Quarry School

Angle sizes and two ways to fill a corner

Explain it like I am five

Picture opening a laptop. Barely open is an acute angle: less than 90° (90 degrees, a square corner). Screen straight up is a right angle: exactly 90°. Leaning back past that, but not flat, is obtuse: between 90° and 180°. Lying flat open is a straight angle: exactly 180°.

Two positive angles that fill a square corner together are complementary: they add to 90°. Two that fill a straight line are supplementary: they add to 180°. Example with 27°: the complement is 90° − 27° = 63°, and the supplement is 180° − 27° = 153°. Check by adding back: 27° + 63° = 90° and 27° + 153° = 180°.

Why subtraction works: the missing piece β (beta) makes 27° + β = 90°. Take 27° away from both sides of the equals sign and β = 63° is left. Now try 124°: 90° − 124° = −34°, a negative leftover, so 124° has no complement; it is already wider than a square corner. Its supplement is 180° − 124° = 56°.

In plain words

Picture opening a door from its closed position. A small opening narrower than a square corner is an Acute angle. Exactly a square corner is a Right angle. A wider opening that has not reached a straight line is an Obtuse angle. Exactly a half turn is a Straight angle. Now picture splitting one square corner into two smaller openings. Those two positive angles are Complementary angles because they fill 90° together. Split a straight corner into two positive openings and they are Supplementary angles, filling 180°. The words describe how much room the angles occupy, so the shared total tells you how to find a missing piece.

32°58°sum = 90°
The missing opening is the amount of the 90° corner left after using 32°.
Reminder
  • Solving for a missing addend. In 55° + β = 90°, subtract 55° from both sides to leave the missing β alone: β = 35°. Adding 55° back verifies the total.
  • Strict boundaries. 0° < θ < 90° excludes both 0° and 90°; a boundary angle receives a different classification.
Why it works. If two positive pieces fill a right-angle opening, their sizes total 90°. In α + β = 90°, subtract α from both sides: α + β − α = 90° − α. The known piece cancels on the left, leaving β = 90° − α. This finds the missing piece. Add α back to check the total. Replacing 90° with 180° gives the supplement. The endpoints matter: exactly 90° is right, not acute or obtuse. A positive complement exists only for 0° < α < 90°.
RuleAcute: 0° < θ < 90°. Right: θ = 90°. Obtuse: 90° < θ < 180°. Straight: θ = 180°. Positive complements sum to 90°; positive supplements sum to 180°.
The same idea, five ways
Say it

Say acute, right, obtuse, or straight for the opening; say complement to fill 90 degrees and supplement to fill 180 degrees.

Write it

Complementary angles are two positive angles totaling 90°; supplementary angles are two positive angles totaling 180°.

In math
  • acute: 0° < θ < 90°
  • right: θ = 90°
  • obtuse: 90° < θ < 180°
  • straight: θ = 180°
  • complement: β = 90° − α, when 0° < α < 90°
  • supplement: β = 180° − α, when 0° < α < 180°
Like

Two fitted pieces fill a square corner or a straight corner; the missing piece is what remains after removing the known one.

See it
104°76°sum = 180°
The two positive pieces fill a straight angle totaling 180°.
The same idea, other ways
Compare with a square corner

Open a door from closed. Before a square corner, its positive opening is acute. At the corner, it is right. Beyond the corner but before a straight line, it is obtuse. At the straight line, it is straight. Exact boundary values receive their own names.

104° is obtuse
This opening has passed 90° but has not reached 180°.
Find the unfilled piece

A complement fills the unused part of a 90° corner. A supplement fills the unused part of a 180° corner. Subtract the piece you already have from the whole corner, then add the pieces back together to check.

55°35°sum = 90°
After a 55° piece, 35° remains to complete 90°.
Read the drawn table

Read a type of angle or pair in the top row, then its size condition and example below. The column under Acute angle excludes 0° and 90°; the column under Complementary angles requires two positive pieces totaling 90°.

input Nameoutput Size or total; A concrete instanceAcute angleSize or total: 0° < θ < 90° A concrete instance: 32°Right angleSize or total: 90° A concrete instance: A square cornerObtuse angleSize or total: 90° < θ < 180° A concrete instance: 113°Straight angleSize or total: 180° A concrete instance: Half a revolutionComplementary anglesSize or total: α + β = 90°, α > 0°, β > 0° A concrete instance: 25° and 65°Supplementary anglesSize or total: α + β = 180°, α > 0°, β > 0° A concrete instance: 110° and 70°
Choose a column in the top row, then read the matching information directly below it.
NameSize or totalA concrete instance
Acute angle0° < θ < 90°32°
Right angle90°A square corner
Obtuse angle90° < θ < 180°113°
Straight angle180°Half a revolution
Complementary anglesα + β = 90°, α > 0°, β > 0°25° and 65°
Supplementary anglesα + β = 180°, α > 0°, β > 0°110° and 70°
.1Right angle

A Right angle is the corner of a square sheet of paper. Its turn is exactly one quarter of a full revolution. The small square drawn in a corner marks this exact size, even if the drawing is tilted.

  • θ = 90°.
  • Four right-angle turns make 360°.
90°
The little square marks an exact quarter turn.
Reminder
  • Quarter turn. 360° ÷ 4 = 90°, so four right-angle turns make a full revolution.
Worked exampleFour square corners

You are adding four quarter-turn sizes to find their total. What is the total of four right angles?

4 of 4 wedges, each 90°
Four quarter turns fill all four wedges of a 360° revolution.
  1. 4 × 90° = 360°.Each right angle is a quarter of a revolution.
Answer
360°.
Check Four quarters make one complete turn.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Any angle drawn with one side vertical is a right angle.
An angle's size depends on the opening between both rays. One ray's direction alone does not determine that opening.
✓ Instead: A right angle has a 90° opening, often marked with a small square.
Tips and tricks
  • Use the square mark or the stated 90° value, rather than estimating from a tilted drawing.
.2Straight angle

A Straight angle is like turning around to face exactly behind you. Its two rays make one straight line through their common vertex. It is twice a right angle and half a full revolution.

  • θ = 180°.
  • The two sides point in opposite directions.
180°
The ending ray points directly opposite the starting ray.
Reminder
  • Half of a number. Half of 360° is 360° ÷ 2 = 180°.
Worked exampleHalf a revolution

You are finding the size of one of two equal pieces of a full turn. How many degrees are in half a full revolution?

180°
The ending ray points directly opposite the starting ray.
  1. 360° ÷ 2 = 180°.Half means divide the full turn into two equal parts.
Answer
180°, a straight angle.
Check 90° + 90° = 180°, so two right-angle turns make the same half turn.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A full turn of 360° is a straight angle.
A full turn returns to the starting direction. A straight angle points in the opposite direction.
✓ Instead: A straight angle is 180°, one half of a revolution.
Tips and tricks
  • A straight angle's two rays lie on one line and point opposite ways.
.3Acute angle

An Acute angle is a positive opening smaller than the corner of a square. Think of the door only partly opened. The definition excludes both a completely closed opening and the full square corner.

  • 0° < θ < 90°.
  • 0° and 90° are excluded.
32°
The opening is positive but smaller than a quarter turn.
Reminder
  • Strict bounds. The symbol < excludes equality. θ < 90° does not include θ = 90°.
Worked exampleAn opening below 90°

You are choosing the size name that fits the stated opening. Classify 32°.

32°
The opening is positive but smaller than a quarter turn.
  1. 32° is acute.0° < 32° < 90°, so it meets both strict boundaries.
Answer
Acute angle.
Check 90° − 32° = 58° remains before reaching a square corner.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 0° is acute because it is smaller than 90°.
Acute angles must satisfy both strict bounds. Zero has no positive opening.
✓ Instead: An acute angle obeys 0° < θ < 90°; 0° and 90° are excluded.
Tips and tricks
  • Check greater than zero and less than a right angle, in that order.
.4Obtuse angle

An Obtuse angle is a positive opening wider than a square corner but narrower than a straight line. Imagine a door swung past its right-angle opening without reaching a half turn.

  • 90° < θ < 180°.
  • 90° and 180° are excluded.
113°
The opening has passed a quarter turn but has not reached a half turn.
Reminder
  • Two required inequalities. 113° is obtuse because 90° < 113° and 113° < 180° are both true.
Worked exampleAn opening beyond 90°

You are choosing the size name that fits the stated opening. Classify 113°.

113°
The opening has passed a quarter turn but has not reached a half turn.
  1. 113° is obtuse.90° < 113° < 180°, so it lies strictly between the two boundary sizes.
Answer
Obtuse angle.
Check 113° is 23° past a square corner and 67° short of a straight angle.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 180° is obtuse because it is wider than 90°.
The definition also requires an opening smaller than 180°.
✓ Instead: 90° < θ < 180° is obtuse; exactly 180° is straight.
Tips and tricks
  • An obtuse angle sits between a square corner and a straight opening, excluding both.
.5Complementary angles

Complementary angles are two positive pieces that fill a square corner. Picture two slices fitted inside one quarter of a pizza. The slices need not be equal, and they do not need to sit next to each other in the problem's drawing. Their sizes must add to 90°.

  • α + β = 90° with α > 0° and β > 0°.
  • The complement of α is 90° − α when 0° < α < 90°.
25°65°sum = 90°
25° and 65° fill the one right-angle corner.
Reminder
  • Finding a missing part. If 25° + β = 90°, subtract 25° from both sides to find the remaining β = 65°.
Worked exampleFill the square corner

You are finding the positive opening still needed to complete 90°. Find the complement of 25°.

25°65°sum = 90°
25° and 65° fill the one right-angle corner.
  1. 90° − 25° = 65°.25° has used that much of the total 90° opening.
Answer
65°.
Check 25° + 65° = 90°, with both pieces positive.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 90° and 0° are complementary because their sum is 90°.
Both members of a complementary pair must be positive, and 0° fails that condition.
✓ Instead: 25° and 65° are complementary because both are positive and 25° + 65° = 90°.
Tips and tricks
  • Complement fills a square corner. After subtracting from 90°, check that both angles are positive.
.6Supplementary angles

Supplementary angles are two positive pieces that fill a straight corner. Picture two slices filling half a pizza. One piece may be acute and the other obtuse, or both may be right angles. The total, rather than how they look separately, determines the name.

  • α + β = 180° with α > 0° and β > 0°.
  • The supplement of α is 180° − α when 0° < α < 180°.
110°70°sum = 180°
The two positive openings fill a straight angle.
Reminder
  • Checking by addition. To check a missing partner found by subtraction, add it to the original angle: 110° + 70° = 180°.
Worked exampleFill the straight corner

You are finding the positive opening still needed to complete 180°. Find the supplement of 110°.

110°70°sum = 180°
The two positive openings fill a straight angle.
  1. 180° − 110° = 70°.A supplement is the positive part remaining in a straight angle.
Answer
70°.
Check 110° + 70° = 180°.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: 110° has supplement 20° because 110 − 90 = 20.
A supplement fills 180°, whereas 90° is the total used for complements.
✓ Instead: The supplement is 180° − 110° = 70°, and 110° + 70° = 180°.
Tips and tricks
  • Supplement fills a straight opening. Write the total 180° before subtracting.
Strategy: step by step
  1. Compare the angle to 0°, 90° and 180° to classify its size.
  2. For a complement, subtract from 90° and require the result to be positive.
  3. For a supplement, subtract from 180° and require the result to be positive.
  4. Add the proposed pair to check the total. Do not use a negative leftover as a positive complement.
  5. In the table picture, read a column under Acute angle, then compare the matching information below it. Each column preserves one complete row of the reference table.
Strategy
Classify the opening and complete its corner
1
Is the requested partner a complement?
YesUse 90° − α and require 0° < α < 90°.
NoFor a supplement use 180° − α and require 0° < α < 180°.
↓
2
Is the leftover piece positive?
YesAdd it to the original angle and check the requested total.
NoState that no positive partner of the requested kind exists.
  1. Compare the angle with 0°, 90°, and 180°; equality identifies a right or straight angle.
  2. If a complement is requested, subtract from 90° to find the unused piece.
  3. If a supplement is requested, subtract from 180° to find the unused piece.
  4. Reject a zero or negative piece when the notes require a positive pair.
  5. Add the original angle and the proposed missing piece to verify 90° or 180°.
Worked exampleA small angle and an angle too large for a complement

You are comparing openings and finding the positive pieces needed to complete 90° and 180° where possible. Find the complement and supplement of 32°. Then decide whether 113° has a positive complement and find its supplement.

32°58°sum = 90°
32° and its positive complement fill the right-angle corner.
113°67°sum = 180°
The obtuse 113° opening has a positive supplement of 67°.
  1. For 32°, complement = 90° − 32° = 58°.Complementary angles fill a right angle together.
  2. For 32°, supplement = 180° − 32° = 148°.Supplementary angles fill a straight angle together.
  3. For 113°, 90° − 113° = −23°, so there is no positive complement.113° already exceeds the whole square corner; the leftover would violate the positivity requirement.
  4. For 113°, supplement = 180° − 113° = 67°.67° is a positive amount completing the straight angle.
Answer
  • 32° complement: 58°.
  • 32° supplement: 148°.
  • 113° complement: no positive complement.
  • 113° supplement: 67°.
Check 32° + 58° = 90°, 32° + 148° = 180°, and 113° + 67° = 180°. The invalid complement −23° fails the required positive-angle condition.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The complement of 104° is −14°.
The subtraction is arithmetically correct, but complementary angles in these notes must both be positive. An angle already larger than 90° has no positive leftover in a 90° corner.
✓ Instead: 104° has no positive complement. Its positive supplement is 180° − 104° = 76°.
✗ Not this: An angle measuring exactly 90° is acute.
The acute definition uses a strict upper boundary and excludes 90°.
✓ Instead: Exactly 90° is right; an acute angle must be greater than 0° and smaller than 90°.
Tips and tricks
  • Memory cue: C comes before S, and Complement fills the smaller 90° total before Supplement fills the larger 180° total.
  • After subtracting, check that both pieces are positive and their sum is the requested corner.
Trap. Interchanging complement and supplement. Remember C comes before S: Complement fills the smaller 90° corner; Supplement fills the larger 180° corner.
Keep in mind
  • Complements need both pieces positive, so only angles between 0° and 90° have one: 89° has complement 1°, while 90° has none.
  • The edges are exact: 90° is a right angle, not acute or obtuse, and 180° is straight, not obtuse.
  • Both sides means the left-hand side (LHS) and the right-hand side (RHS) of the equals sign: taking 27° from both sides of 27° + β = 90° leaves β = 63°.
  • Complementary angles do not need to touch: 25° and 65° drawn on different pages are still complementary, since 25° + 65° = 90°.
Memory hookC comes before S, and 90 comes before 180: Complement makes a Corner (90°), Supplement makes a Straight line (180°).
Flash cards: say the answer out loud, then flip
What are complementary angles?
Two positive angles that add to 90°.
What are supplementary angles?
Two positive angles that add to 180°.
What is an acute angle?
An angle between 0° and 90°: 0° < θ < 90°.
Classify 91°.
Obtuse: between 90° and 180°.
Find the complement and supplement of 38°.
  • Complement: 52°
  • Supplement: 142°
Find the complement of 143°.
  • None, since 90° − 143° = −53° is not positive
  • its supplement is 37°.