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MK · lesson 9 of 14 · 40 min

Angles and Lines

Geometry on Mathematics Knowledge often starts with angles, and this lesson builds that foundation carefully. It begins with points, lines, rays and segments, what an angle is, how to name one unambiguously, and degree measure. From there you classify angles and work with adjacent angles, linear pairs, vertical angles, perpendicular lines, complementary and supplementary pairs, and angle bisectors.

Reading · 40 minVideo · 11 minPractice questions (44)

Video lesson: Angles and Lines

Video transcript

0:00 Welcome to this explainer. If you're prepping for the ASVAB and maybe feeling a little less than confident about the math section, you are in exactly the right place. We're going to patiently walk through all the essential geometry concepts you need for angles and lines, and the best part, we're setting this up so you can confidently tackle these questions entirely by hand. That's right, absolutely no calculator needed. Okay, let's dive right in. Here's our game plan for today. We'll start with lines and basic angles. Move into important angle relationships and then cover parallel lines and transversals. After that, we'll get into solving angle equations. Flag some common mistakes to avoid and finish up with some check yourself practice. All right, part one, lines and basic angles. So what's really interesting here is how simple an angle actually is. It's literally just two rays sharing a common endpoint, which we call the vertex. Now, an angle is measured in degrees, right? But it doesn't measure how long the lines are drawn. It only measures how far one side is turned away from the other. I think of an angle kind of like a door on its hinge.

1:05 The hinge is your vertex, and the size of the angle just measures how far that door has swung open. It goes anywhere from those sharp, acute angles under 90 degrees, all the way to flat, straight 180 degree angles. Important angle relationships. Okay, this is a massive concept for the ASVAB. Complementary angles add up to exactly 90 degrees. Supplementary angles add up to exactly 180 degrees. You can picture them like a linear pair sitting completely flat on a straight line. Now, here is a mental trick that is an absolute lifesaver. C comes before S in the alphabet, just like 90 comes before 180 numerically. Complementary 90. Supplementary 180. Lock that alphabet trick in and you'll never mix them up on a test day. Let's step it up. Part three. Parallel lines and transversals. When a slanted line called a transversal crosses parallel lines, it creates eight separate angles. But here's the secret to making this easy. It really only creates two sizes of angles. It makes four equal acute angles and four equal obtuse angles.

2:10 That means if you know just one single angle, you can easily find all eight. Corresponding, alternate interior and alternate exterior angles are always completely equal to their partners. The only odd balls you have to watch out for are the same side interior angles. Those are supplementary, meaning they add up to 180. Alright, part four. Solving angle equations. Let's put this into practice. We are going to use a totally bulletproof four-step routine here. First, identify the angles you're looking at. Second, pair them up by naming their relationship. You know, ask yourself, are they vertical, supplementary, corresponding? Third, write out an equation, setting them either equal to each other, or adding them up to 90, 180, or 360. And finally, step four. Check your work. Since we aren't using calculators, lean on friendly mental math steps. For instance, if you need to subtract 137 from 180, just think, well, 137 plus 3 gets me to a clean 140, and 140 plus 40 gets me to 180. Add the 3 and 40 together, and boom, your answer is 43. Simple, right? Let's look at our first worked example.

3:16 We've got intersecting lines forming an angle, PTR, which is 57 degrees. To find the rest of the angles, we just trace our way around the intersection. Its vertical partner, directly across from it, QTS, is automatically also 57. The angle right next to it, PTS, forms a straight linear pair. So, we just do 180 minus 57, which gives us 123. The vertical angle across from that one, QTR, is also 123. To check our work, add them all up, 57 plus 123 plus 57 plus 123, perfectly equals a full circle of 360 degrees. Easy. This next one brilliantly illustrates our equation step. In worked example two, we have two algebra expressions representing complementary angles. Now, because they are complementary, we know they sum to 90. So, we simply write them out on one line and add them together. 4x plus 5 plus x plus 10 equals 90. Combine those x's to get 5x plus 15 equals 90. Subtract 15 from both sides, leaving 5x equals 75.

4:20 Divide by 5, and we find that x equals 15. In worked example three, we're dealing with corresponding angles on parallel lines. Remember, corresponding angles essentially slide right on top of each other, meaning they are perfectly equal. So, we just set our two expressions equal to each other. 6x minus 10 equals 4x plus 18. We'll subtract 4x from both sides to get 2x minus 10 equals 18. Add 10 to both sides to get 2x equals 28, cut that in half, and x equals 14. Moving to worked example four. Remember how we said same side interior angles are the odd ones out? They aren't equal. They are supplementary, meaning they add to 180. So, we add the two expressions together. 3x plus 22 plus 2x plus 8 equals 180. Combine your like terms to get 5x plus 30 equals 180. Subtract 30 to leave 5x equals 150. Divide by 5, which gives us a nice clean x equals 30. No matter how many angles sit side by side along a straight line, their grand total is always exactly 180 degrees.

5:24 Here we have three angles. Let's add them up. 2x plus x plus 30 plus 3x minus 6, all equals 180. Grouping our terms gives us 6x plus 24 equals 180. Subtract that 24 to get 6x equals 156. Divide by 6, mentally, and x equals 26. In worked example six, things get a tiny bit more complex. We have a bisected angle, which simply means it was cut into two equal halves. One half measures x plus 9. To get the whole angle, we just double that half. So, two times the group x plus 9 gives us 2x plus 18. Then, we're told this whole angle forms a linear pair with an angle of 2x plus 42. Since a linear pair equals 180, we just add them up. 2x plus 18 plus 2x plus 42 equals 180. That simplifies down to 4x plus 60 equals 180. Subtract 60 to get 4x equals 120, making x equal to 30. Let's look at worked examples seven. Vertical angles sit directly across an intersection from each other.

6:29 That specific position guarantees they are exactly equal, which gives us a very straightforward algebraic setup. We just set them equal. 7x minus 15 equals 4x plus 18. Subtract 4x from both sides to get 3x minus 15 equals 18. Add 15 over to the other side to get 3x equals 33, meaning x equals 11. Finally, worked example eight. If you travel in a full circle around a single point, your angle equations will always add up to a complete 360 degrees. Here, we'll add our four given angles. 2x plus 3x plus 95 plus 115 equals 360. Combine your terms to get 5x plus 210 equals 360. Subtract 210 to get 5x equals 150, which brings us to x equals 30. Okay, part five, common mistakes to avoid. The ASVAB loves to lay traps, so let's outsmart them. First, pitfall, mixing up complementary and supplementary. Remember our trick?

7:30 C before S, 90 before 180. Second, assuming same side interior angles are equal. Remember, they are the only ones between parallel lines that add to 180 instead. Third, trusting a drawing's appearance. Unless it has a little square for 90 degrees or matching arrows for parallel lines, do not assume anything based on how it looks. But honestly, the absolute biggest trap of all is doing all that hard algebra to find x and immediately selecting that as your answer. Completely forgetting the question actually asked you to calculate the final angle measurement. Always, always read the final question. Let's wrap this up with part six. Check yourself practice. What is the complement of a 34-degree angle? Is it A, 326 degrees, B, 34 degrees, C, 56 degrees, or D, 146 degrees? Read this carefully. Recall our alphabet trick and take a brief mental pause to solve it in your head. Okay, the correct answer is C, 56 degrees. Because complementary angles must total exactly 90 degrees, we simply subtract 34 from 90 to get 56.

8:37 Now, if you picked 146, you accidentally found the supplement. See how easy it is to fall into that trap? Let's try question two. Four angles surround a point without overlapping. Three of them measure 140 degrees, 85 degrees, and 60 degrees. What is the measure of the fourth angle? Is it A, 65, B, 85, C, 75, or D, 135? Think about what a full rotation around a point actually equals. Combine the given numbers and take a quick pause to calculate your final step. And the answer is C, 75 degrees. The full circle around a point always totals 360. Our three known angles add up to 285. A quick mental subtraction of 360, minus 285, leaves us with exactly 75 degrees for that fourth and final angle. You've got the rules down, you've mastered the algebra, and you know exactly which traps to step over. You know, if a few simple intersecting lines can completely lock down the dimensions of a room, imagine what you can build when you master the rest of geometry.

9:44 Are you ready to build that foundation?

Contents · 38 sections
  1. Lesson Outcomes and Exam Context
  2. Points, Lines, Rays, and Segments
  3. What Is an Angle?
  4. Naming Angles Without Ambiguity
  5. Degree Measure and Full Rotation
  6. Angle Classification
  7. Adjacent Angles
  8. Linear Pairs and Straight Lines
  9. Vertical Angles
  10. Perpendicular Lines
  11. Complementary Angles
  12. Supplementary Angles
  13. Angle Bisectors
  14. Parallel Lines and Transversals
  15. Labeling the Eight Transversal Angles
  16. Corresponding Angles
  17. Alternate Interior Angles
  18. Alternate Exterior Angles
  19. Same-Side Interior Angles
  20. Every Transversal Angle Is Either Acute or Obtuse in the Nonperpendicular Case
  21. Converse Tests for Parallel Lines
  22. Combining Vertical and Parallel-Line Relationships
  23. Solving Algebraic Angle Equations
  24. Several Adjacent Angles on a Line
  25. Angles Around a Point
  26. Right-Angle and Parallel Constraints in Drawings
  27. Geometry Model: Identify → Pair → Equation → Check
  28. Mental Arithmetic and Answer-Choice Checks
  29. Worked Example 1 — Intersecting Lines
  30. Worked Example 2 — Complementary Expressions
  31. Worked Example 3 — Corresponding Angles
  32. Worked Example 4 — Same-Side Interior Angles
  33. Worked Example 5 — Three Angles on One Line
  34. Worked Example 6 — Bisector and Supplement
  35. Worked Example 7 — Vertical Angles Algebra
  36. Worked Example 8 — Point Rotation with Four Regions
  37. Frequent ASVAB-Style Angle Mistakes
  38. Rapid Review and Final Checklist
Visual lesson

Lesson slides

15 slides
Angles and Lines – slide 1 of 15

Lesson Outcomes and Exam Context

By the end of this lesson you should be able to classify angles by size and use the key totals: complementary angles make 90 degrees, supplementary angles and linear pairs make 180 degrees, and angles around a point make 360 degrees. You will apply vertical angles and angle bisectors, name the angle pairs formed when a transversal crosses parallel lines, and solve equations built from these relationships. ASVAB Mathematics Knowledge includes basic geometry like this, though no particular question is promised.

Points, Lines, Rays, and Segments

A point marks an exact location and has no size; it is named with a capital letter such as P. A line is perfectly straight, has no thickness, and extends forever in both directions; the line through points A and B is called line AB. A ray starts at an endpoint and continues forever in one direction. Ray AB begins at A and passes through B, so ray BA is a different ray. A segment is the part of a line between two endpoints, so segment AB has a measurable length, unlike a line or a ray.

Four rows: point P as a dot; line AB with arrows at both ends; ray AB starting at endpoint A and passing through B; segment AB with endpoints A and B only.

What Is an Angle?

An angle is formed by two rays that share a common endpoint, called the vertex. The rays are the sides of the angle. The size of an angle measures how far one side is turned from the other, not how long the sides are drawn, so extending the sides on a diagram does not change the angle. Think of a door on its hinge: the hinge is the vertex, and how far the door has swung open is the angle, whether the door is wide or narrow.

The same angle drawn twice, once with short sides and once with long sides, with the vertex and sides labelled and the same arc: side length does not change the angle.

Naming Angles Without Ambiguity

An angle can be named by its vertex alone, such as angle Q, only when exactly one angle sits at that vertex. When several rays meet at a point, use three letters with the vertex in the middle: angle PQR has its vertex at Q and sides passing through P and R. Angle RQP names the same angle. Diagrams may also label angles with numbers, such as angle 3.

Angle PQR with vertex Q and sides through P and R; the vertex is in the middle, so angle PQR is the same as angle RQP.

Degree Measure and Full Rotation

Angles are measured in degrees. One full turn around a point is 360 degrees, a half turn along a straight line is 180 degrees, and a quarter turn, like a square corner, is 90 degrees. A protractor measures an angle by placing its center on the vertex and one side along zero. A clock face shows the idea: the minute hand sweeps 360 degrees per hour, or 6 degrees per minute.

Quarter turn of 90° (square corner), half turn of 180° (straight line) and full turn of 360° (full circle).

Angle Classification

Angles are classified by measure. An acute angle is greater than 0 and less than 90 degrees. A right angle is exactly 90 degrees. An obtuse angle is greater than 90 but less than 180 degrees. A straight angle is exactly 180 degrees. Angles between 180 and 360 degrees are called reflex angles. For example, 41 degrees is acute and 152 degrees is obtuse.

Five angle types: acute 41°, right 90° with a square, obtuse 152°, straight angle of exactly 180°, and a reflex angle between 180° and 360°.

Adjacent Angles

Adjacent angles share a vertex and one common side but do not overlap. If ray OB lies between rays OA and OC, then angles AOB and BOC are adjacent, and their measures add to give angle AOC. For instance, if angle AOB is 35 degrees and angle BOC is 50 degrees, then angle AOC is 85 degrees. Angles that touch only at the vertex, without a shared side, are not adjacent.

Rays OA, OB and OC from point O: adjacent angles AOB = 35° and BOC = 50° share side OB and together make angle AOC = 85°.

Linear Pairs and Straight Lines

When two adjacent angles have outer sides that form a straight line, they are called a linear pair, and their measures always add to 180 degrees. If one angle of a linear pair is 64 degrees, the other must be 180 minus 64, or 116 degrees. This is one of the most useful facts in diagram problems: whenever a ray stands on a straight line, the two angles on either side of it total 180 degrees.

A ray standing on a straight line forms a linear pair of 64° and 116°, which add to 180°.

Vertical Angles

When two lines cross, they form four angles. The angles directly across from each other are vertical angles, and vertical angles are always equal. The reason is that each of them forms a linear pair with the same neighboring angle. If one angle is 38 degrees, its vertical partner is also 38 degrees, and each of the other two angles is 142 degrees.

Two crossing lines form vertical angles of 38° and 38° opposite each other, and 142° and 142° in the other pair.

Perpendicular Lines

Perpendicular lines meet at a right angle, so all four angles they form are 90 degrees. Diagrams show this with a small square drawn in the corner. If you know that one angle at an intersection is 90 degrees, the other three must also be 90 degrees, because of vertical angles and linear pairs. A wall meeting a floor and the corners of a sheet of paper are everyday examples.

Two perpendicular lines with a right-angle square in one corner; all four angles are 90°.

Complementary Angles

Two angles are complementary when their measures add to 90 degrees. The complement of a 28-degree angle is 90 minus 28, or 62 degrees. Complementary angles do not need to be adjacent; they only need to total 90. Only acute angles have complements.

A right angle split into adjacent angles of 28° and 62°, showing complementary angles: 28° + 62° = 90°.

Supplementary Angles

Two angles are supplementary when their measures add to 180 degrees. The supplement of a 141-degree angle is 39 degrees. Every linear pair is supplementary, but supplementary angles can also sit apart on a diagram. A memory aid: C comes before S in the alphabet, just as 90 comes before 180.

Two separate angles of 141° and 39° that do not touch; 141° + 39° = 180°, so they are supplementary.

Angle Bisectors

An angle bisector is a ray that splits an angle into two equal angles. If ray OM bisects angle LON, which measures 128 degrees, then angles LOM and MON are each 64 degrees. Working backward, if each half is given, double it to get the whole angle. Bisector problems often combine with linear pairs or algebra.

Ray OM bisects angle LON of 128° into angles LOM and MON of 64° each.

Parallel Lines and Transversals

A transversal is a line that crosses two or more other lines at different points. When it crosses two lines, it forms eight angles, four at each intersection. If the two lines are parallel, meaning they lie in the same plane and never meet, those eight angles follow strict patterns described in the next sections. If the lines are not parallel, the angles at each intersection are still linked by vertical angles and linear pairs, but the two intersections are not tied to each other. Diagrams mark parallel lines with matching arrowheads.

Labeling the Eight Transversal Angles

To keep track of the eight angles, number them. At the upper intersection, call the angles 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right). At the lower intersection, use 5, 6, 7 and 8 in the same positions. Angles 3, 4, 5 and 6 lie between the two lines and are interior angles. Angles 1, 2, 7 and 8 lie outside the lines and are exterior angles. Angles in matching positions, like 1 and 5, are corresponding angles.

Two parallel lines cut by a transversal, with angles 1 (top left), 2 (top right), 3 (bottom left), 4 (bottom right) at the upper crossing and 5 to 8 in the same positions at the lower crossing; the shaded band between the lines is the interior, outside is exterior.

Corresponding Angles

Corresponding angles occupy the same relative position at each intersection, such as angles 2 and 6 in the numbering above. When the lines are parallel, corresponding angles are equal. Picture sliding the upper intersection down the transversal until it sits on the lower one: every angle lands on its match. If angle 2 is 71 degrees, angle 6 is 71 degrees.

Parallel lines cut by a transversal with corresponding angles 2 and 6 highlighted: angle 2 = angle 6 = 71°.

Alternate Interior Angles

Alternate interior angles lie between the two lines and on opposite sides of the transversal, such as angles 3 and 6 or angles 4 and 5. When the lines are parallel, alternate interior angles are equal. A quick way to spot them is the Z shape, or backward Z, traced by the lines. If angle 4 measures 108 degrees, then angle 5 also measures 108 degrees.

Parallel lines cut by a transversal with alternate interior angles 4 and 5 highlighted inside a Z shape: angle 4 = angle 5 = 108°.

Alternate Exterior Angles

Alternate exterior angles lie outside the two lines and on opposite sides of the transversal, such as angles 1 and 8 or angles 2 and 7. When the lines are parallel, alternate exterior angles are equal. So if angle 1 measures 57 degrees, angle 8 also measures 57 degrees. The reason: each one equals a vertical angle that is corresponding to the other.

Parallel lines cut by a transversal with alternate exterior angles 1 and 8 highlighted: angle 1 = angle 8 = 57°.

Same-Side Interior Angles

Same-side interior angles, also called consecutive interior angles, lie between the two lines and on the same side of the transversal, such as angles 3 and 5 or angles 4 and 6. When the lines are parallel, these angles are supplementary, adding to 180 degrees; they are not equal. If angle 3 is 113 degrees, angle 5 is 67 degrees. Same-side exterior angles, such as 1 and 7, are also supplementary.

Parallel lines cut by a transversal with same-side interior angles 3 and 5 highlighted: 113° + 67° = 180°.

Every Transversal Angle Is Either Acute or Obtuse in the Nonperpendicular Case

When a transversal crosses two parallel lines and is not perpendicular to them, the eight angles come in only two sizes: four equal acute angles and four equal obtuse angles. Any acute angle plus any obtuse angle equals 180 degrees, so once you know one angle, you know all eight. If one angle is 49 degrees, every other angle is either 49 or 131 degrees. If the transversal is perpendicular, all eight angles are 90 degrees.

Parallel lines cut by a transversal at 49°: the four acute angles are 49° and the four obtuse angles are 131°, and 49° + 131° = 180°.

Converse Tests for Parallel Lines

The parallel-line facts also work in reverse. Two lines cut by a transversal are parallel if any one of these is true: a pair of corresponding angles is equal, a pair of alternate interior angles is equal, a pair of alternate exterior angles is equal, or a pair of same-side interior angles is supplementary. Facts that hold for every pair of crossing lines, such as equal vertical angles or a linear pair totaling 180 degrees, prove nothing about parallel lines.

Combining Vertical and Parallel-Line Relationships

Harder problems chain relationships together. Suppose angle 2 is 66 degrees. Its vertical angle, angle 3, is also 66 degrees. Angles 3 and 6 are alternate interior, so angle 6 is 66 degrees. Angle 5 forms a linear pair with angle 6, so angle 5 is 114 degrees. Name the relationship at each step, and you can travel from any angle to any other.

Numbered transversal figure showing the chain: angle 2 = 66° (given), angle 3 = 66° (vertical), angle 6 = 66° (alternate interior), angle 5 = 114° (linear pair).

Solving Algebraic Angle Equations

When angle measures are written as expressions, decide which relationship applies, then write the matching equation. Equal angles (vertical, corresponding, alternate) give one expression equal to the other. Complementary angles sum to 90; supplementary angles and linear pairs sum to 180. If vertical angles are (5x - 8) and (3x + 12) degrees, then

5x - 8 = 3x + 12

so x = 10 and each angle is 42 degrees. Finish by finding the quantity asked.

Several Adjacent Angles on a Line

When several adjacent angles fill a straight line, their measures total 180 degrees. If angles of 47, 68 and x degrees sit side by side along a line, then

x = 180 - 47 - 68 = 65 degrees

The same idea works when the angles are given as algebraic expressions.

Three adjacent angles on a straight line: 47°, 68° and x = 65°, with 47° + 68° + x = 180°.

Angles Around a Point

Angles that completely surround a point without overlapping add up to 360 degrees. If five angles around a point measure 70, 85, 60, 90 and x degrees, then

x = 360 - 305 = 55 degrees

Before using 360, make sure the angles neither overlap nor leave a gap.

Five angles around a point measuring 70°, 85°, 60°, 90° and x = 55°, filling the full 360°.

Right-Angle and Parallel Constraints in Drawings

Diagram markings carry information that the drawing alone does not. A small square means 90 degrees, matching arrowheads mean parallel lines, and matching tick marks or arcs mean equal segments or angles. Do not assume two lines are parallel or perpendicular just because they look that way; test figures may not be drawn to scale. Use only the facts that are stated or marked.

Diagram markings: a small square means 90°, matching arrowheads mean parallel lines, matching tick marks mean equal segments, matching arcs mean equal angles.

Geometry Model: Identify → Pair → Equation → Check

Use a four-step routine. Identify: find the angles named in the question and mark what is known. Pair: name the relationship that connects them, such as vertical, linear pair, corresponding, or same-side interior. Equation: for equal-measure relationships, set one expression equal to the other; for sum relationships, set the total to 90, 180 or 360. Check: substitute your value back, confirm each angle is sensibly acute or obtuse, and answer the exact quantity asked.

Mental Arithmetic and Answer-Choice Checks

Subtract from 90 or 180 in friendly steps: 180 - 137 is 43 because

137 + 3 = 140
and
140 + 40 = 180

Estimate before solving: if an angle is clearly obtuse, discard acute answer choices. On algebra items you can plug each choice for x into both expressions and keep the one that fits the relationship. If the question asks for an angle rather than x, finish that last step.

Worked Example 1 — Intersecting Lines

Problem: Lines PQ and RS cross at point T, and angle PTR measures 57°. Find the other three angles.

Lines PQ and RS cross at T; the given angle PTR = 57° is marked, and angles QTS, PTS and QTR are marked with question marks.

Step 1: Find angle QTS. It sits directly across from angle PTR, so the two are vertical angles. Vertical angles are always equal.

Step 2: Find angle PTS. It sits next to angle PTR along line RS, so together they make a straight line (a linear pair). A straight line measures 180°, so subtract.

Step 3: Find angle QTR. It is across from angle PTS, so it is a vertical angle and equals it.

Answer: Angle QTS = 57°, angle PTS = 123°, and angle QTR = 123°.

Solved: lines PQ and RS cross at T with angle PTR = 57° and its vertical angle QTS = 57°; angles PTS and QTR are 123° each.

Check: All four angles around point T must add to 360°.

Worked Example 2 — Complementary Expressions

Problem: Two complementary angles measure (4x + 5)° and (x + 10)°. Find x and both angles.

Step 1: Complementary angles add up to 90°, so write an equation.

Step 2: Combine like terms. The x-terms give

4x + x = 5x

and the numbers give

5 + 10 = 15

Step 3: Subtract 15 from both sides to get the x-term alone.

Step 4: Divide both sides by 5.

Step 5: Substitute x = 15 back into each expression.

Answer: x = 15, and the angles are 65° and 25°.

Check:

65 + 25 = 90

so the angles really are complementary.

Worked Example 3 — Corresponding Angles

Problem: Two parallel lines are cut by a transversal. A pair of corresponding angles measures (6x − 10)° and (4x + 18)°. Find x and the angle measure.

Two parallel lines cut by a transversal; the corresponding angles at the top right of each crossing are labelled (6x − 10)° and (4x + 18)°.

Step 1: When lines are parallel, corresponding angles are equal. Set the expressions equal.

Step 2: Subtract 4x from both sides so x appears on one side only.

Step 3: Add 10 to both sides.

Step 4: Divide both sides by 2.

Step 5: Substitute x = 14 into either expression.

Answer: x = 14, and each angle measures 74°.

Check: Use the other expression:

4(14) + 18 = 56 + 18 = 74

Both give 74°, so they match.

Worked Example 4 — Same-Side Interior Angles

Problem: Two parallel lines are cut by a transversal. Same-side interior angles measure (3x + 22)° and (2x + 8)°. Find both angles.

Two parallel lines cut by a transversal; same-side interior angles on the right of the transversal are labelled (3x + 22)° and (2x + 8)°.

Step 1: Same-side interior angles between parallel lines are supplementary, which means they add to 180°.

Step 2: Combine like terms:

3x + 2x = 5x
and
22 + 8 = 30

Step 3: Subtract 30 from both sides.

Step 4: Divide both sides by 5.

Step 5: Substitute x = 30 into each expression.

Answer: The angles are 112° and 68°.

Check:

112 + 68 = 180

so they are supplementary as required.

Worked Example 5 — Three Angles on One Line

Problem: Three angles sit side by side along a straight line. They measure 2x°, (x + 30)° and (3x − 6)°. Find each angle.

Three adjacent angles along a straight line, 2x°, (x + 30)° and (3x − 6)°, together making 180°.

Step 1: Angles that together form a straight line add to 180°.

Step 2: Combine like terms. The x-terms:

2x + x + 3x = 6x

The numbers:

30 − 6 = 24

Step 3: Subtract 24 from both sides.

Step 4: Divide both sides by 6.

Step 5: Substitute x = 26 into each expression.

Answer: The angles are 52°, 56° and 72°.

Check:

52 + 56 = 108
and
108 + 72 = 180

They fill the straight line exactly.

Worked Example 6 — Bisector and Supplement

Problem: Ray OK bisects angle JOL, and each half measures (x + 9)°. Angle JOL forms a linear pair with an angle of (2x + 42)°. Find angle JOL.

Ray OK bisects angle JOL into two equal halves marked (x + 9)° each; angle JOL forms a linear pair with an angle of (2x + 42)° along the straight line; angle JOL = ?.

Step 1: Bisect means cut into two equal halves. So the whole angle JOL is two halves.

Step 2: A linear pair forms a straight line, so the two angles add to 180°.

Step 3: Combine like terms:

2x + 2x = 4x
and
18 + 42 = 60

Step 4: Subtract 60 from both sides, then divide by 4.

Step 5: Substitute x = 30 into the expression for angle JOL.

Answer: Angle JOL measures 78°.

Check: The partner angle is

2(30) + 42 = 60 + 42 = 102

°, and

78 + 102 = 180

Each half is

30 + 9 = 39

°, and

39 + 39 = 78

Worked Example 7 — Vertical Angles Algebra

Problem: Two lines intersect. A pair of vertical angles measures (7x − 15)° and (4x + 18)°. Find all four angles.

Two intersecting lines; one pair of vertical angles is labelled (7x − 15)° and (4x + 18)°, and the other two angles are marked with question marks.

Step 1: Vertical angles are equal, so set the expressions equal.

Step 2: Subtract 4x from both sides.

Step 3: Add 15 to both sides, then divide by 3.

Step 4: Substitute x = 11 to find the vertical angles.

Step 5: Each neighboring angle forms a straight line with a 62° angle, so subtract from 180°.

Answer: The vertical angles are 62° each, and the other two angles are 118° each.

Check:

4(11) + 18 = 44 + 18 = 62

which matches. Also

62 + 118 + 62 + 118 = 360

Worked Example 8 — Point Rotation with Four Regions

Problem: Four angles surround a point without overlapping. They measure 2x°, 3x°, 95° and 115°. Find the two unknown angles.

Four angles around a point, labelled 2x°, 3x°, 95° and 115°, filling a full 360°.

Step 1: Angles all the way around a point add to 360°.

Step 2: Combine like terms:

2x + 3x = 5x
and
95 + 115 = 210

Step 3: Subtract 210 from both sides, then divide by 5.

Step 4: Substitute x = 30.

Answer: The unknown angles are 60° and 90°.

Check:

60 + 90 = 150
and
95 + 115 = 210
then
150 + 210 = 360

Frequent ASVAB-Style Angle Mistakes

Mixing up complement and supplement: subtracting from 180 when the question says complementary gives an answer 90 degrees too large, so tie complementary to 90 and supplementary to 180. Assuming same-side interior angles are equal: between parallel lines they add to 180 instead. Stopping at x: many items ask for an angle, and the value of x is often planted among the choices. Trusting appearance: lines are parallel only when stated or marked. Citing a fact that is always true, such as equal vertical angles, as proof of parallel lines. Forgetting to double a bisected half when the whole angle is asked.

Rapid Review and Final Checklist

Acute angles are under 90 degrees, right angles are 90, obtuse angles fall between 90 and 180, and straight angles are 180. Complementary angles total 90; supplementary angles, linear pairs and angles along a line total 180; angles around a point total 360. Vertical angles are equal, and a bisector creates two equal halves. With parallel lines and a transversal, corresponding, alternate interior and alternate exterior angles are equal, while same-side interior angles are supplementary. Identify the relationship, write one equation, solve, and answer the exact quantity asked.

Quick review

How are angles classified by measure?
Acute: more than 0° and less than 90°. Right: exactly 90°. Obtuse: between 90° and 180°. Straight: exactly 180°. Reflex: between 180° and 360°.
Complementary vs. supplementary angles?
Complementary angles total 90°; supplementary angles total 180°. The complement of 34° is 56°, and its supplement is 146°. Memory aid: C comes before S, as 90 comes before 180.
Two lines cross: how are the four angles related?
Opposite (vertical) angles are equal, and side-by-side angles form linear pairs that total 180°. If one angle is 47°, the four angles are 47°, 133°, 47° and 133°.
Angle totals along a straight line and around a point?
Adjacent angles filling a straight line total 180°; non-overlapping angles that surround a point total 360°. Angles of 100°, 75°, 120° and x around a point give x = 65°.
Parallel lines cut by a transversal: which angle pairs are equal?
Corresponding angles, alternate interior angles and alternate exterior angles are all equal. If one angle of a corresponding pair is 64°, its partner is also 64°.
Are same-side interior angles equal when the lines are parallel?
No. Same-side (consecutive) interior angles are supplementary, adding to 180°. If one is 105°, the other is 75°.
Parallel lines and a slanted transversal: how many angle sizes appear?
Only two: four equal acute angles and four equal obtuse angles, with acute + obtuse = 180°. If one angle is 52°, every angle is either 52° or 128°.
Can you assume lines are parallel or perpendicular from a drawing?
No; figures may not be to scale. Trust only stated facts and markings: a small square for 90°, matching arrows for parallel lines, tick marks for equal parts. Equal vertical angles never prove lines parallel.
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About this lesson

A large section covers parallel lines cut by a transversal: labeling the eight angles, corresponding, alternate interior, alternate exterior and same-side interior angles, and the converse tests that prove lines parallel. You will then combine relationships, solve algebraic angle equations, handle several angles on a line or around a point, and follow a four-step model of identify, pair, equation and check.

Eight worked examples and a list of frequent ASVAB-style angle mistakes reinforce the skills. Because MK is an AFQT subtest, geometry points matter. The lesson takes about 40 minutes, and a 20-question lesson test follows.

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