📐 Angles, Triangles and Polygons

Part 1 of 7 — Angles on the ASVAB

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the fifth subtest on the computer version (CAT-ASVAB): 15 scored questions in 31 minutes, so you have about 2 minutes per question. Every question has four answer choices, and there is no calculator. At MEPS you get two sheets of scratch paper and two pencils, and you can ask for more.

The official description of MK includes "geometry and measurement," so expect several questions about shapes: a missing angle, a side of a right triangle, an angle inside a polygon. The numbers are usually friendly. What the question really tests is whether you know which rule to use.

MK is one of the four subtests in your AFQT (AR + MK + 2·VE), the score the military uses to check whether you are eligible to enlist. Every geometry rule you lock in here is points toward that score.

Facts to Know by Heart

Plan on having no formulas given to you: learn these by heart. This topic uses every one of them.

Don't try to memorize the whole table today. By Part 7 you will have used each line many times.

📐 Angles, Triangles and Polygons

Part 2 of 7 — Vertical Angles and Parallel Lines

When Two Lines Cross

Two crossing lines make four angles, and two rules tell you all four from just one:

• Vertical angles are the angles across from each other. They are always equal.
• Angles next to each other (sharing a side) sit on a straight line, so they add to $180^\circ$. This pair is called a linear pair.

Worked Example

Two lines cross and one angle is $38^\circ$.

• The angle across from it (vertical) is also $38^\circ$.
• Each angle next to it is $180 - 38 = 142^\circ$.
• The four angles are $38^\circ$, $142^\circ$, $38^\circ$, $142^\circ$. Check: they add to $360^\circ$. ✓

Key Fact: Crossing lines make only two sizes of angle: a pair of equal vertical angles and the other equal pair, and the two sizes add to $180^\circ$.

📐 Angles, Triangles and Polygons

Part 3 of 7 — Triangles: Types, Angle Sums and Side Rules

Naming Triangles

Triangles get two kinds of names: one from their sides and one from their angles.

A few facts follow right away:

• An equilateral triangle also has three equal angles: $180 \div 3 = 60^\circ$ each.
• A triangle can have at most one right angle or obtuse angle. Two of them would already use up $180^\circ$ or more.
• In a right triangle, the two smaller angles add to $90^\circ$ (they are complementary).

📐 Angles, Triangles and Polygons

Part 4 of 7 — The Pythagorean Theorem and Similar Triangles

The Pythagorean Theorem

In a right triangle, the two sides that form the right angle are the legs ($a$ and $b$). The side across from the right angle is the hypotenuse ($c$). It is always the longest side.

$a^2 + b^2 = c^2$

It works only for right triangles.

Worked Examples

1. Find the hypotenuse. Legs $6$ and $8$:

$6^2 + 8^2 = 36 + 64 = 100 \quad\Rightarrow\quad c = \sqrt{100} = 10$

2. Find a leg. Hypotenuse $13$, one leg $5$. Now you subtract:

$5^2 + b^2 = 13^2 \quad\Rightarrow\quad b^2 = 169 - 25 = 144 \quad\Rightarrow\quad b = 12$

3. Not a whole number. Legs $1$ and $2$: $c^2 = 1 + 4 = 5$, so $c = \sqrt{5}$. Leave it as a square root unless the choices show a decimal.

Key Fact: Looking for the hypotenuse? Add the squares. Looking for a leg? Subtract the smaller square from the larger one. Either way, take the square root at the end.

📐 Angles, Triangles and Polygons

Part 5 of 7 — Quadrilaterals and Other Polygons

Quadrilaterals

A quadrilateral is any four-sided figure. Its four angles always add to $360^\circ$ (two triangles' worth).

The family tree

• Every square is a rectangle, a rhombus and a parallelogram.
• A rectangle is a square only when its sides are all equal.
• A trapezoid is not a parallelogram: it has only one pair of parallel sides.

Key Fact: In a parallelogram, the angles next to each other (consecutive angles) add to $180^\circ$, and the angles across from each other are equal. One angle tells you all four.

📐 Angles, Triangles and Polygons

Part 6 of 7 — Test Strategy and Common Traps

How Geometry Items Play Out

MK gives you 31 minutes for 15 questions, about 2 minutes each. Most angle and triangle questions take well under that once you know the rule, which banks time for the longer algebra and word-style items.

The CAT-ASVAB is adaptive: the first question is of average difficulty, and a right answer usually brings a harder one. You also can't go back to a question after you answer it, so check before you confirm.

A four-step routine

1. Sketch and label. Copy the figure onto your scratch paper and write every given number on it, including the $90^\circ$ that a small square stands for.
2. Name the rule. Straight line $180$? Triangle $180$? Parallel lines (equal or add to $180$)? Pythagorean theorem? Similar triangles?
3. Write one equation and solve it.
4. Sanity-check. Does an angle that looks obtuse come out over $90^\circ$? Is the hypotenuse the longest side? Do the angles of your triangle add to $180^\circ$?

Use what is marked, not what it looks like

Work from the numbers and marks given (labels, tick marks, parallel arrows, right-angle squares). Two lines that look parallel only count as parallel if the figure marks them or the question says so. Your eyes are for the sanity check, not for the answer.

When time runs short

Unfinished questions are scored as if you had guessed at random, and the official advice is to keep reading and genuinely answering rather than clicking randomly: the CAT applies a relatively large penalty when several wrong answers come at the end of a subtest. Geometry rules are fast, so a quick honest answer usually beats a blind guess. Some questions may be unscored tryouts, but you can't tell which, so treat every one as scored.

📐 Angles, Triangles and Polygons

Part 7 of 7 — Mixed Mastery Set

Put It All Together

Eleven questions, mixed the way the test mixes them: a few quick ones, several medium, and a few that take two or three steps. Give yourself about 2 minutes each, use scratch paper, and no calculator.

For every question, run the routine: sketch and label → name the rule → one equation → sanity check.