Lines and Angles · Lesson 8 of 9
Classifying Angles and Solving Angle Problems
“Use precise angle ranges and reference turns to solve connected geometry problems.”
• Classify angles by their degree measures, including boundary values. • Distinguish the smaller and reflex openings between the same arms. • Use known right, straight, and full turns to find missing measures. • Construct reflex angles and figures with specified openings. • Solve equally spaced spoke problems and test the conditions of an angle puzzle.
Precise Boundaries for Angle Types
You already compared acute and obtuse openings with a right angle. Degree measures let us state their boundaries exactly. An acute angle is greater than 0° and less than 90°. An obtuse angle is greater than 90° and less than 180°. Exactly 90° is right, and exactly 180° is straight, so those boundary values do not belong to the neighbouring categories.
A larger turn can go beyond a half turn without completing a full turn. This is a reflex angle. For example, 230° is larger than 180° but smaller than 360°, so it is reflex. A full 360° turn returns to the starting direction; it is not included in the reflex range.
An angle greater than 180° and less than 360°. It represents a turn larger than a half turn but smaller than a full turn.
| Type | Degree measure | Turn comparison |
|---|---|---|
| Acute | Greater than 0° and less than 90° | Less than a quarter turn |
| Right | Exactly 90° | One quarter turn |
| Obtuse | Greater than 90° and less than 180° | Between quarter and half turns |
| Straight | Exactly 180° | One half turn |
| Reflex | Greater than 180° and less than 360° | Between half and full turns |
Problem
Classify 35°, 90°, 140°, 180°, 195°, and 360°.
- 1.35° lies above 0° and below 90°, so it is acute.
- 2.90° is exactly right. 140° lies between 90° and 180°, so it is obtuse.
- 3.180° is exactly straight. 195° lies between 180° and 360°, so it is reflex.
- 4.360° is a full turn. It is not reflex because the reflex range stops before 360°.
Do not call 90° acute or obtuse, or 180° obtuse or reflex. The words “less than” and “greater than” exclude the boundary itself.
The Same Arms Can Indicate Different Turns
Two ray directions can bound a small opening on one side and a larger sweep around the other side. A curve tells the reader which opening is intended. For arms 100° apart through the smaller opening, the remaining sweep is 260°. The arms and vertex are the same, but the specified amount of turn differs.
The two turn measures together complete a full turn. Therefore the larger one is the full turn minus the smaller opening. Naming letters in reverse does not, by itself, decide that one name is small and the other reflex. Follow the curve or explicit words such as “reflex angle”.
Problem
A reflex angle is indicated by a large curve. The other opening between its arms measures 100°. Find the reflex measure.
- 1.Identify the two openings around the vertex: the measured small opening and the indicated large opening.
- 2.Their turns fill 360° together.
- 3.Subtract the measured part: 360° − 100° = 260°.
- 4.260° is between 180° and 360°, confirming that the result is reflex.
To draw a reflex angle of 195°, first calculate the other opening: 360° − 195° = 165°. Construct the 165° opening with a semicircular protractor, then draw the curved indicator around the remaining side and label it 195°. Simply drawing the arms without this indicator leaves the intended turn unclear.
Find a Missing Part of a Known Turn
You do not always need to measure an unknown angle directly. If adjoining parts fill a known right or straight opening, subtract the known part from that whole. First identify the whole and confirm that the parts cover it once without overlap. This geometric relationship is the reason the subtraction works.
Problem
In the diagram, ER and EB form a straight angle, ∠RES is right, and ∠TER = 80°. Find ∠BET and ∠SET.
- 1.The straight opening ∠REB measures 180°. ET splits it into the 80° opening to ER and the remaining opening to EB.
- 2.Therefore ∠BET = 180° − 80° = 100°.
- 3.The right opening ∠RES measures 90°. ET lies within it and uses 80° from ER.
- 4.Therefore ∠SET = 90° − 80° = 10°. This smaller angle is the gap between ET and ES, not the entire 90° opening.
Problem
How can you draw an M with two 40° top-side openings and a 60° middle opening, and a Y with openings 150°, 60°, and 150°?
- 1.For M, begin with a V-shaped pair of segments meeting at the lower middle vertex with an opening of 60°.
- 2.At each upper end of the V, construct an outward-and-downward arm that makes 40° with the segment toward the middle. Choose convenient lengths to complete the M shape. Measure the three specified openings again.
- 3.For Y, draw two upward arms separated by 60° from one vertex.
- 4.Add a downward arm so that it makes a 150° opening with each upward arm. The openings around the vertex add to 150° + 60° + 150° = 360°, so they fit a full turn.
Equally Spaced Spokes
The Ashoka Chakra has 24 equally spaced spokes from one centre. The spokes divide a full turn into 24 equal openings. This repeats the circle-division idea, but now the spacing helps you reason about angles involving several spoke intervals. Count the intervals between spokes, rather than counting both endpoint spokes as intervals.
Problem
Find the angle between neighbouring spokes and the largest acute angle between any two spokes of the Ashoka Chakra.
- 1.Twenty-four equal intervals fill 360°, so one interval is 360° ÷ 24 = 15°.
- 2.Possible smaller turns are multiples of 15°: 15°, 30°, 45°, 60°, 75°, 90°, and so on.
- 3.An acute angle must remain below 90°. Five intervals give 5 × 15° = 75°.
- 4.Six intervals give 90°, which is right rather than acute. Therefore the largest acute opening is 75°.
Check Every Condition in a Puzzle
An angle puzzle can impose several conditions at once. Do not stop when one condition works. The angle in this puzzle remains acute when multiplied by 2, 3, or 4, but becomes obtuse when multiplied by 5. Start with the strongest acute condition and the lower boundary for the obtuse condition, then test all the candidates.
Problem
Find the possible whole-number degree measures of an acute angle if four times it is still acute but five times it is obtuse.
- 1.Four times the angle must be less than 90°. Dividing the boundary by 4 gives 90° ÷ 4 = 22.5°, so the original measure must be below 22.5°.
- 2.Five times the angle must be greater than 90°. Dividing by 5 gives 90° ÷ 5 = 18°, so the original measure must be above 18°.
- 3.The whole-number measures strictly between 18° and 22.5° are 19°, 20°, 21°, and 22°.
- 4.Check the largest candidate: 2 × 22° = 44°, 3 × 22° = 66°, and 4 × 22° = 88°, all acute; 5 × 22° = 110°, obtuse.
- 5.Check the smallest: 4 × 19° = 76° and 5 × 19° = 95°, satisfying the limiting conditions. The intermediate candidates work as well.
- 6.If fractional degree measures are allowed, every value greater than 18° and less than 22.5° also works. The four listed answers are the whole-number possibilities.
Try the classification tasks on dot grids as well: keep one arm fixed, choose a second grid point, and indicate an acute, obtuse, or reflex turn with a curve. A grid helps you draw straight directions, but it does not replace degree measurement when an exact number is required.
Quiz
Which measure is obtuse?
A smaller opening measures 75°. The remaining reflex turn is:
Which description is correct for a reflex angle?
A straight opening contains adjoining parts of 80° and an unknown angle. The unknown is:
Twenty-four equal spoke intervals fill a circle. The largest acute opening between spokes is:
Which original whole-number angle satisfies “four times is acute, five times is obtuse”?
To construct a reflex angle of 230° using a semicircular protractor, first construct the other opening of:
Practice Problems
- Classify 12°, 82°, 90°, 140°, 180°, 195°, 270°, and 359°. Explain the boundary cases.
- Draw acute, obtuse, and reflex openings using a dot grid. Indicate the intended turn with a curve.
- Construct angles of 140°, 82°, 195°, 70°, and 35°. Explain how you construct the reflex one.
- A marked reflex turn has an unmarked opening of 68°. Find the marked measure and check its type.
- A right opening is split into 37° and an unknown part. A straight opening is split into 112° and an unknown part. Find both missing measures.
- Make a figure containing three acute openings, one right opening, and two obtuse openings. Label the six intended angles.
- Construct the M and Y described in the lesson. Check the specified measures and explain why the Y openings fill a full turn.
- For 24 equally spaced spokes, find the openings spanning 2, 4, 5, and 6 intervals. Classify each.
- Solve the whole-number angle puzzle again using a table of candidates. Test all the multiplication conditions.
- Explain why the names ∠ABC and ∠CBA alone do not specify which of two possible sweeps is intended.
Key Takeaways
• Angle types have precise degree boundaries: acute below 90°, obtuse between 90° and 180°, and reflex between 180° and 360°. • Exactly 90° is right, exactly 180° is straight, and 360° is a full turn. • A curved indicator distinguishes the intended opening between the same arms. • Use subtraction only after identifying a whole right, straight, or full turn and its parts. • Equal spoke intervals and angle puzzles connect turn measures with counting and careful boundary checks.