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Types of Angles and Special Angle Relationships

Capítulo 6

Estimated reading time: 5 minutes

Classifying Angles by Measure

Angles are often classified by their measure (in degrees). This vocabulary helps you recognize angle types quickly and decide which relationships might apply.

Type	Measure range	Quick visual cue
Acute	`0° < m∠ < 90°`	Small opening
Right	`m∠ = 90°`	Corner; square mark
Obtuse	`90° < m∠ < 180°`	Wide opening (but not flat)
Straight	`m∠ = 180°`	Flat line
Reflex	`180° < m∠ < 360°`	“Outside” turn more than a straight angle

Example sketches (text-only diagrams)

Use these as mental pictures. In diagrams, angle arcs indicate the interior being measured; a small square indicates a right angle.

Acute:            Right:             Obtuse:            Straight:          Reflex (measured outside):

   /                |                 \                 --------            
  /                 |                  \                    
 /____              |____               \____                                  (large arc around)
  arc               [ ] square           arc               arc on line

Step-by-step: How to classify an angle by measure

Step 1: Identify the given measure (or estimate from the diagram).
Step 2: Compare it to 90° and 180°.
Step 3: Choose the label: acute (<90°), right (=90°), obtuse (between 90° and 180°), straight (=180°), reflex (between 180° and 360°).

Practice: Classify each angle.

m∠A = 42° → acute
m∠B = 90° → right
m∠C = 135° → obtuse
m∠D = 180° → straight
m∠E = 225° → reflex

Diagram Conventions You Will See

Geometry diagrams use consistent marks so you can recognize relationships quickly.

Angle arc: a curved mark showing which angle is being discussed.
Matching arc marks: if two angles have the same number/style of arcs, they are congruent (equal measure).
Right-angle square: a small square at the vertex indicates 90°.
Shared side: adjacent angles often share a ray (a common side).

Congruent angles:        Right angle:

  )   )                 |
 ∠1  ∠2                 |_
(two matching arcs)     [ ] indicates 90°

Special Angle Relationships

Adjacent angles

Adjacent angles share a common vertex and a common side, and their interiors do not overlap.

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Two adjacent angles share the middle ray:

   ray 1
     
      \  ∠1
       \)
        \ ray 2 (shared)
        /(
       /  ∠2
      /
   ray 3

Key idea: Adjacent angles are “next to each other,” but they do not automatically add to 90° or 180° unless extra information is given.

Vertical angles

Vertical angles are opposite angles formed by two intersecting lines. Vertical angles always have equal measure.

Intersecting lines form two pairs of vertical angles:

   \  ∠1  /
    \   /
     \ /
     / \
    /   \
   /  ∠2  \

∠1 and ∠2 are vertical (opposite).

Definition-only reasoning: If two angles are vertical, then m∠1 = m∠2.

Complementary angles (sum 90°)

Two angles are complementary if their measures add to 90°.

Example: If m∠A = 35° and ∠A is complementary to ∠B, then m∠B = 90° − 35° = 55°.

Complementary often appears inside a right angle:

|\
| \  (two angles that fill the corner)
|__\
[ ] total is 90°

Supplementary angles (sum 180°)

Two angles are supplementary if their measures add to 180°.

Example: If m∠C = 120° and ∠C is supplementary to ∠D, then m∠D = 180° − 120° = 60°.

Supplementary often appears along a straight line:

------\)------
   ∠1  \( ∠2
(total is 180°)

Linear pairs

A linear pair is a special kind of adjacent angle pair whose non-common sides form a straight line. Linear pairs are always supplementary.

They are adjacent (share a side).
The outer sides are opposite rays (form a straight angle).
So m∠1 + m∠2 = 180°.

Linear pair:

A-----V-----B
      \
       \
        C

∠AVC and ∠CVB share ray VC, and VA and VB form a straight line.

Spot the Relationship (Label the Pairs)

Look at the diagram below. Identify pairs that are: adjacent, vertical, complementary, supplementary, and linear pairs (some pairs may fit more than one category).

Diagram: Two lines intersect at V, and one extra ray goes to the right.

        N
        |
        |
W ------V------ E ---- R
        |
        |
        S

Angles around V:
∠NVW is between VN and VW
∠NVE is between VN and VE
∠EVS is between VE and VS
∠SVW is between VS and VW
∠EVR is between VE and VR (VR is the extra ray to the right)

Your tasks

Vertical angles: Name one vertical pair.
Adjacent angles: Name two adjacent angles that share side VE.
Linear pair: Name a linear pair that forms a straight line along W–V–E.
Supplementary: Find a pair that must sum to 180° because it is a linear pair.
Complementary: If you are told that ∠NVE is a right angle (square mark at V between VN and VE), which two angles in the diagram could be complementary if a ray inside that right angle splits it?

Check (relationship-only answers)

Vertical pair example: ∠NVE and ∠SVW are vertical (opposite angles).
Adjacent sharing VE: ∠NVE and ∠EVS are adjacent (share side VE).
Linear pair along W–V–E: ∠NVW and ∠NVE form a linear pair (their non-common sides VW and VE are opposite rays).
Supplementary example: any linear pair, such as ∠NVW and ∠NVE, is supplementary.
Complementary note: complementary requires a total of 90°; it would occur only if a right angle is given and split into two angles.

Reasoning Tasks (Use Definitions Only)

These problems are designed to be solved with one idea at a time: “vertical angles are equal,” “complements add to 90°,” “supplements add to 180°,” and “linear pairs are supplementary.”

1) Linear pair arithmetic

If two angles form a linear pair and one angle measures 35°, find the other.

Definition used: linear pair ⇒ supplementary ⇒ sum is 180°
Other angle = 180° − 35° = 145°

2) Vertical angles

Two lines intersect. One of the angles measures 112°. Find the measure of its vertical angle.

Definition used: vertical angles have equal measure
Vertical angle = 112°

3) Complementary angles

Angle X is complementary to angle Y. If m∠X = 48°, find m∠Y.

Definition used: complementary ⇒ sum is 90°
m∠Y = 90° − 48° = 42°

4) Supplementary angles

Angle P is supplementary to angle Q. If m∠P = 163°, find m∠Q and classify ∠Q as acute/right/obtuse/straight/reflex.

Definition used: supplementary ⇒ sum is 180°
m∠Q = 180° − 163° = 17°
17° is acute (0° < 17° < 90°)

5) Mixed recognition

In an intersection, one angle is 70°. List the measures of all four angles formed.

Vertical angle is also 70°.
Each adjacent angle forms a linear pair with 70°, so each is 180° − 70° = 110°.
The fourth angle (vertical to 110°) is 110°.