Angles - Tenth Grade Geometry
Introduction to Angles
Angle: Formed by two rays with a common endpoint
Vertex: The common endpoint of the two rays
Sides: The two rays that form the angle
Measure: Usually measured in degrees (°) or radians
Vertex: The common endpoint of the two rays
Sides: The two rays that form the angle
Measure: Usually measured in degrees (°) or radians
1. Angle Vocabulary
Angle: The figure formed by two rays sharing a common endpoint
Vertex: The point where two rays meet
Sides/Rays: The two rays that form the angle
Interior: The region between the two rays
Exterior: The region outside the angle
Vertex: The point where two rays meet
Sides/Rays: The two rays that form the angle
Interior: The region between the two rays
Exterior: The region outside the angle
Angle Notation:
Three ways to name an angle:
1. By vertex: $\angle B$
Use when only one angle at that vertex
2. By three points: $\angle ABC$ or $\angle CBA$
Vertex must be in the middle
Order: point on one side, vertex, point on other side
3. By number: $\angle 1$, $\angle 2$
Use when angles are labeled with numbers
Three ways to name an angle:
1. By vertex: $\angle B$
Use when only one angle at that vertex
2. By three points: $\angle ABC$ or $\angle CBA$
Vertex must be in the middle
Order: point on one side, vertex, point on other side
3. By number: $\angle 1$, $\angle 2$
Use when angles are labeled with numbers
Example 1: Name the angle
Given angle with vertex at B, sides through A and C:
• $\angle B$ (if it's the only angle at B)
• $\angle ABC$ (vertex B in middle)
• $\angle CBA$ (same angle, different order)
Note: $\angle BAC$ would be INCORRECT (B is not the vertex)
Given angle with vertex at B, sides through A and C:
• $\angle B$ (if it's the only angle at B)
• $\angle ABC$ (vertex B in middle)
• $\angle CBA$ (same angle, different order)
Note: $\angle BAC$ would be INCORRECT (B is not the vertex)
2. Angle Measures
Degree (°): Unit of angle measurement
Protractor: Tool used to measure angles
Full rotation: 360°
Straight angle: 180°
Right angle: 90°
Protractor: Tool used to measure angles
Full rotation: 360°
Straight angle: 180°
Right angle: 90°
Classification of Angles by Measure:
Acute Angle: $0° < \theta < 90°$
• Smaller than a right angle
• Example: 45°, 60°
Right Angle: $\theta = 90°$
• Forms a perfect "L" shape
• Marked with a small square
Obtuse Angle: $90° < \theta < 180°$
• Larger than right angle but less than straight
• Example: 120°, 150°
Straight Angle: $\theta = 180°$
• Forms a straight line
• Also called a "linear angle"
Reflex Angle: $180° < \theta < 360°$
• Larger than a straight angle
• Example: 270°, 300°
Full Rotation: $\theta = 360°$
• Complete circle
Acute Angle: $0° < \theta < 90°$
• Smaller than a right angle
• Example: 45°, 60°
Right Angle: $\theta = 90°$
• Forms a perfect "L" shape
• Marked with a small square
Obtuse Angle: $90° < \theta < 180°$
• Larger than right angle but less than straight
• Example: 120°, 150°
Straight Angle: $\theta = 180°$
• Forms a straight line
• Also called a "linear angle"
Reflex Angle: $180° < \theta < 360°$
• Larger than a straight angle
• Example: 270°, 300°
Full Rotation: $\theta = 360°$
• Complete circle
Example 1: Classify angles
Angle of 35°: Acute (less than 90°)
Angle of 90°: Right angle
Angle of 125°: Obtuse (between 90° and 180°)
Angle of 180°: Straight angle
Angle of 270°: Reflex (between 180° and 360°)
Angle of 35°: Acute (less than 90°)
Angle of 90°: Right angle
Angle of 125°: Obtuse (between 90° and 180°)
Angle of 180°: Straight angle
Angle of 270°: Reflex (between 180° and 360°)
3. Identify Complementary, Supplementary, Vertical, Adjacent, and Congruent Angles
Complementary Angles
Complementary Angles: Two angles whose measures add up to 90°
Key Point: Can be adjacent or non-adjacent
Each is called the "complement" of the other
Key Point: Can be adjacent or non-adjacent
Each is called the "complement" of the other
Complementary Angles Formula:
$$\angle A + \angle B = 90°$$
To find complement:
$$\text{Complement of } \theta = 90° - \theta$$
Properties:
• Both angles must be acute (less than 90°)
• If one angle is $x°$, the other is $(90-x)°$
$$\angle A + \angle B = 90°$$
To find complement:
$$\text{Complement of } \theta = 90° - \theta$$
Properties:
• Both angles must be acute (less than 90°)
• If one angle is $x°$, the other is $(90-x)°$
Example 1: Complementary angles
If $\angle A = 35°$, find its complement.
Complement $= 90° - 35° = 55°$
Check: $35° + 55° = 90°$ ✓
If $\angle A = 35°$, find its complement.
Complement $= 90° - 35° = 55°$
Check: $35° + 55° = 90°$ ✓
Supplementary Angles
Supplementary Angles: Two angles whose measures add up to 180°
Linear Pair: Adjacent supplementary angles that form a straight line
Each is called the "supplement" of the other
Linear Pair: Adjacent supplementary angles that form a straight line
Each is called the "supplement" of the other
Supplementary Angles Formula:
$$\angle A + \angle B = 180°$$
To find supplement:
$$\text{Supplement of } \theta = 180° - \theta$$
Properties:
• Can be two acute, two right, or one acute and one obtuse
• If one angle is $x°$, the other is $(180-x)°$
$$\angle A + \angle B = 180°$$
To find supplement:
$$\text{Supplement of } \theta = 180° - \theta$$
Properties:
• Can be two acute, two right, or one acute and one obtuse
• If one angle is $x°$, the other is $(180-x)°$
Example 2: Supplementary angles
If $\angle C = 110°$, find its supplement.
Supplement $= 180° - 110° = 70°$
Check: $110° + 70° = 180°$ ✓
If $\angle C = 110°$, find its supplement.
Supplement $= 180° - 110° = 70°$
Check: $110° + 70° = 180°$ ✓
Vertical Angles
Vertical Angles: Opposite angles formed when two lines intersect
Key Property: Vertical angles are ALWAYS congruent
Also called: Vertically opposite angles
Key Property: Vertical angles are ALWAYS congruent
Also called: Vertically opposite angles
Vertical Angles Theorem:
When two lines intersect, vertical angles are congruent.
$$\angle 1 \cong \angle 3$$
$$\angle 2 \cong \angle 4$$
In terms of measures:
$$m\angle 1 = m\angle 3$$
$$m\angle 2 = m\angle 4$$
When two lines intersect, vertical angles are congruent.
$$\angle 1 \cong \angle 3$$
$$\angle 2 \cong \angle 4$$
In terms of measures:
$$m\angle 1 = m\angle 3$$
$$m\angle 2 = m\angle 4$$
Example 3: Vertical angles
Two lines intersect. If $\angle 1 = 65°$, find $\angle 3$ (vertical to $\angle 1$).
By Vertical Angles Theorem:
$\angle 3 = \angle 1 = 65°$
Answer: $\angle 3 = 65°$
Two lines intersect. If $\angle 1 = 65°$, find $\angle 3$ (vertical to $\angle 1$).
By Vertical Angles Theorem:
$\angle 3 = \angle 1 = 65°$
Answer: $\angle 3 = 65°$
Adjacent Angles
Adjacent Angles: Two angles that share a common vertex and a common side, but have no interior points in common
Requirements: Same vertex, common side, non-overlapping
Note: Adjacent angles do NOT have to be congruent
Requirements: Same vertex, common side, non-overlapping
Note: Adjacent angles do NOT have to be congruent
Characteristics of Adjacent Angles:
1. Share a vertex: Same corner point
2. Share a side: Common ray between them
3. Non-overlapping: No interior points in common
4. Side-by-side: Next to each other
Linear Pair: Special adjacent angles that are supplementary
1. Share a vertex: Same corner point
2. Share a side: Common ray between them
3. Non-overlapping: No interior points in common
4. Side-by-side: Next to each other
Linear Pair: Special adjacent angles that are supplementary
Example 4: Adjacent angles
$\angle ABD$ and $\angle DBC$ share vertex B and side BD.
They are adjacent angles.
If $\angle ABD = 40°$ and $\angle DBC = 50°$:
$\angle ABC = \angle ABD + \angle DBC = 40° + 50° = 90°$
$\angle ABD$ and $\angle DBC$ share vertex B and side BD.
They are adjacent angles.
If $\angle ABD = 40°$ and $\angle DBC = 50°$:
$\angle ABC = \angle ABD + \angle DBC = 40° + 50° = 90°$
Congruent Angles
Congruent Angles: Angles with equal measures
Symbol: $\cong$ (is congruent to)
Note: Shape doesn't matter, only the measure
Symbol: $\cong$ (is congruent to)
Note: Shape doesn't matter, only the measure
Congruent Angles Definition:
$$\angle A \cong \angle B \iff m\angle A = m\angle B$$
Properties:
• Reflexive: $\angle A \cong \angle A$
• Symmetric: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$
• Transitive: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$
$$\angle A \cong \angle B \iff m\angle A = m\angle B$$
Properties:
• Reflexive: $\angle A \cong \angle A$
• Symmetric: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$
• Transitive: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$
Example 5: Congruent angles
If $m\angle X = 75°$ and $m\angle Y = 75°$:
Then $\angle X \cong \angle Y$
Note: Even if angles look different in size on paper, if they have the same measure, they are congruent.
If $m\angle X = 75°$ and $m\angle Y = 75°$:
Then $\angle X \cong \angle Y$
Note: Even if angles look different in size on paper, if they have the same measure, they are congruent.
4. Find Measures of Complementary, Supplementary, Vertical, and Adjacent Angles
Example 1: Complementary angles
Two angles are complementary. One angle is 3 times the other. Find both angles.
Let smaller angle = $x$
Then larger angle = $3x$
$x + 3x = 90°$
$4x = 90°$
$x = 22.5°$
Smaller angle: $22.5°$
Larger angle: $3(22.5°) = 67.5°$
Check: $22.5° + 67.5° = 90°$ ✓
Two angles are complementary. One angle is 3 times the other. Find both angles.
Let smaller angle = $x$
Then larger angle = $3x$
$x + 3x = 90°$
$4x = 90°$
$x = 22.5°$
Smaller angle: $22.5°$
Larger angle: $3(22.5°) = 67.5°$
Check: $22.5° + 67.5° = 90°$ ✓
Example 2: Supplementary angles
Two supplementary angles differ by 40°. Find both angles.
Let smaller angle = $x$
Then larger angle = $x + 40$
$x + (x + 40) = 180°$
$2x + 40 = 180°$
$2x = 140°$
$x = 70°$
Smaller angle: $70°$
Larger angle: $110°$
Check: $70° + 110° = 180°$ ✓ and $110° - 70° = 40°$ ✓
Two supplementary angles differ by 40°. Find both angles.
Let smaller angle = $x$
Then larger angle = $x + 40$
$x + (x + 40) = 180°$
$2x + 40 = 180°$
$2x = 140°$
$x = 70°$
Smaller angle: $70°$
Larger angle: $110°$
Check: $70° + 110° = 180°$ ✓ and $110° - 70° = 40°$ ✓
Example 3: Vertical and adjacent angles
Two lines intersect. $\angle 1 = (2x + 10)°$ and $\angle 3 = (3x - 15)°$ are vertical. Find $x$ and the angle measures.
Vertical angles are congruent:
$2x + 10 = 3x - 15$
$10 + 15 = 3x - 2x$
$25 = x$
$\angle 1 = 2(25) + 10 = 60°$
$\angle 3 = 3(25) - 15 = 60°$ ✓
Adjacent angles $\angle 2$:
$\angle 1 + \angle 2 = 180°$ (linear pair)
$60° + \angle 2 = 180°$
$\angle 2 = 120°$
Two lines intersect. $\angle 1 = (2x + 10)°$ and $\angle 3 = (3x - 15)°$ are vertical. Find $x$ and the angle measures.
Vertical angles are congruent:
$2x + 10 = 3x - 15$
$10 + 15 = 3x - 2x$
$25 = x$
$\angle 1 = 2(25) + 10 = 60°$
$\angle 3 = 3(25) - 15 = 60°$ ✓
Adjacent angles $\angle 2$:
$\angle 1 + \angle 2 = 180°$ (linear pair)
$60° + \angle 2 = 180°$
$\angle 2 = 120°$
Example 4: Linear pair
$\angle ABC$ and $\angle CBD$ form a linear pair. If $m\angle ABC = (5x - 20)°$ and $m\angle CBD = (3x + 40)°$, find $x$ and both angle measures.
Linear pair means supplementary:
$(5x - 20) + (3x + 40) = 180$
$8x + 20 = 180$
$8x = 160$
$x = 20$
$m\angle ABC = 5(20) - 20 = 80°$
$m\angle CBD = 3(20) + 40 = 100°$
Check: $80° + 100° = 180°$ ✓
$\angle ABC$ and $\angle CBD$ form a linear pair. If $m\angle ABC = (5x - 20)°$ and $m\angle CBD = (3x + 40)°$, find $x$ and both angle measures.
Linear pair means supplementary:
$(5x - 20) + (3x + 40) = 180$
$8x + 20 = 180$
$8x = 160$
$x = 20$
$m\angle ABC = 5(20) - 20 = 80°$
$m\angle CBD = 3(20) + 40 = 100°$
Check: $80° + 100° = 180°$ ✓
5. Angle Diagrams: Solve for the Variable
Steps to Solve Angle Diagram Problems:
Step 1: Identify the angle relationship (complementary, supplementary, vertical, linear pair)
Step 2: Write an equation based on the relationship
Step 3: Solve for the variable
Step 4: Substitute back to find angle measures
Step 5: Check that your answer makes sense
Step 1: Identify the angle relationship (complementary, supplementary, vertical, linear pair)
Step 2: Write an equation based on the relationship
Step 3: Solve for the variable
Step 4: Substitute back to find angle measures
Step 5: Check that your answer makes sense
Example 1: Complementary angles in diagram
Two complementary angles: $\angle A = (2x + 15)°$ and $\angle B = (3x - 5)°$
$(2x + 15) + (3x - 5) = 90$
$5x + 10 = 90$
$5x = 80$
$x = 16$
$\angle A = 2(16) + 15 = 47°$
$\angle B = 3(16) - 5 = 43°$
Answer: $x = 16$, $\angle A = 47°$, $\angle B = 43°$
Two complementary angles: $\angle A = (2x + 15)°$ and $\angle B = (3x - 5)°$
$(2x + 15) + (3x - 5) = 90$
$5x + 10 = 90$
$5x = 80$
$x = 16$
$\angle A = 2(16) + 15 = 47°$
$\angle B = 3(16) - 5 = 43°$
Answer: $x = 16$, $\angle A = 47°$, $\angle B = 43°$
Example 2: Three angles on a line
Three angles form a straight line: $(x + 20)°$, $(2x)°$, and $(x - 10)°$
Sum must equal 180°:
$(x + 20) + 2x + (x - 10) = 180$
$4x + 10 = 180$
$4x = 170$
$x = 42.5$
Angles: $62.5°$, $85°$, $32.5°$
Check: $62.5 + 85 + 32.5 = 180$ ✓
Three angles form a straight line: $(x + 20)°$, $(2x)°$, and $(x - 10)°$
Sum must equal 180°:
$(x + 20) + 2x + (x - 10) = 180$
$4x + 10 = 180$
$4x = 170$
$x = 42.5$
Angles: $62.5°$, $85°$, $32.5°$
Check: $62.5 + 85 + 32.5 = 180$ ✓
6. Angle Bisectors
Angle Bisector: A ray that divides an angle into two congruent angles
Bisect: To divide into two equal parts
Result: Two angles with equal measures
Bisect: To divide into two equal parts
Result: Two angles with equal measures
Angle Bisector Properties:
If ray $\overrightarrow{BD}$ bisects $\angle ABC$, then:
$$\angle ABD \cong \angle DBC$$
$$m\angle ABD = m\angle DBC$$
$$m\angle ABD = m\angle DBC = \frac{1}{2}m\angle ABC$$
$$m\angle ABC = 2 \cdot m\angle ABD = 2 \cdot m\angle DBC$$
If ray $\overrightarrow{BD}$ bisects $\angle ABC$, then:
$$\angle ABD \cong \angle DBC$$
$$m\angle ABD = m\angle DBC$$
$$m\angle ABD = m\angle DBC = \frac{1}{2}m\angle ABC$$
$$m\angle ABC = 2 \cdot m\angle ABD = 2 \cdot m\angle DBC$$
Example 1: Find angles with bisector
Ray BD bisects $\angle ABC$. If $m\angle ABC = 80°$, find $m\angle ABD$ and $m\angle DBC$.
Since BD bisects the angle:
$m\angle ABD = m\angle DBC = \frac{80°}{2} = 40°$
Answer: Both angles measure 40°
Ray BD bisects $\angle ABC$. If $m\angle ABC = 80°$, find $m\angle ABD$ and $m\angle DBC$.
Since BD bisects the angle:
$m\angle ABD = m\angle DBC = \frac{80°}{2} = 40°$
Answer: Both angles measure 40°
Example 2: Solve for variable with bisector
Ray QS bisects $\angle PQR$. If $m\angle PQS = (3x + 12)°$ and $m\angle SQR = (5x - 8)°$, find $x$ and $m\angle PQR$.
Since QS bisects the angle:
$3x + 12 = 5x - 8$
$12 + 8 = 5x - 3x$
$20 = 2x$
$x = 10$
$m\angle PQS = 3(10) + 12 = 42°$
$m\angle SQR = 5(10) - 8 = 42°$ ✓
$m\angle PQR = 42° + 42° = 84°$
Answer: $x = 10$, $m\angle PQR = 84°$
Ray QS bisects $\angle PQR$. If $m\angle PQS = (3x + 12)°$ and $m\angle SQR = (5x - 8)°$, find $x$ and $m\angle PQR$.
Since QS bisects the angle:
$3x + 12 = 5x - 8$
$12 + 8 = 5x - 3x$
$20 = 2x$
$x = 10$
$m\angle PQS = 3(10) + 12 = 42°$
$m\angle SQR = 5(10) - 8 = 42°$ ✓
$m\angle PQR = 42° + 42° = 84°$
Answer: $x = 10$, $m\angle PQR = 84°$
7. Construct an Angle Bisector
Construction: Drawing with compass and straightedge only
No measuring: Cannot use protractor or ruler measurements
Goal: Create ray that divides angle into two equal parts
No measuring: Cannot use protractor or ruler measurements
Goal: Create ray that divides angle into two equal parts
Construction: Bisect Angle ABC
Step 1: Place compass point on vertex B
Step 2: Draw an arc that intersects both sides of the angle
• Mark intersection points as D (on BA) and E (on BC)
Step 3: Place compass point on D
Step 4: Set compass width to more than half of DE
Step 5: Draw an arc in the interior of the angle
Step 6: Keep same compass width, place compass on E
Step 7: Draw an arc that intersects the first arc
• Mark intersection point as F
Step 8: Draw ray from B through F
Result: Ray BF is the angle bisector of $\angle ABC$
Step 1: Place compass point on vertex B
Step 2: Draw an arc that intersects both sides of the angle
• Mark intersection points as D (on BA) and E (on BC)
Step 3: Place compass point on D
Step 4: Set compass width to more than half of DE
Step 5: Draw an arc in the interior of the angle
Step 6: Keep same compass width, place compass on E
Step 7: Draw an arc that intersects the first arc
• Mark intersection point as F
Step 8: Draw ray from B through F
Result: Ray BF is the angle bisector of $\angle ABC$
Why This Works:
• Points D and E are equidistant from B (same arc radius)
• Point F is equidistant from both D and E (same compass width)
• By symmetry, ray BF divides the angle into two congruent parts
• Can be proven using triangle congruence (SSS)
• Points D and E are equidistant from B (same arc radius)
• Point F is equidistant from both D and E (same compass width)
• By symmetry, ray BF divides the angle into two congruent parts
• Can be proven using triangle congruence (SSS)
8. Construct a Congruent Angle
Goal: Copy an angle to a new location
Result: New angle with same measure as original
Tools: Compass and straightedge only
Result: New angle with same measure as original
Tools: Compass and straightedge only
Construction: Copy Angle ABC to Create Congruent Angle
Given: Angle ABC (original)
Goal: Create angle DEF congruent to angle ABC
Step 1: Draw a ray with endpoint D (this will be DE)
Step 2: Place compass on B (vertex of original angle)
Step 3: Draw an arc intersecting both sides of $\angle ABC$
• Mark intersections as P and Q
Step 4: Keep same compass width, place compass on D
Step 5: Draw an arc intersecting ray DE
• Mark intersection as E
Step 6: Measure distance PQ with compass
• Place compass on P, adjust to reach Q
Step 7: Keep that compass width, place compass on E
Step 8: Draw an arc intersecting the first arc from Step 5
• Mark intersection as F
Step 9: Draw ray DF
Result: $\angle EDF \cong \angle ABC$
Given: Angle ABC (original)
Goal: Create angle DEF congruent to angle ABC
Step 1: Draw a ray with endpoint D (this will be DE)
Step 2: Place compass on B (vertex of original angle)
Step 3: Draw an arc intersecting both sides of $\angle ABC$
• Mark intersections as P and Q
Step 4: Keep same compass width, place compass on D
Step 5: Draw an arc intersecting ray DE
• Mark intersection as E
Step 6: Measure distance PQ with compass
• Place compass on P, adjust to reach Q
Step 7: Keep that compass width, place compass on E
Step 8: Draw an arc intersecting the first arc from Step 5
• Mark intersection as F
Step 9: Draw ray DF
Result: $\angle EDF \cong \angle ABC$
9. Proofs Involving Angles
Proof: Logical argument that uses definitions, postulates, and theorems
Given: Information provided in the problem
Prove: Statement to be proven true
Two-column proof: Statements in left column, reasons in right column
Given: Information provided in the problem
Prove: Statement to be proven true
Two-column proof: Statements in left column, reasons in right column
Key Theorems for Angle Proofs
Angle Theorems and Postulates:
1. Vertical Angles Theorem:
Vertical angles are congruent
2. Linear Pair Postulate:
If two angles form a linear pair, they are supplementary
3. Congruent Supplements Theorem:
If two angles are supplementary to the same angle (or congruent angles), then they are congruent
4. Congruent Complements Theorem:
If two angles are complementary to the same angle (or congruent angles), then they are congruent
5. Angle Addition Postulate:
If point D is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$
6. Right Angle Congruence Theorem:
All right angles are congruent
1. Vertical Angles Theorem:
Vertical angles are congruent
2. Linear Pair Postulate:
If two angles form a linear pair, they are supplementary
3. Congruent Supplements Theorem:
If two angles are supplementary to the same angle (or congruent angles), then they are congruent
4. Congruent Complements Theorem:
If two angles are complementary to the same angle (or congruent angles), then they are congruent
5. Angle Addition Postulate:
If point D is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$
6. Right Angle Congruence Theorem:
All right angles are congruent
Example 1: Two-column proof
Given: $\angle 1 \cong \angle 2$, $\angle 2 \cong \angle 3$
Prove: $\angle 1 \cong \angle 3$
Given: $\angle 1 \cong \angle 2$, $\angle 2 \cong \angle 3$
Prove: $\angle 1 \cong \angle 3$
| Statements | Reasons |
|---|---|
| 1. $\angle 1 \cong \angle 2$ | 1. Given |
| 2. $\angle 2 \cong \angle 3$ | 2. Given |
| 3. $\angle 1 \cong \angle 3$ | 3. Transitive Property of Congruence |
Example 2: Proof using supplementary angles
Given: $\angle 1$ and $\angle 2$ are supplementary, $\angle 3$ and $\angle 2$ are supplementary
Prove: $\angle 1 \cong \angle 3$
Given: $\angle 1$ and $\angle 2$ are supplementary, $\angle 3$ and $\angle 2$ are supplementary
Prove: $\angle 1 \cong \angle 3$
| Statements | Reasons |
|---|---|
| 1. $\angle 1$ and $\angle 2$ are supplementary | 1. Given |
| 2. $\angle 3$ and $\angle 2$ are supplementary | 2. Given |
| 3. $\angle 1 \cong \angle 3$ | 3. Congruent Supplements Theorem |
Example 3: Proof using vertical angles
Given: Lines AB and CD intersect at E
Prove: $\angle AEC \cong \angle BED$
Given: Lines AB and CD intersect at E
Prove: $\angle AEC \cong \angle BED$
| Statements | Reasons |
|---|---|
| 1. Lines AB and CD intersect at E | 1. Given |
| 2. $\angle AEC$ and $\angle BED$ are vertical angles | 2. Definition of vertical angles |
| 3. $\angle AEC \cong \angle BED$ | 3. Vertical Angles Theorem |
Angle Classifications Summary
| Angle Type | Measure Range | Example | Visual Description |
|---|---|---|---|
| Acute | $0° < \theta < 90°$ | 45°, 30°, 60° | Sharp, less than right angle |
| Right | $\theta = 90°$ | 90° | Perfect "L" shape, square corner |
| Obtuse | $90° < \theta < 180°$ | 120°, 135°, 150° | Wide, more than right angle |
| Straight | $\theta = 180°$ | 180° | Straight line |
| Reflex | $180° < \theta < 360°$ | 270°, 300° | More than straight, less than full |
Angle Pairs Summary
| Angle Pair | Definition | Sum Formula | Key Property |
|---|---|---|---|
| Complementary | Two angles that add to 90° | $\angle A + \angle B = 90°$ | Both must be acute |
| Supplementary | Two angles that add to 180° | $\angle A + \angle B = 180°$ | Can be any combination |
| Vertical | Opposite angles when lines intersect | N/A | Always congruent |
| Adjacent | Share vertex and side, non-overlapping | N/A (can vary) | Side by side |
| Linear Pair | Adjacent angles on straight line | $\angle A + \angle B = 180°$ | Adjacent AND supplementary |
| Congruent | Angles with equal measures | N/A | $m\angle A = m\angle B$ |
Formulas Quick Reference
| Concept | Formula | Use |
|---|---|---|
| Complement | $90° - \theta$ | Find complementary angle |
| Supplement | $180° - \theta$ | Find supplementary angle |
| Angle Bisector | $\frac{1}{2}\angle ABC$ | Each half angle measure |
| Vertical Angles | $\angle 1 = \angle 3$ | Opposite angles are equal |
| Linear Pair | $\angle 1 + \angle 2 = 180°$ | Adjacent angles on line |
| Angle Addition | $m\angle ABC = m\angle ABD + m\angle DBC$ | When D is in interior |
Key Theorems for Proofs
| Theorem/Postulate | Statement |
|---|---|
| Vertical Angles Theorem | Vertical angles are congruent |
| Linear Pair Postulate | Linear pairs are supplementary |
| Congruent Supplements Theorem | Angles supplementary to same angle are congruent |
| Congruent Complements Theorem | Angles complementary to same angle are congruent |
| Right Angle Congruence | All right angles are congruent |
| Angle Addition Postulate | $m\angle ABD + m\angle DBC = m\angle ABC$ (if D in interior) |
Success Tips for Angles:
✓ Complementary angles add to 90°; Supplementary angles add to 180°
✓ Vertical angles are ALWAYS congruent (equal measures)
✓ Adjacent angles share a vertex and a side but don't overlap
✓ Linear pair = Adjacent + Supplementary
✓ Angle bisector creates TWO congruent angles
✓ To find complement: 90° - angle; To find supplement: 180° - angle
✓ In proofs, always state reasons for each statement
✓ Use variables to set up equations for unknown angles
✓ Check your answer: Do the angles add up correctly?
✓ Practice constructions to understand angle relationships!
✓ Complementary angles add to 90°; Supplementary angles add to 180°
✓ Vertical angles are ALWAYS congruent (equal measures)
✓ Adjacent angles share a vertex and a side but don't overlap
✓ Linear pair = Adjacent + Supplementary
✓ Angle bisector creates TWO congruent angles
✓ To find complement: 90° - angle; To find supplement: 180° - angle
✓ In proofs, always state reasons for each statement
✓ Use variables to set up equations for unknown angles
✓ Check your answer: Do the angles add up correctly?
✓ Practice constructions to understand angle relationships!