14.2 Lines, Angles, and Triangles

14.2 Lines, Angles, and Triangles

Quick Answer: The second PSAT 8/9 Geometry testing point is lines, angles, and triangles, including right triangles. The interior angles of a triangle sum to 180°. Vertical angles are equal; complementary angles sum to 90°; supplementary angles sum to 180°. Parallel lines cut by a transversal give equal corresponding and alternate interior angles. In a right triangle, a² + b² = c² with c the hypotenuse. Sine, cosine, and tangent are not assessed on the PSAT 8/9.

Together with area and volume, this testing point makes up Geometry’s ≈12.5% / 4–6 operational questions. The Assessment Framework claim for PSAT 8/9 Geometry is explicit: students apply theorems such as triangle sum and solve problems using the Pythagorean theorem, from middle-school and first-year algebra courses—not from a high-school trigonometry unit. This OpenExamPrep section is independent study material for the PSAT 8/9. It is not an official College Board publication and does not claim partnership, approval, or exact equivalence.

Angle sum in a triangle

The Bluebook reference sheet states: the sum of the measures in degrees of the angles of a triangle is 180. If two angles are known, subtract from 180 to get the third.

Worked example. In triangle ABC, angle A is 42° and angle B is 85°. Angle C = 180 − 42 − 85. First 42 + 85 = 127, then 180 − 127 = 53°.

A right triangle already uses 90° at one vertex, so the two acute angles are complementary: they sum to 90°. If one acute angle is 34°, the other is 56°.

An equilateral triangle has three equal sides and three equal angles, so each angle is 180 ÷ 3 = 60°. An isosceles triangle has two equal sides; the angles opposite those sides (the base angles) are equal. If the vertex angle is 50°, the two base angles share the remaining 130°, so each is 65°.

Complementary, supplementary, and vertical angles

Worked example. Two lines intersect. One angle measures 112°. The vertical (opposite) angle is also 112°. Each adjacent angle on the straight line is 180 − 112 = 68°.

Worked example. An angle of 37° is complementary to 53° because 37 + 53 = 90. The same 37° is supplementary to 143° because 37 + 143 = 180. Complementary does not require the angles to share a ray; it is a number relationship. A linear pair does require adjacency on a straight line.

Parallel lines and a transversal

When two parallel lines are cut by a transversal (a line that crosses both), several pairs of angles are locked together. At grade 8–9, master two equal pairs and one supplementary pair:

• Corresponding angles sit in the same position at each intersection (for example, both upper-left). They are equal.
• Alternate interior angles sit between the parallel lines, on opposite sides of the transversal. They are equal.
• Consecutive (same-side) interior angles sit between the parallel lines, on the same side of the transversal. They are supplementary (sum to 180°).

Worked example. Lines m and n are parallel. A transversal crosses them. One corresponding angle measures 65°. Then every corresponding copy is 65°, every alternate interior angle is 65°, and each consecutive interior angle is 180 − 65 = 115°. Vertical angles at the same vertex stay equal: if 65° is at a vertex, the opposite angle there is also 65°, and the adjacent linear pair is 115°.

If a question does not state that the lines are parallel, do not assume corresponding angles are equal. Parallelism is the extra fact that makes those theorems apply.

Right triangles and the Pythagorean theorem

A right triangle has one 90° angle. The side opposite that angle is the hypotenuse—always the longest side. The other two sides are legs. The Pythagorean theorem, printed on the reference sheet as c² = a² + b², says:

leg² + leg² = hypotenuse².

You may use it to find a missing side. You may not use it on an acute or obtuse triangle. Look for a right-angle mark, a square corner, or a statement that a side is perpendicular to another (a ladder standing on flat ground against a vertical wall is the classic story).

Worked example: find the hypotenuse. Legs 9 and 12. 9² + 12² = 81 + 144 = 225, and 225 = 15², so the hypotenuse is 15. This is a 3-4-5 triangle scaled by 3.

Worked example: find a leg. Hypotenuse 13, one leg 5. 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12. This is the 5-12-13 triple.

Worked example: 8-15-17. Legs 8 and 15: 8² + 15² = 64 + 225 = 289, and 289 = 17², so the hypotenuse is 17. If you stop at 289, you have c², not c. Taking the square root is part of the theorem.

Worked example: ladder. A 13-foot ladder leans against a vertical wall. The base is 5 feet from the wall. How high up the wall does the ladder reach? 5² + h² = 13² → 25 + h² = 169 → h² = 144 → h = 12 feet. Desmos scientific can compute √144, but recognizing 5-12-13 is faster.

Worked example: ramp. A ramp rises 3 feet over a 4-foot horizontal run. The ramp length is the hypotenuse: 3² + 4² = 9 + 16 = 25 = 5², so the ramp is 5 feet long. Do not reach for tangent or sine; those ratios are trigonometry and are not assessed on the PSAT 8/9.

Useful integer triples to recognize (and their multiples):

If the numbers are not a triple, still square, add, and take a square root. The embedded calculator is allowed on every Math question.

Special right triangles—optional geometry, not trigonometry

The reference sheet shows two special right triangles:

• 45°-45°-90°: legs s, s; hypotenuse s√2 (an isosceles right triangle).
• 30°-60°-90°: short leg x (opposite 30°), long leg x√3 (opposite 60°), hypotenuse 2x (opposite 90°).

Treat these as side-length facts if a figure is clearly 45-45-90 or 30-60-90. Example: a square with side 5 has diagonal 5√2 because the diagonal splits the square into two 45-45-90 triangles. Example: an equilateral triangle of side 6 splits into two 30-60-90 triangles with short leg 3, so the altitude is 3√3. That is not “sin 60°.” Do not spend PSAT 8/9 hours drilling sine, cosine, or tangent. Those skills are assessed on later SAT Suite tests, not on this one.

What to skip, and how to check

• No SOHCAHTOA, no unit-circle trigonometry, no inverse sine.
• No circle theorems (central angle, inscribed angle, arc, tangent-radius). Circles are SAT-only.
• After Pythagorean arithmetic, sanity-check: the hypotenuse must be longer than either leg, and a figure drawn to scale should look that way.
• If a² + b² = c² for the three sides, the triangle is right (converse). If you needed a right angle and the numbers fail that check, you used the wrong method.

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