Angles, Triangles and Polygons

📐 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.

Warm-up. 🔥 Try these before we teach anything. Missing one is fine: it tells you what to watch for.

Angle Basics

An angle is made by two rays that start at the same point, called the vertex. We measure angles in degrees ($^\circ$). A full turn is $360^\circ$.

Key Fact: The little square in a corner means exactly $90^\circ$. If a question shows that square, you have been handed a number even though no degrees are written.

Angles are named in two common ways: by the vertex alone ($\angle B$), or by three points with the vertex in the middle ($\angle ABC$).

Angle Pairs: Complementary, Supplementary and All the Way Around

Most angle questions come down to three sums:

Memory trick: Complementary = Corner ($90^\circ$). Supplementary = Straight ($180^\circ$).

Worked Examples

1. The complement of $48^\circ$. Complementary means the pair makes $90^\circ$: $90 - 48 = 42^\circ$.

2. The supplement of $75^\circ$. Supplementary means the pair makes $180^\circ$: $180 - 75 = 105^\circ$.

3. Two supplementary angles, one is 3 times the other. Call the smaller one $x$. Then the larger is $3x$, and together they make a straight line:

$x + 3x = 180 \quad\Rightarrow\quad 4x = 180 \quad\Rightarrow\quad x = 45$

The angles are $45^\circ$ and $135^\circ$. Check: $45 + 135 = 180$. ✓

4. Angles around a point. Three angles fill a full turn: $90^\circ$, $150^\circ$ and $x$.

$90 + 150 + x = 360 \quad\Rightarrow\quad x = 120$

Key Fact: Write the sum first ("these make $180$"), then the equation. Naming the rule is the step that keeps you out of the traps.

Use the figure above. 🎯

Use the figure above. 🎯 The square in the corner matters.

Use the figure above. 🎯

Name the angle. 🔽 Choose the type for each measure.

Work it out. ✍️ Enter numbers only (degrees).

1) The complement of a $17^\circ$ angle:

2) The supplement of a $109^\circ$ angle:

3) Two complementary angles: one is 4 times the other. The larger angle:

Key Takeaways — Part 1

• MK: 15 questions, 31 minutes (about 2 minutes each), four choices, no calculator
• Acute $< 90^\circ$, right $= 90^\circ$, obtuse between $90^\circ$ and $180^\circ$, straight $= 180^\circ$
• Complementary = $90^\circ$ (corner), supplementary = $180^\circ$ (straight), around a point = $360^\circ$
• A small square in a corner means exactly $90^\circ$
• Name the sum first, then write the equation

Up Next: Part 2 — what happens when two lines cross, and the shortcut that solves every parallel-lines question.

📐 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 $180^\circ$.

Use the figure above. 🎯

Parallel Lines Cut by a Transversal

Parallel lines never meet; a figure shows them with matching arrowheads. A transversal is a line that crosses both. That makes eight angles, and the pairs have names:

The Shortcut: Only Two Sizes

You don't need to memorize the names to get the answer. With parallel lines, all eight angles come in just two sizes:

• every acute angle in the figure is equal ($\angle 2, \angle 3, \angle 6, \angle 7$),
• every obtuse angle is equal ($\angle 1, \angle 4, \angle 5, \angle 8$),
• and an acute one plus an obtuse one is $180^\circ$.

So for any two angles, ask: do they look the same size? Then they are equal. One big, one small? Then they add to $180^\circ$.

Key Fact: The rules work only when the lines are parallel. If a figure doesn't mark them parallel (or the question doesn't say so), the angles need not match.

Worked Examples

1. One angle gives you all eight. In the figure, $\angle 2 = 52^\circ$.

• The acute angles $\angle 3$, $\angle 6$ and $\angle 7$ are also $52^\circ$.
• The obtuse angles $\angle 1$, $\angle 4$, $\angle 5$ and $\angle 8$ are $180 - 52 = 128^\circ$.

2. Angles written with $x$. Suppose $\angle 1 = (3x + 8)^\circ$ and $\angle 5 = (4x - 32)^\circ$. They are corresponding angles (both obtuse), so they are equal:

$3x + 8 = 4x - 32 \quad\Rightarrow\quad 40 = x$

Then $\angle 1 = 3(40) + 8 = 128^\circ$. Check the other expression: $4(40) - 32 = 128$. ✓

3. A same-side pair. If $\angle 4 = (2x)^\circ$ and $\angle 6 = (x + 30)^\circ$, the angles are one obtuse and one acute, so they add to $180^\circ$:

$2x + x + 30 = 180 \quad\Rightarrow\quad 3x = 150 \quad\Rightarrow\quad x = 50$

So $\angle 4 = 100^\circ$ and $\angle 6 = 80^\circ$.

Key Fact: Before writing an equation, decide equal or add to $180$. Getting that choice right is most of the problem.

Use the figure above. 🎯 Lines $m$ and $n$ are parallel, and $\angle 2 = 52^\circ$.

Name the pair. 🔽 Use the numbered angles in the figure.

Solve with algebra. ✍️ Lines $m$ and $n$ are parallel. Enter numbers only.

1) The value of $x$:

2) The measure, in degrees, of each labeled angle:

3) The measure, in degrees, of the angle that forms a straight line with the $(3x + 10)^\circ$ angle along line $m$:

Know the rule. 🎯

Key Takeaways — Part 2

• Vertical angles (across from each other) are equal; neighbors on a line add to $180^\circ$
• Parallel lines + transversal: corresponding, alternate interior and alternate exterior angles are equal; same-side interior angles add to $180^\circ$
• Shortcut: all eight angles come in two sizes: same-looking angles are equal, a big one and a small one add to $180^\circ$
• With $x$ in the labels: decide equal or add to 180, then solve
• The rules need the lines to be parallel

Up Next: Part 3 — triangles: the $180^\circ$ rule, the exterior-angle shortcut and which side lengths can even make a triangle.

📐 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).

The $180^\circ$ Rule and the Exterior-Angle Shortcut

The angles of every triangle add to $180^\circ$. That one rule solves most triangle questions.

An exterior angle is made by extending one side of the triangle. It sits on a straight line with the interior angle next to it, and that leads to a shortcut:

Key Fact: An exterior angle equals the sum of the two interior angles across from it (the two that don't touch it). In the figure, the exterior angle is $a + b$.

Why it works: $c$ and the exterior angle make $180^\circ$, and so do $c$ and $a + b$ (the triangle rule). So the exterior angle and $a + b$ must be equal.

Worked Examples

1. Missing angle. Two angles of a triangle are $48^\circ$ and $67^\circ$. The third is $180 - 48 - 67 = 65^\circ$.

2. Exterior angle. The two interior angles across from an exterior angle are $40^\circ$ and $75^\circ$. The exterior angle is $40 + 75 = 115^\circ$. No need to find the third interior angle first.

3. Isosceles. An isosceles triangle has a top (vertex) angle of $30^\circ$. The two base angles are equal, so each is $(180 - 30) \div 2 = 75^\circ$.

4. Right triangle. One acute angle of a right triangle is $23^\circ$. The other is $90 - 23 = 67^\circ$.

Side Rules

Isosceles triangles

The two equal sides sit across from the two equal angles. Equal sides ↔ equal angles: if you know one, you know the other.

Bigger angle, longer side

In any triangle, the longest side is across from the largest angle, and the shortest side is across from the smallest angle. In a right triangle, the longest side is the one across from the $90^\circ$ angle.

The triangle inequality

Three lengths make a triangle only if the two shorter ones add to more than the longest one. If they are equal or smaller, the sides lie flat or can't meet.

Range for the third side: if two sides are $5$ and $9$, the third side must be more than $9 - 5 = 4$ and less than $9 + 5 = 14$.

Key Fact: Third side: between the difference and the sum of the other two, never equal to either.

Use the figure above. 🎯

Use the figure above. 🎯

Apply the rules. 🎯

Classify each triangle. 🔽

Work it out. ✍️ Enter numbers only.

1) The angles of a triangle are in the ratio $1 : 2 : 3$. The smallest angle, in degrees:

2) A triangle has sides of $6$ and $11$. The third side must be greater than:

3) ... and less than:

Key Takeaways — Part 3

• By sides: equilateral (3 equal), isosceles (2 equal), scalene (none); by angles: acute, right, obtuse
• The angles of a triangle add to $180^\circ$; equilateral angles are $60^\circ$ each
• Exterior angle = sum of the two interior angles across from it
• Isosceles: equal sides ↔ equal base angles; longest side is across from the largest angle
• Triangle inequality: the third side is strictly between the difference and the sum of the other two

Up Next: Part 4 — the Pythagorean theorem, the number patterns that make right triangles fast, and similar triangles.

📐 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.

Pythagorean Triples: The Time Savers

Test writers like right triangles with whole-number sides. These sets, called Pythagorean triples, come up again and again. Knowing them lets you skip the squaring.

Spotting a multiple. A rectangular field is $30$ yards by $40$ yards. How long is the diagonal? Divide out the common factor: $30, 40$ is $10 \times (3, 4)$, so the diagonal is $10 \times 5 = 50$ yards. No squaring of big numbers needed.

Is it a right triangle? Square the two shorter sides and compare with the square of the longest. Sides $9, 12, 15$: $81 + 144 = 225 = 15^2$, so yes. Sides $5, 6, 8$: $25 + 36 = 61$, but $8^2 = 64$, so no.

Key Fact: The hypotenuse must be the longest side. If an answer choice for the hypotenuse is shorter than a leg, cross it out.

Use the figure above. 🎯

Use the figure above. 🎯

Right-triangle practice. ✍️ Enter numbers only.

1) Legs $15$ and $20$. The hypotenuse:

2) Hypotenuse $17$, one leg $8$. The other leg:

3) Legs $2$ and $3$. The value of $c^2$:

Right triangle or not? 🔽 Compare the two smaller squares with the largest square.

Similar Triangles

Similar triangles have the same shape but not necessarily the same size: one is a scaled copy of the other.

• Their matching (corresponding) angles are equal.
• Their matching sides are in the same ratio. Every side is multiplied by the same scale factor.

How to solve

1. Match the sides: shortest with shortest, longest with longest (or by position in the figure).
2. Find the scale factor from one known pair.
3. Multiply (or set up a proportion) for the missing side.

Worked example. A triangle has sides $3, 4, 5$. A similar triangle has shortest side $9$. The scale factor is $9 \div 3 = 3$, so the other sides are $4 \times 3 = 12$ and $5 \times 3 = 15$.

As a proportion: $\dfrac{3}{9} = \dfrac{4}{y}$ gives $3y = 36$, so $y = 12$.

Shadows. At the same moment, the sun hits everything at the same angle. A person and their shadow form a right triangle similar to the one a tree and its shadow form. That's how you find the height of something you can't measure directly.

Key Fact: Similar means multiply, not add. If one side grows from $4$ to $10$, that is $\times 2.5$; it is not "$+6$" for every side.

Use the figure above. 🎯 The two triangles are similar.

Use the figure above. 🎯

Key Takeaways — Part 4

• Right triangles only: $a^2 + b^2 = c^2$; the hypotenuse $c$ is across from the right angle and is the longest side
• Hypotenuse: add the squares; leg: subtract; then take the square root
• Know the triples: $3, 4, 5$ · $5, 12, 13$ · $8, 15, 17$ · $7, 24, 25$ and their multiples
• Similar triangles: equal angles, sides multiplied by one scale factor; set up a proportion
• Shadow problems are similar triangles

Up Next: Part 5 — quadrilaterals, polygons and the $(n - 2) \cdot 180^\circ$ rule.

📐 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.

Polygons and Their Angle Sums

A polygon is a closed figure with straight sides. Draw diagonals from one corner and the polygon splits into triangles, always two fewer triangles than sides. Each triangle holds $180^\circ$, so:

$\text{Sum of interior angles} = (n - 2) \cdot 180^\circ \qquad (n = \text{number of sides})$

Regular polygons

A regular polygon has all sides equal and all angles equal. To find one angle, divide the sum by $n$.

Worked example. One interior angle of a regular pentagon: $540 \div 5 = 108^\circ$.

Exterior angles

Walk around any polygon and you turn a full $360^\circ$. So the exterior angles of any polygon add to $360^\circ$, and each exterior angle of a regular polygon is $360 \div n$. Each exterior angle and its interior angle sit on a straight line, so they add to $180^\circ$.

Worked example. A regular polygon has exterior angles of $30^\circ$. It has $360 \div 30 = 12$ sides.

Key Fact: Interior sum $(n - 2) \cdot 180$; exterior sum always $360$. For a regular polygon, divide either sum by $n$ to get one angle.

Use the figure above. 🎯

Use the figure above. 🎯

Polygon angles. ✍️ Enter numbers only (degrees).

1) The sum of the interior angles of a heptagon (7 sides):

2) One interior angle of a regular octagon:

3) One exterior angle of a regular polygon with 20 sides:

Always, sometimes or never? 🔽

Work backward. 🎯

Key Takeaways — Part 5

• Quadrilateral angles add to $360^\circ$
• Parallelogram: opposite angles equal, consecutive angles add to $180^\circ$; rectangle, rhombus and square are special parallelograms; a trapezoid has exactly one pair of parallel sides
• Interior angle sum: $(n - 2) \cdot 180^\circ$; exterior angle sum: always $360^\circ$
• Regular polygon: one interior angle $= \dfrac{(n - 2) \cdot 180}{n}$, one exterior angle $= \dfrac{360}{n}$
• Interior + exterior at the same corner $= 180^\circ$

Up Next: Part 6 — how to work geometry items fast on test day, and the traps the wrong answers are built from.

📐 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.

The Traps the Wrong Answers Are Built From

Wrong answer choices are rarely random. Most come from one of these mistakes, so if your answer matches one of them, double-check.

Key Fact: If your first answer shows up as a choice, that is not proof it is right. The traps are there because they are common answers. Run the sanity check.

Spot the trap. 🎯

Use the figure above. 🎯 Check whether the angles should be equal or add to $180^\circ$.

More trap practice. 🎯

Diagnose the wrong answer. 🔽 Pick the mistake that produces each student answer.

Key Takeaways — Part 6

• About 2 minutes per MK question; geometry items are usually fast, so bank the time
• Routine: sketch and label → name the rule → one equation → sanity check
• Work from what is marked (labels, ticks, arrows, squares), not from how the picture looks
• No going back on the CAT: check before you confirm; if time runs short, keep answering honestly rather than guessing blindly
• Know the trap list: complement/supplement, exterior angle, same-side interior, legs added, missing square root, $n \cdot 180$, forgetting to divide by $n$, equal-sum "triangles"

Up Next: Part 7 — a mixed set that pulls every rule together.

📐 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.

Set A. 🎯

Set B. 🎯 Use the figure above. Lines $m$ and $n$ are parallel.

Set C. 🎯 Use the figure above.

Set D. 🎯

Set E. 🎯 Use the figure above.

Set F. 🎯

Key Takeaways — Part 7

• Most geometry items are one rule plus one equation; the hard ones chain two rules (parallel lines then a straight line, or similar triangles with the whole side)
• Match similar sides carefully: the small triangle's side goes with the whole matching side of the big one
• The hypotenuse is always the longest side, and $x$ is not the answer unless the question asks for $x$
• Missed a question? Go back to its part: angles (1), parallel lines (2), triangles (3), right and similar triangles (4), polygons (5), traps (6)

Next step: take the exit quiz for this topic. Then move on to Area, Volume, Circles and Coordinates, where these shapes get measured.