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SAT Digital Math · SL AI

Geometry and Trigonometry: Lines, Angles, and Triangles

Targeted Practice Sheet 1

Worked Examples: SAT Strategy

Example 1: Easy (Triangle Sum & Similarity)

Question: Right triangles $PQR$ and $STU$ are similar, where $P$ corresponds to $S$. If the measure of angle $Q$ is $18^\circ$, what is the measure of angle $S$?
SAT Strategy: First, remember that the sum of the angles in any triangle is $180^\circ$. Second, "similar triangles" means their corresponding angles are perfectly equal.
  1. We are told the triangles are "right triangles," meaning one of their angles is $90^\circ$.
  2. Let's look at triangle $PQR$. We know angle $R = 90^\circ$ and angle $Q = 18^\circ$.
  3. Find angle $P$: $180^\circ - 90^\circ - 18^\circ = 72^\circ$.
  4. Because the triangles are similar and $P$ corresponds to $S$, angle $S$ must equal angle $P$.
  5. Answer: $72^\circ$.

Example 2: Medium (Parallel Lines & Transversals)

Question: Lines $s$ and $t$ are parallel. A third line intersects both. An angle measuring $110^\circ$ is vertical to an angle measuring $x^\circ$. Another angle inside the parallel lines on the same side of the transversal measures $y^\circ$. What is the value of $y$?
SAT Strategy: Use the rules of transversals. Vertical angles are equal. Same-side interior angles are supplementary (add up to $180^\circ$).
  1. Since the angle measuring $110^\circ$ is vertical to $x^\circ$, $x$ must be $110$.
  2. The angle $x$ and the angle $y$ are same-side interior angles between parallel lines.
  3. Therefore, $x + y = 180^\circ$.
  4. Substitute $110$ for $x$: $110 + y = 180 \implies y = 70$.
  5. Answer: $70$.

Example 3: Hard (Algebraic Similarity)

Question: In triangle $ABC$, point $D$ on side $AB$ is connected to point $E$ on side $AC$ such that line segment $DE$ is parallel to side $BC$. If $CE = 2AE$ and $BC = 162$, what is the length of line segment $DE$?
SAT Strategy: When a line is drawn parallel to the base of a triangle, it creates a smaller "nested" similar triangle ($ADE$) inside the larger one ($ABC$). The ratio of their sides will be proportional.
  1. Because $CE = 2AE$, the total length of side $AC$ is $AE + CE = AE + 2AE = 3AE$.
  2. The ratio of the small triangle's side to the large triangle's side is $\frac{AE}{AC} = \frac{AE}{3AE} = \frac{1}{3}$.
  3. Because the triangles are similar, this $1:3$ ratio applies to all corresponding sides, including the bases.
  4. $\frac{DE}{BC} = \frac{1}{3}$.
  5. Substitute $BC = 162$: $\frac{DE}{162} = \frac{1}{3} \implies 3DE = 162 \implies DE = 54$.
  6. Answer: $54$.

Practice Questions

Level 1: Easy (Foundational SAT)

  1. In a triangle, the measures of two angles are $45^\circ$ and $65^\circ$. What is the measure of the third angle? (Grid-in)
  2. Two lines intersect at a point, forming four angles. If one of the angles measures $105^\circ$, what is the measure of the angle vertically opposite to it?
    • A) $75^\circ$
    • B) $85^\circ$
    • C) $105^\circ$
    • D) $180^\circ$
  3. Triangle $DEF$ is similar to triangle $GHI$, where $D$ corresponds to $G$ and $E$ corresponds to $H$. If the measure of angle $D$ is $40^\circ$ and the measure of angle $H$ is $60^\circ$, what is the measure of angle $F$?
    • A) $40^\circ$
    • B) $60^\circ$
    • C) $80^\circ$
    • D) $100^\circ$
  4. Lines $a$ and $b$ are parallel and are intersected by a transversal line $c$. If one of the corresponding angles measures $132^\circ$, what is the measure of the other corresponding angle? (Grid-in)
  5. In right triangle $ABC$, angle $C$ is a right angle. If the measure of angle $A$ is $27^\circ$, what is the measure of angle $B$? (Grid-in)

Level 2: Medium (Standard SAT)

  1. Lines $l$ and $m$ are parallel. A transversal intersects them, creating a pair of same-side interior angles. One angle measures $(3x - 10)^\circ$ and the other measures $(2x + 40)^\circ$. What is the value of $x$?
    • A) $30$
    • B) $50$
    • C) $75$
    • D) $150$
  2. In triangle $XYZ$, the length of side $XY$ equals the length of side $XZ$. If the measure of angle $Y$ is $55^\circ$, what is the measure of angle $X$? (Grid-in)
  3. The side lengths of a right triangle are $6$, $8$, and $10$. A second triangle is similar to the first, and its longest side has a length of $25$. What is the perimeter of the second triangle?
    • A) $24$
    • B) $50$
    • C) $60$
    • D) $150$
  4. In the $xy$-plane, an angle has a measure of $x^\circ$ and another angle has a measure of $y^\circ$. The two angles are supplementary. If $x = 4y - 20$, what is the value of $y$? (Grid-in)
  5. The measure of an exterior angle of a triangle is $115^\circ$. The two opposite interior angles measure $4x^\circ$ and $(2x + 25)^\circ$. What is the value of $x$?
    • A) $15$
    • B) $18$
    • C) $22.5$
    • D) $45$

Level 3: Hard (Advanced SAT)

  1. The measures of the interior angles of a triangle are in the ratio $2 : 3 : 4$. What is the measure of the largest angle in the triangle? (Grid-in)
  2. In triangle $PQR$, line segment $ST$ is drawn parallel to side $PR$, with $S$ on side $PQ$ and $T$ on side $QR$. If $PQ = 24$, $SQ = 8$, and $PR = 36$, what is the length of line segment $ST$?
    • A) $9$
    • B) $12$
    • C) $16$
    • D) $24$
  3. Two intersecting lines form four angles. The measure of one angle is $(5x + 12)^\circ$ and the measure of an adjacent angle is $(3x + 8)^\circ$. What is the measure of the smaller of the two angles?
    • A) $20^\circ$
    • B) $68^\circ$
    • C) $112^\circ$
    • D) $180^\circ$
  4. Right triangles $ABC$ and $DEF$ are similar, with $A$ corresponding to $D$. The lengths of the legs of triangle $ABC$ are $5$ and $12$. If the length of the hypotenuse of triangle $DEF$ is $39$, what is the length of the shortest leg of triangle $DEF$? (Grid-in)
  5. Lines $p$ and $q$ are parallel. A transversal intersects them. The measure of an alternate exterior angle is $7x + 5$, and the corresponding alternate exterior angle on the other line is $4x + 32$. What is the value of $x$? (Grid-in)

Level 4: Very Hard (Slightly beyond SAT)

  1. Triangle $ABC$ is similar to triangle $XYZ$. The area of triangle $ABC$ is $45$ square inches. If the lengths of the sides of triangle $XYZ$ are $3$ times the lengths of the corresponding sides of triangle $ABC$, what is the area of triangle $XYZ$ in square inches?
    • A) $15$
    • B) $135$
    • C) $405$
    • D) $1215$
  2. In triangle $MNO$, the measure of angle $M$ is $(3x + 10)^\circ$, the measure of angle $N$ is $(5x - 20)^\circ$, and the measure of angle $O$ is $(x + 10)^\circ$. Which of the following accurately describes triangle $MNO$?
    • A) It is an equilateral triangle.
    • B) It is an isosceles right triangle.
    • C) It is a scalene right triangle.
    • D) It is an isosceles acute triangle.
  3. A line intersects parallel lines $l_1$ and $l_2$. A second line intersects $l_1$ and $l_2$ such that the two transversals intersect exactly on $l_1$, forming a triangle between the transversals and $l_2$. If the two angles formed on $l_1$ outside the triangle measure $110^\circ$ and $130^\circ$, what is the measure of the third angle inside the triangle resting on $l_2$? (Grid-in)
  4. In right triangle $PQR$, an altitude is drawn from the right angle $Q$ to the hypotenuse $PR$, intersecting $PR$ at point $S$. If $PQ = 9$ and $QR = 12$, what is the length of the altitude $QS$? (Express as a decimal). (Grid-in)
  5. In a regular polygon with $n$ sides, the measure of each interior angle is $156^\circ$. What is the value of $n$? (Grid-in)

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