$
   • C) $135^\circ$
   • D) $180^\circ$
3. Two intersecting lines form a pair of vertical angles with measures $(3x + 10)^\circ$ and $(5x - 20)^\circ$. What is the value of $x$? (Grid-in)
4. Triangles $ABC$ and $DEF$ are similar. The lengths of the sides of $\triangle ABC$ are $5$, $12$, and $13$. If the shortest side of $\triangle DEF$ is $15$, what is the length of the longest side of $\triangle DEF$?
   • A) $26$
   • B) $36$
   • C) $39$
   • D) $60$
5. The measure of an exterior angle of a triangle is $110^\circ$. If one of the remote interior angles is $40^\circ$, what is the measure of the other remote interior angle? (Grid-in)

Level 2: Medium (Standard SAT)

1. In an isosceles triangle, the vertex angle (the angle between the two equal sides) measures $50^\circ$. What is the measure of one of the base angles?
   • A) $50^\circ$
   • B) $65^\circ$
   • C) $80^\circ$
   • D) $130^\circ$
2. Three angles form a straight line. Their measures are $x^\circ$, $2x^\circ$, and $3x^\circ$. What is the measure of the largest angle?
   • A) $30^\circ$
   • B) $60^\circ$
   • C) $90^\circ$
   • D) $120^\circ$
3. Triangle $XYZ$ contains a line segment $AB$ parallel to $YZ$, with $A$ on $XY$ and $B$ on $XZ$. If $XA = 5$, $AY = 3$, and $AB = 10$, what is the length of $YZ$? (Grid-in)
4. The interior angles of a pentagon (5-sided polygon) measure $100^\circ$, $110^\circ$, $120^\circ$, $100^\circ$, and $x^\circ$. What is the value of $x$?
   • A) $90$
   • B) $100$
   • C) $110$
   • D) $120$
5. Lines $j$ and $k$ are parallel and cut by a transversal. Two consecutive interior angles measure $(4y + 5)^\circ$ and $(5y - 14)^\circ$. What is the measure, in degrees, of the larger of the two angles? (Grid-in)

Level 3: Hard (Advanced SAT)

1. Two sides of a triangle have lengths of $7$ and $10$. Which of the following could be the length of the third side?
   • A) $3$
   • B) $15$
   • C) $17$
   • D) $20$
2. A person who is $6$ feet tall casts a shadow that is $8$ feet long. At the exact same time of day, a nearby tree casts a shadow that is $40$ feet long. What is the height of the tree in feet? (Grid-in)
3. In a right triangle, the two acute angles measure $2a^\circ$ and $(3a - 10)^\circ$. What is the measure of the smaller of these two acute angles?
   • A) $20^\circ$
   • B) $40^\circ$
   • C) $50^\circ$
   • D) $90^\circ$
4. Lines $l_1$ and $l_2$ are parallel. A third line $l_3$ intersects $l_1$ at point $A$ and intersects $l_2$ at point $B$, forming acute angles of $40^\circ$ at the intersections. A fourth line $l_4$ intersects $l_1$ at point $C$ and intersects $l_2$ at point $D$, forming acute angles of $70^\circ$. If $l_3$ and $l_4$ intersect each other at point $P$ directly between the parallel lines, what is the measure of $\angle AP C$?
   • A) $30^\circ$
   • B) $70^\circ$
   • C) $110^\circ$
   • D) $140^\circ$
5. Triangle $ABC$ is similar to Triangle $DEF$. The perimeter of $\triangle ABC$ is $24$. If every side of $\triangle DEF$ is exactly $3$ times the length of the corresponding side of $\triangle ABC$, what is the perimeter of $\triangle DEF$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

1. Triangle $ABC$ is an isosceles triangle with $AB = AC$. The line segment $BC$ is extended past $C$ to a point $D$, creating an exterior angle $\angle ACD$. If $\angle ACD = 130^\circ$, what is the measure of $\angle A$? (Grid-in)
2. Lines $p$ and $q$ are parallel. Transversal line $t$ is perpendicular to a second transversal line $m$. If line $m$ intersects line $p$ at a $30^\circ$ angle, what is the measure of the acute angle formed by the intersection of line $t$ and line $q$?
   • A) $30^\circ$
   • B) $60^\circ$
   • C) $90^\circ$
   • D) $120^\circ$
3. A triangle has side lengths of $8$, $15$, and $x$. If $x$ must be an integer, how many possible values are there for $x$? (Grid-in)
4. In $\triangle PQR$, the length of $PQ$ equals the length of $PR$. A line segment $QS$ is drawn such that it bisects $\angle PQR$. If the measure of $\angle P$ is $36^\circ$, what is the measure of $\angle PQS$?
   • A) $18^\circ$
   • B) $36^\circ$
   • C) $72^\circ$
   • D) $144^\circ$
5. Two parallel lines are intersected by a transversal. The bisectors of two consecutive interior angles are drawn, and they intersect at a point $P$ between the parallel lines. What is the measure of the angle formed at $P$ by these two bisectors? (Grid-in)