25 hard SAT Lines, angles & triangles questions

In the xy-plane, line m is parallel to line n. A transversal intersects line m at point P, forming an angle of measure x°. The transversal intersects line n at point Q, forming an angle of measure $(2y + 15)$°, where these two angles are alternate interior angles. If a third line intersects the transversal at point P, forming a vertical angle to the angle at P of measure $(3y - 10)$°, what is the value of y?

• A$5$
• B$15$
• C$25$
• D$65$

In triangle ABC, the measure of angle A is 42° and the measure of angle B is $(2y - 6)$°. An exterior angle at vertex C has measure 108°. What is the value of y?

• A$57$
• B$36$
• C$33$
• D$54$

Triangle JKL is similar to triangle MNP such that J, K, and L correspond to M, N, and P, respectively. The measure of angle K is 58°. In triangle MNP, the measure of angle M is y° and the measure of angle P is 3y°. What is the value of y?

• A$19$
• B$29$
• C$30.5$
• D$58$

Lines a and b are parallel, and line c is a transversal. Angle 7 measures 73°. A line segment connects a point on line b to a point above line a, forming triangle MNO where angle M is at the intersection of c and b, angle N is on the segment, and angle O is 58°. If angle M is supplementary to an angle formed by the transversal, what is the value of angle N?

• A$49$
• B$73$
• C$107$
• D$131$

Triangle PQR is similar to triangle STU such that P, Q, and R correspond to S, T, and U, respectively. In triangle PQR, the measure of angle Q is 53 degrees and PQ = 12. In triangle STU, ST = 18. If the measure of angle P is w degrees and the measure of angle S is 2w degrees, what is the value of w?

• A$31.75$
• B$42.33$
• C$53$
• D$127$

In the xy-plane, line p has a slope of 2 and passes through the origin. Line q is perpendicular to line p and intersects line p at the point (3, 6). What is the y-coordinate of the point where line q intersects the y-axis?

• A$4.5$
• B$7.5$
• C$9$
• D$12$

Line m is parallel to line n. A transversal intersects line m at point P, forming an angle measuring x degrees. The transversal intersects line n at point Q, forming an angle measuring 3y degrees on the same side of the transversal as the x-degree angle. If the angle adjacent to the x-degree angle on line m measures 112 degrees, what is the value of y?

• A$68$
• B$112$
• C$204$
• D$22.67$

Triangle ABC is similar to triangle DEF, where A, B, and C correspond to D, E, and F, respectively. The measure of angle A is 44° and the measure of angle E is 71°. What is the measure of angle C?

• A$27^{\circ}$
• B$44^{\circ}$
• C$65^{\circ}$
• D$109^{\circ}$

In triangle GHI, the measure of angle H is 90° and HJ is an altitude from H to side GI. The length of GH is 18 and the length of GI is 32 more than the length of GH. If the length of HJ is d, what is the value of HI divided by d?

• A$\displaystyle \frac{18}{50}$
• B$\displaystyle \frac{32}{18}$
• C$\displaystyle \frac{40}{18}$
• D$\displaystyle \frac{50}{18}$

In triangle GHI, angle H is a right angle. Point J lies on side HI such that segment GJ divides the triangle into two smaller triangles. In triangle GHJ, the measure of angle HGJ is 34°. What is the measure of angle GJI?

• A$56$
• B$124$
• C$34$
• D$214$

In triangle DEF, the measure of angle D is 52° and the measure of angle E is 74°. Triangle DEF is similar to triangle JKL, where D, E, and F correspond to J, K, and L, respectively. An exterior angle at vertex L has measure $(7m + 5)$°. What is the value of m?

• A$17$
• B$25$
• C$7$
• D$10$

Line p is parallel to line q. Lines r and s intersect line p at points A and B, respectively, and intersect line q at points C and D, respectively. The measure of angle ABD is 65°. If the measure of angle CAB is x°, what is the value of x?

• A$25$
• B$65$
• C$90$
• D$115$

Triangle LMN is similar to triangle XYZ such that L, M, and N correspond to X, Y, and Z, respectively. In triangle LMN, the measure of angle L is twice the measure of angle M, and the measure of angle N is 84°. What is the measure of angle Y?

• A$32^{\circ}$
• B$48^{\circ}$
• C$64^{\circ}$
• D$96^{\circ}$

In triangle PQR, the measure of angle Q is 90° and QS is an altitude of the triangle. The length of PQ is 24 and the length of PR is 26. What is the value of the ratio of QR to QS?

• A$\displaystyle \frac{5}{12}$
• B$\displaystyle \frac{10}{24}$
• C$\displaystyle \frac{13}{12}$
• D$\displaystyle \frac{24}{13}$

Quadrilateral RSTV has vertices with interior angles as follows: angle R is $(2k + 15)$°, angle S is $(3k - 10)$°, angle T is $(k + 25)$°, and angle V is $(4k - 10)$°. What is the value of k?

• A$18$
• B$32$
• C$34$
• D$38$

In triangle ABC, the measure of angle A is 48°. An exterior angle at vertex C has measure $(5y - 12)$°. If the measure of angle B is twice the measure of angle A, what is the value of y?

• A$22$
• B$31.2$
• C$38.4$
• D$28.8$

Lines m and n are parallel. A transversal intersects line m at point P, forming an angle that measures 68°. The transversal intersects line n at point Q. At point Q, an angle is formed on the same side of the transversal as the 68° angle at point P. A line from Q makes an angle of 34° with line n, measured on the opposite side of line n from the transversal. What is the measure, in degrees, of the angle formed by this line from Q and the transversal?

• A$34$
• B$78$
• C$102$
• D$112$

In triangle DEF, the measure of angle D is $(2n + 15)$° and the measure of angle E is $(3n - 10)$°. An exterior angle at vertex F measures 125°. What is the value of n?

• A$11$
• B$18$
• C$24$
• D$34$

In triangle PQR, angle P measures 48° and angle Q measures 67°. Triangle PQR is similar to triangle XYZ, where P, Q, and R correspond to X, Y, and Z, respectively. If the ratio of PQ to XY is 3 to 5, what is the measure of angle Z?

• A$39^{\circ}$
• B$65^{\circ}$
• C$115^{\circ}$
• D$131^{\circ}$

Lines p and q are parallel. Line r intersects both lines, creating angle 5 measuring 112° on line p. A triangle is formed below line q with one vertex at the intersection of r and q. If two angles of this triangle measure 68° and x°, and the third angle is supplementary to angle 5, what is the value of x?

• A$44$
• B$68$
• C$112$
• D$136$

In triangle UVW, the measure of the exterior angle at vertex W is 124°. If the measure of angle U is three times the measure of angle V, what is the measure of angle V?

• A$14^{\circ}$
• B$31^{\circ}$
• C$56^{\circ}$
• D$93^{\circ}$

Two parallel lines are cut by a transversal. One of the angles formed measures 5p degrees. The angle vertically opposite to an alternate interior angle to the 5p-degree angle measures $3p + 36$ degrees. What is the value of p?

• A$9$
• B$18$
• C$28.8$
• D$72$

Triangle DEF is similar to triangle GHI such that D, E, and F correspond to G, H, and I, respectively. In triangle DEF, the measure of angle D is $(3k + 10)$° and the measure of angle E is $(2k + 20)$°. In triangle GHI, the measure of angle I is 70°. What is the value of k?

• A$10$
• B$16$
• C$22$
• D$35$

Triangle ABC is similar to triangle DEF such that A, B, and C correspond to D, E, and F, respectively. In triangle ABC, the measure of angle A is 53° and the measure of angle B is 71°. If the length of side DE is 3 times the length of side AB, what is the measure of angle F?

• A$19$
• B$37$
• C$56$
• D$109$

Lines j and k are parallel. Line m intersects both lines. At the intersection of m and j, angle 8 measures 118°. A triangle is formed with one vertex at the intersection of m and k, and angle measurements of y° and 56° for two of its angles. If the third angle of the triangle is an alternate interior angle to angle 8, what is the value of y?

• A$6$
• B$62$
• C$118$
• D$174$