SAT lines, angles and triangles: review and practice

• Use vertical, supplementary, corresponding and alternate interior angles to find a missing angle.
• Use the fact that a triangle's angles add to 180°.
• Set up ratios between matching sides of similar triangles.
• Decide whether the given facts are enough to show two triangles are similar or congruent.

• Using a parallel-line rule when the lines aren't stated to be parallel. Corresponding angles are equal only if the problem says the lines are parallel or marks them that way.
• Solving for x and stopping. If an angle measures (3x + 10)° and x = 20, the angle is 70°, not 20°.
• Pairing sides that don't match. If triangle ABC is similar to triangle DEF, AB matches DE and BC matches EF, even when the drawing is rotated.
• Scaling angles along with sides. Similar triangles have equal angles. If one triangle's sides are three times as long, its 50° angle is still 50°.
• Using the side ratio for area. If two similar figures have a side ratio of 2:5, their area ratio is 4:25.
• Using part of a side as the whole side. If point P is on side JK, the similar-triangle ratio uses JP to JK, not JP to PK.

1. Question 1 · Easy

   In triangle ABC, angle A measures 31° and angle B measures 47°. What is the measure, in degrees, of angle C?

   • A16
   • B39
   • C78
   • D102
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   D. The interior angles sum to 180°, so angle C measures 180 − 31 − 47 = 102°. The other values are the difference of the given measures (16), their average (39), and their sum (78).

2. Question 2 · Easy

   Triangle KLM is similar to triangle UST, such that vertices K, L, and M correspond to vertices U, S, and T, respectively. Angle K measures 39°, and the corresponding lengths satisfy ST = LM/3. What is the measure, in degrees, of angle U?

   • A13
   • B39
   • C117
   • D141
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   B. Vertices K and U correspond, so similarity gives ∠U = ∠K = 39°. Choice A divides the angle by 3, choice C multiplies it by 3, and choice D takes its supplement.

3. Question 3 · Easy

   Triangle ABC is congruent to triangle FDE, with corresponding vertices listed in the same order. Angle D measures 90°, and angle E measures 64°. What is the measure, in degrees, of angle A?

   • A26
   • B64
   • C90
   • D116
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   A. The correspondence is A ↔ F, B ↔ D, and C ↔ E, so angles B and C measure 90° and 64°, and angle A measures 180 − 90 − 64 = 26°. The values 64 and 90 copy angles at different corresponding vertices, while 116 is the supplement of 64° and omits the right angle.

4. Question 4 · Medium

   In triangle JKL, angles J, K, and L measure 57°, x°, and (4x − 7)°, respectively. Point M lies in the interior of side LJ, and point N lies in the interior of side LK, such that MN is parallel to JK. What is the measure, in degrees, of angle LNM?

   • A26
   • B57
   • C97
   • D154
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   A. From 57 + x + (4x − 7) = 180, it follows that 5x = 130 and x = 26. Angle LNM corresponds to angle LKJ because MN is parallel to JK, so it measures 26°. The values 57 and 97 belong to the other two vertices, while 154 is the supplement of the requested angle.

5. Question 5 · Medium

   In triangles ABC and DEF, angles A and D each measure 41°, AC = 18, and DF = 54. Which additional information is sufficient to prove that the triangles are similar?

   • ABC = 15 and EF = 45
   • BAB = 12 and DE = 12
   • CAngle B measures 62°, and angle F measures 77°.
   • DAngle B measures 41°, and angle E measures 62°.
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   C. If ∠B = 62° and ∠F = 77°, then ∠C = 77° and ∠E = 62°, so AA establishes similarity. The side pairs in choice B do not share the factor 54/18, choice A gives a nonincluded angle with proportional sides and permits two different shapes, and choice D produces different angle sets.

6. Question 6 · Hard

   In triangle JKL, point P lies on side JK, and point Q lies on side JL. It is given that JP/PK = JQ/QL = t, PQ = 18, and KL = 30. What is the value of t?

   • A3/5
   • B2/3
   • C3/2
   • D5/3
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   C. The equal part ratios imply JP/JK = JQ/JL = t/(1 + t), and the included angle at J is common, so △JPQ ∼ △JKL by SAS. Hence t/(1 + t) = 18/30 = 3/5, giving 5t = 3 + 3t and t = 3/2; choice A stops at the part-to-whole ratio, choice B inverts the requested ratio, and choice D inverts the part-to-whole ratio.

Write the angle facts as one equation, such as x + (3x + 10) + 50 = 180, type it into Desmos and click the vertical line to read x. Then type the angle expression the question asks for with that value. For similar triangles, type the proportion with the unknown, like 12/18 = x/30, and read x the same way.

Ashore's SAT course starts with a 15-minute check that finds the question types costing you the most points, then gives you lessons and practice for each.

• Linear Equations and Inequalities
• Linear Functions and Systems
• Ratios, Rates, Percentages, and Units
• One- and Two-Variable Data
• Probability and Statistical Inference
• Equivalent Expressions and Quadratics
• Nonlinear Functions and Exponentials
• Right Triangles and Trigonometry
• Circles, Area, and Volume
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