SAT: Geometry & Trig - Lines, Angles, Triangles

Triangle $ABC$ is similar to triangle $XYZ$, where $A$, $B$, and $C$ correspond to $X$, $Y$, and $Z$, respectively. In triangle $ABC$, the length of $\overline{AB}$ is 170 and the length of $\overline{BC}$ is 850. In triangle $XYZ$, the length of $\overline{YZ}$ is 60. What is the length of $\overline{XY}$?

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In the figure, $RT = TU$, the measure of angle $\angle VST$ is $29^\circ$, and the measure of angle $\angle RVS$ is $41^\circ$. What is the value of $x$?

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In the figure shown, $\overline{WZ}$ and $\overline{XY}$ intersect at point $Q$. $YQ = 63$, $WQ = 70$, $WX = 60$, and $XQ = 120$. What is the length of $\overline{YZ}$?

In triangles $ABC$ and $DEF$, angles $B$ and $E$ each have measure $27^\circ$ and angles $C$ and $F$ each have measure $41^\circ$. Which additional piece of information is sufficient to determine whether triangle $ABC$ is congruent to triangle $DEF$?

In triangles $LMN$ and $RST$, angles $L$ and $R$ each have measure $60^\circ$, $LN = 10$, and $RT = 30$. Which additional piece of information is sufficient to prove that triangle $LMN$ is similar to triangle $RST$?

Each side of equilateral triangle S is multiplied by a scale factor of $k$ to create equilateral triangle T. The length of each side of triangle T is greater than the length of each side of triangle S. Which of the following could be the value of $k$?

Triangles $ABC$ and $DEF$ are shown above. Which of the following is equal to the ratio $\dfrac{BC}{AB}$?

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In the figure shown, triangle $CAE$ is similar to triangle $CBD$. The measure of angle $CBD$ is $57^\circ$, and $AE = 26(BD)$. What is the measure of angle $CAE$?

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In the figure shown, line $\ell$ intersects lines $j$ and $k$. Which additional piece of information is sufficient to prove that lines $j$ and $k$ are parallel?

In right triangle $ABC$, angle $C$ is the right angle and $BC = 162$. Point $D$ on side $AB$ is connected by a line segment with point $E$ on side $AC$ such that line segment $DE$ is parallel to side $BC$ and $CE = 2AE$. What is the length of line segment $DE$?

In the figure above, what is the length of $NQ$?

Triangle $XYZ$ is similar to triangle $RST$ such that $X$, $Y$, and $Z$ correspond to $R$, $S$, and $T$, respectively. The measure of $\angle Z$ is $20^\circ$ and $2XY = RS$. What is the measure of $\angle T$?

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In the figure shown, lines $r$ and $s$ are parallel, and line $m$ intersects both lines. If $y < 65$, which of the following must be true?

Triangle $ABC$ is similar to triangle $XYZ$, such that $A$, $B$, and $C$ correspond to $X$, $Y$, and $Z$ respectively. The length of each side of triangle $XYZ$ is 2 times the length of its corresponding side in triangle $ABC$. The measure of side $AB$ is 16. What is the measure of side $XY$?

In $\triangle XYZ$, the measure of $\angle X$ is $23^\circ$ and the measure of $\angle Y$ is $66^\circ$. What is the measure of $\angle Z$?

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In the figure, line $m$ is parallel to line $n$, and line $t$ intersects both lines. What is the value of $x$?

In triangle $JKL$, the measures of $\angle K$ and $\angle L$ are each $48^\circ$. What is the measure of $\angle J$, in degrees? (Disregard the degree symbol when entering your answer.)

In $\triangle RST$, the measure of $\angle R$ is $63^\circ$. Which of the following could be the measure, in degrees, of $\angle S$?

In triangle $DEF$, the measure of angle $D$ is $47^\circ$ and the measure of angle $E$ is $97^\circ$. In triangle $RST$, the measure of angle $R$ is $47^\circ$ and the measure of angle $S$ is $97^\circ$. Which of the following additional pieces of information is needed to determine whether triangle $DEF$ is similar to triangle $RST$?

Quadrilateral $P'Q'R'S'$ is similar to quadrilateral $PQRS$, where $P, Q, R$, and $S$ correspond to $P', Q', R'$, and $S'$, respectively. The measure of angle $P$ is $30^\circ$, the measure of angle $Q$ is $50^\circ$, and the measure of angle $R$ is $70^\circ$. The length of each side of $P'Q'R'S'$ is 3 times the length of each corresponding side of $PQRS$. What is the measure of angle $P'$?

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these eight angles is $(7x - 250)^\circ$. The sum of the measures of four of the eight angles is $k^\circ$. Which of the following could NOT be equivalent to $k$, for all values of $x$?

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In the right triangle shown, what is the value of $a$?

At a certain time and day, the Washington Monument in Washington, DC, casts a shadow that is 300 feet long. At the same time, a nearby cherry tree casts a shadow that is 16 feet long. Given that the Washington Monument is approximately 555 feet tall, which of the following is closest to the height, in feet, of the cherry tree?

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In the figure above, intersects at C. If , what is the value of $y$?

In triangle $ABC$, the measure of angle $A$ is $50^\circ$. If triangle $ABC$ is isosceles, which of the following is NOT a possible measure of angle $B$?

In the figure above, $RT = TU$.

What is the value of $x$?

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In the figure, lines $m$ and $n$ are parallel. If $x = 6k + 13$ and $y = 8k - 29$, what is the value of $z$?

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In triangle $PQR$, $QR$ is extended to point $S$. The measure of $\angle PQR$ is $132^\circ$, and the measure of $\angle PRS$ is $163^\circ$. What is the measure of $\angle QPR$?

Quadrilaterals $PQRS$ and $WXYZ$ are similar, where $P$, $Q$, and $R$ correspond to $W$, $X$, and $Y$, respectively. The measure of $\angle S$ is $135^\circ$, $PS = 45$, and $WZ = 9$. What is the measure of $\angle Z$?

The area of triangle $ABC$ above is at least $48$ but no more than $60$. If $y$ is an integer, what is one possible value of $x$?

In the triangle above, . What is the value of $b$?

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In the figure, two lines intersect at a point. Angle 1 and angle 2 are vertical angles. The measure of angle 1 is $72^\circ$. What is the measure of angle 2?

In the figure, three lines intersect at point $P$. If $x = 65$ and $y = 75$, what is the value of $z$?

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In the figure shown, line $m$ is parallel to line $n$. What is the value of $x$?

In the figure above, lines $\ell$ and $k$ are parallel.

What is the value of $a$?

In triangle $\triangle ABC$, the measure of angle $B$ is $52^\circ$ and the measure of angle $C$ is $17^\circ$. What is the measure of angle $A$?

Triangle $ABC$ and triangle $DEF$ are shown. The relationship between the side lengths of the two triangles is such that

If the measure of angle $BAC$ is $20^\circ$, what is the measure, in degrees, of angle $EDF$? (Disregard the degree symbol when gridding your answer.)

ID: 48fb6483

In triangle $XYZ$, angle $Y$ is a right angle, point $P$ lies on $\overline{XZ}$, and point $Q$ lies on $\overline{YZ}$ such that $\overline{PQ}$ is parallel to $\overline{XY}$. If the measure of angle $XZY$ is $63^\circ$, what is the measure, in degrees, of angle $XPQ$?

Triangles $\triangle PQR$ and $\triangle LMN$ are graphed in the xy-plane. Triangle $\triangle PQR$ has vertices $P, Q,$ and $R$ at $(4,5)$, $(4,7)$, and $(6,5)$, respectively. Triangle $\triangle LMN$ has vertices $L, M,$ and $N$ at $(4,5)$, $(4,7+k)$, and $(6+k,5)$, respectively, where $k$ is a positive constant. If the measure of $\angle Q$ is $t^\circ$, what is the measure of $\angle N$?

In the given figure, $\overline{AC}$ extends to point D. If the measure of $\angle BAC$ is equal to the measure of $\angle BCA$, what is the value of $x$?

Triangles $ABC$ and $DEF$ are congruent, where $A$ corresponds to $D$, and $B$ and $E$ are right angles. The measure of angle $A$ is $69^\circ$. What is the measure, in degrees, of angle $F$?

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In the figure, line $j$ is parallel to line $k$. What is the value of $x$?

In triangle $RST$, angle $T$ is a right angle, point $L$ lies on $\overline{RS}$, point $K$ lies on $\overline{ST}$, and $\overline{LK}$ is parallel to $\overline{RT}$. If the length of $\overline{RT}$ is 72 units, the length of $\overline{LK}$ is 24 units, and the area of triangle $RST$ is 792 square units, what is the length of $\overline{KT}$, in units?

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In the figure shown, line $c$ intersects parallel lines $s$ and $t$. What is the value of $x$?

Triangles $ABC$ and $DEF$ are congruent, where $A$ corresponds to $D$, and $B$ and $E$ are right angles. The measure of angle $A$ is $18^\circ$. What is the measure of angle $F$?

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In the figure, line $m$ is parallel to line $n$, and line $k$ intersects both lines. Which of the following statements is true?

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In the figure, $\overline{LQ}$ intersects $\overline{MP}$ at point $R$, and $\overline{LM}$ is parallel to $\overline{PQ}$. The lengths of $\overline{MR}$, $\overline{LR}$, and $\overline{RP}$ are $6$, $7$, and $11$, respectively. What is the length of $\overline{LQ}$?

In the figure above, segments $\overline{AE}$ and $\overline{BD}$ are parallel. If angle $\angle BDC$ measures $58^\circ$ and angle $\angle ACE$ measures $62^\circ$, what is the measure of angle $\angle CAE$?

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 10 feet tall and has a shadow that is 5 feet long. At the same time, the shadow of the other tree is 2 feet long. How tall, in feet, is the other tree?

In $\triangle XYZ$, the measure of $\angle X$ is $24^\circ$ and the measure of $\angle Y$ is $98^\circ$. What is the measure of $\angle Z$?

In triangle $ABC$, the measure of angle $B$ is $90^\circ$ and $\overline{BD}$ is an altitude of the triangle. The length of $\overline{AB}$ is $15$ and the length of $\overline{AC}$ is $23$ greater than the length of $\overline{AB}$. What is the value of $\dfrac{BC}{BD}$?

In the figure above, $\overline{AF}$, $\overline{BE}$, and $\overline{CD}$ are parallel. Points B and E lie on $\overline{AC}$ and $\overline{FD}$, respectively. If $AB = 9$, $BC = 18.5$, and $FE = 8.5$, what is the length of $\overline{ED}$, to the nearest tenth?

In the figure above, lines $\ell$ and $m$ are parallel, $y = 20$, and $z = 60$. What is the value of $x$?

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In the figure, line $q$ is parallel to line $r$, and both lines are intersected by line $s$. If $y = 2x + 8$, what is the value of $x$?

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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 $S$?

In the figure above, and intersect at point P, , and . What is the measure, in degrees, of ? (Disregard the degree symbol when gridding your answer.)

In the given triangle, and has a measure of . What is the value of $x$?

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In the figure, line $m$ is parallel to line $n$. What is the value of $w$?

Intersecting lines $r$, $s$, and $t$ are shown below.

What is the value of $x$?

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In the figure shown, points $Q$, $R$, $S$, and $T$ lie on line segment $PV$, and line segment $RU$ intersects line segment $SX$ at point $W$. The measure of $\angle SQX$ is $48^\circ$, the measure of $\angle SXQ$ is $86^\circ$, the measure of $\angle SWU$ is $85^\circ$, and the measure of $\angle VTU$ is $162^\circ$. What is the measure, in degrees, of $\angle TUR$?

Triangles $EFG$ and $JKL$ are congruent, where $E$, $F$, and $G$ correspond to $J$, $K$, and $L$, respectively. The measure of angle $E$ is $45^\circ$ and the measure of angle $F$ is $20^\circ$. What is the measure of angle $J$?

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In the figure, $AC = CD$. The measure of angle $EBC$ is $45^\circ$, and the measure of angle $ACD$ is $104^\circ$. What is the value of $x$?

In convex pentagon $ABCDE$, segment $\overline{AB}$ is parallel to segment $\overline{DE}$. The measure of angle $B$ is $139^\circ$, and the measure of angle $D$ is $174^\circ$. What is the measure, in degrees, of angle $C$?

In the figure above, lines $m$ and $n$ are parallel. What is the value of $b$?

In triangle $ABC$, $AB = 4{,}680$ millimeters (mm) and $BC = 4{,}680$ mm. Which statement is sufficient to prove that triangle $ABC$ is equilateral?

Right triangles $LMN$ and $PQR$ are similar, where $L$ and $M$ correspond to $P$ and $Q$, respectively. Angle $M$ has a measure of $53^\circ$. What is the measure of angle $Q$?

In a right triangle, the measure of one of the acute angles is $51^\circ$. What is the measure, in degrees, of the other acute angle?

In the figure above, two sides of a triangle are extended. What is the value of $x$?

In triangle $ABC$, the measure of angle $A$ is $54^\circ$, the measure of angle $B$ is $90^\circ$, and the measure of angle $C$ is $\left(\dfrac{k}{2}\right)^\circ$. What is the value of $k$?

In the figure above, $BD = 6$ and $AD = 8$.

What is the length of $\overline{DC}$?

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is $(9x - 560)^\circ$. The sum of the measures of one of the acute angles and three of the obtuse angles is $(-18x + w)^\circ$. What is the value of $w$?