Complementary and Supplementary Angles
Pointwise AISSEE Class 6 Mathematics guide for Complementary and Supplementary Angles (Topic 13). Master 90° vs 180° definitions, finding unknown complements/supplements, and algebraic angle modeling.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Complementary and Supplementary Angles (Topic 13). Master 90° vs 180° definitions, finding unknown complements/supplements, and algebraic angle modeling.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 13 | Definition and pairs of complementary angles, supplementary angles, finding unknown complement or supplement. | Master complementary (90°) and supplementary (180°) angle properties, algebraic linear setups, and ratio division of angle pairs for AISSEE Class VI. |
1. Complementary Angles (Sum = 90°)
- Definition of Complementary Angles: Two angles are called Complementary Angles if the sum of their degree measures equals $90^\circ$ (a right angle): $$\angle A + \angle B = 90^\circ$$
- Complement of an Angle: Each angle in a complementary pair is called the complement of the other: $$\text{Complement of } \theta = 90^\circ - \theta$$ Example: The complement of $35^\circ$ is $90^\circ - 35^\circ = 55^\circ$.
- Equal Complement Angle: An angle that is equal to its own complement is $45^\circ$ ($45^\circ + 45^\circ = 90^\circ$).
- Acute Angle Constraint: Since their sum is $90^\circ$, both angles in a complementary pair must strictly be acute angles ($< 90^\circ$). An obtuse angle can never have a complement.
| Given Angle ($\theta$) | Its Complement ($90^\circ - \theta$) | Verification |
|---|---|---|
| $20^\circ$ | $90^\circ - 20^\circ = 70^\circ$ | $20^\circ + 70^\circ = 90^\circ$ |
| $45^\circ$ | $90^\circ - 45^\circ = 45^\circ$ | $45^\circ + 45^\circ = 90^\circ$ (Equal to complement) |
| $63^\circ$ | $90^\circ - 63^\circ = 27^\circ$ | $63^\circ + 27^\circ = 90^\circ$ |
| $1^\circ$ | $90^\circ - 1^\circ = 89^\circ$ | $1^\circ + 89^\circ = 90^\circ$ |
2. Supplementary Angles (Sum = 180°)
- Definition of Supplementary Angles: Two angles are called Supplementary Angles if the sum of their degree measures equals $180^\circ$ (a straight angle): $$\angle A + \angle B = 180^\circ$$
- Supplement of an Angle: Each angle in a supplementary pair is called the supplement of the other: $$\text{Supplement of } \theta = 180^\circ - \theta$$ Example: The supplement of $75^\circ$ is $180^\circ - 75^\circ = 105^\circ$.
- Equal Supplement Angle: An angle that is equal to its own supplement is $90^\circ$ (a right angle, $90^\circ + 90^\circ = 180^\circ$).
- Angle Types in Supplementary Pair:
• Two right angles are always supplementary ($90^\circ + 90^\circ = 180^\circ$).
• If one angle is acute ($< 90^\circ$), its supplement must be obtuse ($> 90^\circ$).
• Two acute angles can never be supplementary (max sum $< 180^\circ$).
• Two obtuse angles can never be supplementary (sum $> 180^\circ$).
Cadet Mnemonic • Alphabetical Degree Anchor
Remember the degree values by alphabetical order:
- C comes before S (C < S)
- 90° comes before 180° (90° < 180°)
- Therefore: Complementary = 90°; Supplementary = 180°
AISSEE Trap Alert • The Complement vs Supplement Mix-up
Always double-check question phrasing! 'Find the supplement of 40°' is 180° − 40° = 140°. Writing 90° − 40° = 50° loses 3 marks immediately.
3. Algebraic Modeling & Ratio Division in Angle Pairs
- Angle with Difference / Excess: If an angle exceeds its complement by $d^\circ$: $$\theta = \frac{90 + d}{2}$$ Example: Exceeds complement by $20^\circ \implies \theta = \frac{90 + 20}{2} = 55^\circ$.
- Ratio of Supplementary Angles: If two supplementary angles are in the ratio $a : b$: $$\text{First Angle} = \frac{a}{a + b} \times 180^\circ, \quad \text{Second Angle} = \frac{b}{a + b} \times 180^\circ$$
- Ratio of Complementary Angles: If two complementary angles are in ratio $a : b$: $$\text{First Angle} = \frac{a}{a + b} \times 90^\circ, \quad \text{Second Angle} = \frac{b}{a + b} \times 90^\circ$$
Exam Trick • Ratio Angle One-Part Multiplier
For supplementary ratio 2:3, sum of parts = 5. Value of 1 part = 180° ÷ 5 = 36°. Angles are 2 × 36 = 72° and 3 × 36 = 108° in seconds!
Visual Concept Blueprint • Vector Architecture
Speed Shortcuts & Mental Calculation Strategies
Master these high-yield time-saving heuristics curated by top AISSEE scorers to eliminate calculation bottlenecks and answer questions in under 45 seconds.
Shortcut Technique 1: Supplement Minus Complement Law
For any angle $\theta$, the difference between its supplement and its complement is ALWAYS equal to $90^\circ$: $(180 - \theta) - (90 - \theta) = 90^\circ$.
Shortcut Technique 2: Multiple Angle Equating
If an angle is $k$ times its complement: $\theta = \frac{k}{k + 1} \times 90^\circ$. If 4 times complement: $\frac{4}{5} \times 90 = 72^\circ$.
Shortcut Technique 3: Obtuse Complement Impossibility
Option elimination: If asked to find the complement of an angle, any option $\ge 90^\circ$ is immediately disqualified.
Interactive 3D Revision Flashcards Deck
Click any card to flip and reveal the key exam rule or topper secret.
REVISION FLASHCARD
What is the core rule & definition for Complementary and Supplementary Angles?
CADET MEMORY KEY
Complementary Rule: \angle A + \angle B = 90^\circ
REVISION FLASHCARD
Cadet Shortcut: How to solve Complementary and Supplementary Angles in under 30 seconds?
CADET MEMORY KEY
Fast MCQ Elimination: Check the last digit (unit digit) and estimate the order of magnitude to eliminate 2 distractors in under 10 seconds.
REVISION FLASHCARD
Exam Trap Alert: What pitfall do most candidates fall into?
CADET MEMORY KEY
Angle Equal to Complement: 45° is the unique angle that is equal to its own complement (90 - 45 = 45).
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Angle Equal to Supplement: 90° is the unique angle that is equal to its own supplement (180 - 90 = 90).
REVISION FLASHCARD
What is the difference between the supplement and complement of an angle of 35°?
CADET MEMORY KEY
Supplement = 180 − 35 = 145°. Complement = 90 − 35 = 55°. Difference = 145 − 55 = 90°. (The gap is always 90° for ANY angle!).
Common Traps & Examination Pitfalls
Avoid the Speed-Guess Trap: Under AISSEE pressure, candidates frequently misread units (e.g., mixing metres and centimetres, or per hour and per second) and confuse complementary ($90^\circ$) with supplementary ($180^\circ$) angles. Read every stem carefully and check units before selecting your OMR option.
Key Identities & Formulas
Complementary Angles
$$\angle A + \angle B = 90^\circ$$
Sum of two complementary angles.
Supplementary Angles
$$\angle A + \angle B = 180^\circ$$
Sum of two supplementary angles.
Supplement-Complement Constant Gap
$$\text{Supplement}(\theta) - \text{Complement}(\theta) = 90^\circ$$
Universal constant difference.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Find the measure of an angle which is $24^\circ$ more than its complement.
Detailed Solution:
Let the complement be $x$. The angle is $(x + 24^\circ)$.
Sum of complementary angles = $90^\circ$:
$x + (x + 24) = 90 \implies 2x = 66 \implies x = 33^\circ$.
Angle = $33^\circ + 24^\circ = 57^\circ$.
Solved Example 2
Model Problem
Two supplementary angles are in the ratio $7 : 11$. Find the measure of both angles.
Detailed Solution:
Sum of ratio parts = $7 + 11 = 18$.
Sum of supplementary angles = $180^\circ$.
Each part = $180^\circ / 18 = 10^\circ$.
Angles: $7 \times 10^\circ = 70^\circ$ and $11 \times 10^\circ = 110^\circ$.
Solved Example 3
Model Problem
An angle is $30^\circ$ more than one-half of its complement. Find the measure of the angle.
Detailed Solution:
1. Let the angle be $x$. Its complement is $(90 - x)$.
2. According to the condition: $x = \frac{1}{2}(90 - x) + 30$.
3. Multiply the entire equation by 2 to clear fractions: $2x = (90 - x) + 60$.
4. $2x = 150 - x \implies 3x = 150 \implies x = 50^\circ$.
5. Check: Complement of $50^\circ$ is $40^\circ$. Half of 40 is 20. $20 + 30 = 50^\circ$ (matches). Correct Option is (A).
2. According to the condition: $x = \frac{1}{2}(90 - x) + 30$.
3. Multiply the entire equation by 2 to clear fractions: $2x = (90 - x) + 60$.
4. $2x = 150 - x \implies 3x = 150 \implies x = 50^\circ$.
5. Check: Complement of $50^\circ$ is $40^\circ$. Half of 40 is 20. $20 + 30 = 50^\circ$ (matches). Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
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Question 1 of 10
Exam Standard
What is the complement of $38^\circ$?
Correct Answer: Option (A)
Complement = $90^\circ - 38^\circ = 52^\circ$.
Question 2 of 10
Exam Standard
What is the supplement of $115^\circ$?
Correct Answer: Option (A)
Supplement = $180^\circ - 115^\circ = 65^\circ$.
Question 3 of 10
Exam Standard
Find the angle which is equal to its own complement.
Correct Answer: Option (A)
$x + x = 90^\circ \implies 2x = 90^\circ \implies x = 45^\circ$.
Question 4 of 10
Exam Standard
Find the angle which is equal to its own supplement.
Correct Answer: Option (A)
$x + x = 180^\circ \implies 2x = 180^\circ \implies x = 90^\circ$.
Question 5 of 10
Exam Standard
If two complementary angles are in the ratio $2 : 3$, find the smaller angle.
Correct Answer: Option (A)
$2x + 3x = 90^\circ \implies 5x = 90^\circ \implies x = 18^\circ$. Smaller angle = $2 \times 18^\circ = 36^\circ$.
Question 6 of 10
Exam Standard
The difference between two supplementary angles is $40^\circ$. What is the measure of the smaller angle?
Correct Answer: Option (A)
Let angles be $x$ and $x + 40^\circ$. $x + (x + 40) = 180 \implies 2x = 140 \implies x = 70^\circ$.
Question 7 of 10
Exam Standard
An angle is double its complement. What is the angle?
Correct Answer: Option (A)
Angle = $2x$, complement = $x$. $2x + x = 90^\circ \implies 3x = 90^\circ \implies x = 30^\circ$. The angle is $2 \times 30^\circ = 60^\circ$.
Question 8 of 10
Exam Standard
Can two acute angles be supplementary?
Correct Answer: Option (A)
An acute angle is strictly less than $90^\circ$. The sum of two angles each $< 90^\circ$ must be $< 180^\circ$, so they can never be supplementary.
Question 9 of 10
Exam Standard
Can two obtuse angles be complementary?
Correct Answer: Option (A)
An obtuse angle is $> 90^\circ$. Their sum is already greater than $180^\circ$, so they can never be complementary (sum must equal $90^\circ$).
Question 10 of 10
Exam Standard
If an angle is $1/5$ of its supplement, what is the measure of the angle?
Correct Answer: Option (A)
Angle = $x$, supplement = $5x$. $x + 5x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ$.
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Last-Minute Cadet Revision Matrix
Rapid recall checkpoints to review 24 hours before stepping into the examination hall.
| Concept Checkpoint | High-Yield Exam Rule & Memory Anchor |
|---|---|
| Complement Sum | Sum of complementary angles is strictly 90°. |
| Supplement Sum | Sum of supplementary angles is strictly 180°. |
| Equal Angle Complements | Angle equal to its complement is 45°; Angle equal to its supplement is 90°. |
| Acute Constraint | Both complementary angles must be acute. Two obtuse angles cannot be supplementary. |
Frequently Asked Questions
How many questions from Complementary and Supplementary Angles appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Complementary and Supplementary Angles in the 50-question Mathematics paper, contributing 6 to 9 marks towards the 150-mark section total.
Is there any negative marking for incorrect answers in AISSEE Class VI?
No. The National Testing Agency (NTA) does not apply negative marking in the AISSEE Class VI entrance examination. Candidates should attempt every single question.
What is the best way to revise Complementary and Supplementary Angles before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.