Lines, Angles, and Triangles: SAT Practice Questions & Study Guide
Questions covering angle relationships formed by parallel lines, properties of triangles, congruence, similarity, and the triangle inequality.
Understanding Lines, Angles, and Triangles on the SAT
Angle relationship questions on the SAT rest on a small set of rules that you need to recognize instantly. Vertical angles are equal. Supplementary angles add to 180°. Angles on a straight line add to 180°. Angles around a point add to 360°. When a transversal crosses two parallel lines, alternate interior angles are equal, corresponding angles are equal, and co-interior (same-side interior) angles are supplementary. These rules let you fill in most angles in a diagram from a single given angle measure.
Triangle properties include the angle sum rule (interior angles always sum to 180°), the exterior angle theorem (an exterior angle equals the sum of the two non-adjacent interior angles), and the triangle inequality (the sum of any two sides must exceed the third). The SAT tests these both directly and in multi-step problems where you compute one angle from a parallel-lines relationship and then use it inside a triangle.
Similar triangles have the same shape but different sizes: all three pairs of corresponding angles are equal, and corresponding side lengths are proportional. They arise in SAT problems involving shadows, scale models, or diagrams where a line parallel to one side of a triangle cuts the other two sides. Congruent triangles are identical copies: SSS, SAS, ASA, and AAS are the standard congruence criteria. The SAT rarely asks you to name a congruence theorem by name; instead, it asks you to use the equal side lengths or angles that congruence implies.
Special triangles—the 30-60-90 and 45-45-90 triangles—have side ratios worth knowing: 1 : √3 : 2 for 30-60-90 and 1 : 1 : √2 for 45-45-90. These appear on the reference sheet but coming up slow on them will cost time.
Key Rules & Formulas
Memorize these rules — they come up directly in SAT questions.
Triangle angle sum: angles A + B + C = 180°
If two angles of a triangle are 55° and 75°, the third is 180 − 55 − 75 = 50°
Exterior angle theorem: exterior angle = sum of the two non-adjacent interior angles
If the two non-adjacent interior angles are 40° and 65°, the exterior angle is 105°
Parallel lines cut by a transversal: alternate interior angles are equal; co-interior angles are supplementary
If one alternate interior angle is 70°, the other is also 70°; the co-interior angle is 110°
Similar triangles: corresponding sides are proportional; set up a ratio and cross-multiply
If ΔABC ~ ΔDEF with AB = 6, DE = 9, and BC = 4, then EF = 4 × (9/6) = 6
45-45-90 sides: 1 : 1 : √2; 30-60-90 sides: 1 : √3 : 2
30-60-90 triangle with hypotenuse 10: short leg = 5, long leg = 5√3
Lines, Angles, and Triangles Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
In the figure, two parallel lines are cut by a transversal. One of the alternate interior angles measures 65°. What is the measure of the other alternate interior angle?
In triangle PQR, angle P = 48° and angle Q = 73°. What is the measure of angle R?
Two lines intersect, forming vertical angles. One of the angles measures 112°. What is the measure of its vertical angle?
In triangle ABC, the exterior angle at vertex C measures 130°. The interior angle at vertex A measures 55°. What is the measure of the interior angle at vertex B?
Triangle DEF is similar to triangle GHI, with DE corresponding to GH, EF corresponding to HI, and DF corresponding to GI. If DE = 8, EF = 12, and GH = 10, what is the length of HI?
In the figure, a line parallel to the base of triangle ABC passes through point D on side AB and point E on side AC, creating triangle ADE. If AD = 4, DB = 6, and DE = 5, what is the length of BC?
Two parallel lines are cut by a transversal. A co-interior (same-side interior) angle measures (3x + 20)° and its co-interior partner measures (2x + 30)°. What is the value of x?
In triangle XYZ, the measure of angle X is twice the measure of angle Y, and the measure of angle Z is 40° more than angle Y. What is the measure of angle X in degrees?
A 30-60-90 triangle has its shorter leg equal to 7. What is the length of its hypotenuse?
In quadrilateral ABCD, angles A, B, C, and D have measures in the ratio 2:3:4:6. What is the measure of the largest angle?
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Common Mistakes to Avoid
These are the most frequent errors students make on Lines, Angles, and Triangles questions. Knowing them in advance prevents costly point losses.
- !Confusing supplementary (sum = 180°) with complementary (sum = 90°)
- !Using the wrong angle pair in a parallel-lines diagram—treating co-interior angles as equal instead of supplementary
- !Setting up similar triangle ratios with non-corresponding sides in the same fraction
- !Forgetting that the exterior angle theorem applies only to the exterior angle adjacent to one vertex, not any angle outside the triangle
- !Misidentifying the short and long legs in a 30-60-90 triangle (the side opposite 30° is always the short leg)
SAT Strategy Tips: Lines, Angles, and Triangles
Whenever you see parallel lines in a figure, immediately mark all equal angles and supplementary pairs before reading the question
For similar triangle problems, write the correspondence explicitly (A↔D, B↔E, C↔F) before setting up any ratio to avoid swapping sides
If the triangle looks like it might be a 45-45-90 or 30-60-90, check the angle labels—using the special ratios is almost always faster than the Pythagorean theorem
When given an exterior angle and one remote interior angle, find the other remote interior angle by subtraction rather than setting up a full equation
Other Geometry & Trigonometry Subtopics
Area and Volume
Questions asking you to compute or compare areas of 2D figures and surface areas and volumes of 3D solids.
Right Triangles and Trigonometry
Questions applying the Pythagorean theorem, SOH-CAH-TOA, and complementary angle trig identities to find side lengths and angle measures in right triangles.
Circles
Questions covering circle area, circumference, arc length, sector area, central and inscribed angles, and the equation of a circle in the coordinate plane.
Master Lines, Angles, and Triangles on the SAT
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