How to Solve Angle-Based Geometry Problems?

Have you ever used a protractor? It’s that half-circle shaped tool with lots of tiny lines and numbers on it. If you want to draw an angle, a protractor is your best friend. It helps you measure and draw angles accurately.

But what exactly is an angle, and why do we need to measure them? Think of an angle as the amount of turn between two lines that share a common point. Angles are everywhere – in the corners of your room, in the slices of a pizza, even in the hands of a clock!

Angles are especially important in geometry, which is a type of math that deals with shapes and figures. By understanding angles, we can learn a lot about the properties of different shapes.

In this lesson, we’ll dive into the world of angles. You’ll learn what angles are, how to classify them based on their size, and some of their special properties.

Definition of an Angle

An angle is made up of two rays that share the same endpoint. This shared endpoint is called the vertex. The two rays are called the sides of the angle.

Naming Angles

We name angles using the points on the sides and the vertex. For example, in this figure:

We can call the angle ∠ABC or ∠CBA. The important thing is to always put the vertex letter in the middle when naming an angle. So in this case, B is in the middle since it’s the vertex.

Examples

1. In this figure, the angles are ∠PQR, ∠PQS, and ∠SQR:

2. Using the same figure, SRQ is not an angle, because R is not the vertex (endpoint) of two rays.

The key things to remember are:

• An angle has two rays with a shared endpoint
• That endpoint is called the vertex
• Always put the vertex letter in the middle when naming an angle

Measuring Angles

Just like we can measure the length of a line or the size of a shape, we can also measure angles.

The Protractor Postulate

The Protractor Postulate states that the measurement of an angle can be any number between 0 and 180 degrees. We use the symbol ° for degrees.

Using a Protractor

To actually measure an angle, we use a tool called a protractor. Here’s how it works:

1. Place the center of the protractor on the vertex of the angle.
2. Line up one side of the angle with the 0° line on the protractor.
3. Look where the other side of the angle intersects the protractor. That number is the measure of the angle in degrees.

For example, if we measure angle ∠ABC and side AB lines up with 0° while side BC lines up with 120°, then the measure of ∠ABC is 120°.

Notation

We write the measure of an angle using the letter m. So if ∠ABC measures 120°, we write this as m∠ABC = 120°.

In Practice

When you’re solving geometry problems or taking exams, the measure of angles is usually given to you. You probably won’t need to use a protractor. But it’s still good to understand how angle measurement works!

The key points are:

• Angles are measured in degrees (°)
• A protractor is the tool we use to measure angles
• The measure of an angle can be any number from 0° to 180°
• We write angle measures using the letter m, like m∠ABC = 120°

Types of Angles

1. Acute Angle

An angle that measures between 0° and 90°.

Example: If m∠PQR = 45°, then ∠PQR is an acute angle.

2. Right Angle

An angle that measures exactly 90°. It looks like the letter “L”.

Note: The measure must be precisely 90°. Even 90.5° is not a right angle.

3. Obtuse Angle

An angle that measures between 90° and 180°.

Example: If m∠XYZ = 105°, then ∠XYZ is an obtuse angle.

4. Straight Angle

An angle that measures exactly 180°. It forms a straight line.

Examples

1. Classify the following angles:

• m∠ABC = 125° (Obtuse: between 90° and 180°)
• m∠COR = 90.5° (Obtuse: not exactly 90°, but between 90° and 180°)
• m∠RAW = 0.01° (Acute: between 0° and 90°)
• m∠CDO = 179.12° (Obtuse: between 90° and 180°)

2. What type of angle is formed by the hands of a clock at 3 o’clock?

Answer: Right angle (90°), because it looks like the letter “L”.

3. What type of angle is formed by the hands of a clock at 2 o’clock?

Answer: Acute angle, because it’s smaller than the right angle formed at 3 o’clock.

Remember:

• Acute angles: 0° to 90°
• Right angles: Exactly 90°
• Obtuse angles: 90° to 180°
• Straight angles: Exactly 180°

Angle Addition Postulate

If you have an angle ∠ABC, and a point D inside the angle, the Angle Addition Postulate says:

The measure of ∠ABC is equal to the sum of the measures of ∠ABD and ∠DBC.

In symbols: m∠ABC = m∠ABD + m∠DBC

Basically, if you split an angle into two smaller angles, the measure of the big angle is equal to the sum of the measures of the two smaller angles.

Example 1

Let’s say ∠PQR = 25°, ∠PQS = 3x + 10°, and ∠SQR = 2x°. We can find the value of x:

1. The Angle Addition Postulate says: m∠PQR = m∠PQS + m∠SQR
2. Substitute the given values: 25° = (3x + 10°) + 2x°
3. Simplify: 25° = 5x + 10°
4. Solve for x:
   • Subtract 10° from both sides: 15° = 5x
   • Divide both sides by 5: 3° = x

So, x = 3°.

Example 2

In this figure, we know m∠DBC = 52°. We can find m∠ABD:

1. ∠ABC is a straight angle, so m∠ABC = 180°.
2. The Angle Addition Postulate says: m∠ABC = m∠ABD + m∠DBC
3. Substitute the known values: 180° = m∠ABD + 52°
4. Solve for m∠ABD:
   • Subtract 52° from both sides: 128° = m∠ABD

Therefore, m∠ABD = 128°.

Remember, the Angle Addition Postulate is a way to find the measure of a big angle if you know the measures of the smaller angles that make it up.

Congruent Angles

Two angles are congruent if they have the same measure (size). Congruent angles have the same shape and size, but they may be oriented differently.

Euclid’s Postulate on Right Angles

Remember Euclid’s postulate that says all right angles are congruent? Let’s think about why this makes sense.

We know that a right angle is an angle that measures exactly 90°. So, any right angle you draw, no matter how big or small, will always measure 90°.

Since all right angles have the same measure (90°), they are all congruent to each other. This is what Euclid’s postulate tells us.

Notation

We use the symbol ≅ to show that two angles are congruent. For example:

If ∠ABC and ∠DEF both measure 45°, we write: ∠ABC ≅ ∠DEF

This means “angle ABC is congruent to angle DEF”.

Key Points

• Congruent angles have the same measure (size)
• All right angles are congruent because they all measure 90°
• We use the symbol ≅ to show congruent angles

Angle Bisector

An angle bisector is a ray that divides an angle into two equal (congruent) angles.

For example, in this figure:

If ray QS divides ∠PQR into two congruent angles (∠PQS and ∠SQR), then QS is an angle bisector.

Example Problem

Let’s say ∠MNO measures 60°, and ray PN bisects ∠MNO. What is the measure of ∠ONP?

We can solve this in two ways:

Method 1:

1. Since PN bisects ∠MNO, we know ∠MNP and ∠ONP are congruent (equal).
2. So, the measure of ∠ONP is half the measure of ∠MNO.
3. Half of 60° is 30° (60 ÷ 2 = 30).
4. Therefore, m∠ONP = 30°.

Method 2:

1. Let’s say the measure of ∠ONP is x°.
2. Since PN bisects ∠MNO, ∠MNP also measures x°.
3. By the Angle Addition Postulate: m∠ONP + m∠MNP = m∠MNO
4. Substitute the values: x° + x° = 60°
5. Simplify: 2x° = 60°
6. Solve for x: x = 60 ÷ 2 = 30
7. So, m∠ONP = 30°.

Both methods give us the same answer: m∠ONP = 30°.

Remember, an angle bisector splits an angle into two equal parts. If you know the whole angle’s measure, you can find the measure of each part by dividing by 2.

Angle Pairs

Angle pairs are two angles that are related in a specific way. One important type of angle pair is vertical angles.

1. Vertical Angles

Vertical angles are formed when two lines intersect. They are the opposite angles created by the intersecting lines.

In this figure, ∠ABC and ∠DBE are vertical angles. They are opposite each other. ∠ABD and ∠CBE are also vertical angles.

How to Identify Vertical Angles?

Imagine the intersecting lines form an “X”. The angles in the opposite openings of the “X” are vertical angles.

Vertical Angle Theorem

The Vertical Angle Theorem states:

Vertical angles formed by intersecting lines are always congruent.

This means vertical angles always have the same measure.

For example, if m∠ABC = 45°, then m∠DBE is also 45° because they are vertical angles.

Example Problem

Let’s say ∠CBE = 2x + 20° and ∠ABD = 120°. Find the value of x.

1. ∠ABD and ∠CBE are vertical angles, so they are congruent (equal).
2. Set up an equation: m∠ABD = m∠CBE
3. Substitute the values: 120° = 2x + 20°
4. Solve for x:
   • Subtract 20° from both sides: 100° = 2x
   • Divide both sides by 2: 50° = x

So, x = 50°.

Remember:

• Vertical angles are opposite angles formed by intersecting lines
• Vertical angles are always congruent (they have the same measure)
• You can use the Vertical Angle Theorem to solve problems involving vertical angles

2. Complementary Angles

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

Example

If m∠PQR = 30° and m∠RQS = 60°, then ∠PQR and ∠RQS are complementary because:

30° + 60° = 90°

Key Points

• Complementary angles always add up to 90°
• The angles don’t have to be next to each other or share a side
• If you know one angle in a complementary pair, you can find the other by subtracting from 90°

For example, if m∠ABC = 25° and ∠ABC is complementary to ∠DEF, then:

m∠DEF = 90° – 25° = 65°

Notation

If ∠PQR and ∠XYZ are complementary, we can write:

∠PQR + ∠XYZ = 90°

Remember, complementary angles are all about adding up to 90°. This is a useful concept when solving geometry problems involving right angles or right triangles.

3. Supplementary Angles

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

Example

If m∠CAR = 120° and m∠UAV = 60°, then ∠CAR and ∠UAV are supplementary because:

120° + 60° = 180°

Key Points

• Supplementary angles always add up to 180°
• The angles don’t have to be next to each other or share a side
• If you know one angle in a supplementary pair, you can find the other by subtracting from 180°

For example, if m∠ABC = 110° and ∠ABC is supplementary to ∠DEF, then:

m∠DEF = 180° – 110° = 70°

Example Problems

1. An angle is a complement of another angle. If the measure of one angle is twice the other, what is the measure of the smaller angle?
   • Let the smaller angle be x°. The larger angle is 2x°.
   • Since they are complementary: x° + 2x° = 90°
   • Solve for x: 3x = 90, so x = 30
   • The smaller angle measures 30°.
2. Angles 1 and 2 are supplementary. Angle 1 measures 60° more than twice angle 2. What is the measure of the complement of angle 2?
   • Let angle 2 be x°. Angle 1 is 2x + 60°.
   • Since they are supplementary: x° + (2x + 60°) = 180°
   • Solve for x: 3x + 60 = 180, so 3x = 120, and x = 40
   • Angle 2 measures 40°. Its complement is 90° – 40° = 50°.

Remember, supplementary angles add up to 180°. This concept is useful when solving problems involving straight lines or parallel lines cut by a transversal.

4. Adjacent Angles

Two angles are adjacent if they:

1. Share a common vertex (corner point)
2. Share a common side (ray)
3. Do not overlap

In this figure, ∠XYZ and ∠WYZ are adjacent angles. They share vertex Y and side YZ.

Linear Pairs

A linear pair is a special case of adjacent angles. Two angles form a linear pair if:

1. They are adjacent
2. Their non-common sides form a straight line
3. Their measures add up to 180°

In the figure above, ∠XYZ and ∠WYZ form a linear pair because:

• They are adjacent
• XY and WY form a straight line
• m∠XYZ + m∠WYZ = 180°

Example Problem

Let’s say ∠ABC and ∠CBE form a linear pair. If m∠CBE = 70°, find m∠ABC.

1. Since ∠ABC and ∠CBE form a linear pair, they are supplementary.
2. Supplementary angles add up to 180°.
3. So, m∠ABC + 70° = 180°
4. Solve for m∠ABC: m∠ABC = 180° – 70° = 110°

Therefore, m∠ABC = 110°.

Remember:

• Adjacent angles share a vertex and side but don’t overlap
• Linear pairs are adjacent angles that form a line and add up to 180°
• If you know one angle in a linear pair, you can find the other by subtracting from 180°

Angles Formed By Transversal Intersecting Parallel Lines

Parallel lines are lines that never intersect, no matter how far they are extended. We use the symbol || to denote parallel lines.

For example, if lines l1 and l2 are parallel, we write: l1 || l2

A transversal line is a line that intersects two or more parallel lines.

In this figure, l3 is a transversal line because it intersects parallel lines l1 and l2.

When a transversal intersects parallel lines, it creates eight angles, called transversal angles.

We usually number these angles 1 through 8 for easy reference.

These angle pairs have special properties that are useful in solving geometry problems.

Remember:

• Parallel lines never intersect
• A transversal is a line that intersects parallel lines
• A transversal creates eight angles with special relationships

1. Corresponding Angles

When a transversal intersects two parallel lines, it creates pairs of corresponding angles. Corresponding angles:

1. Are on the same side of the transversal
2. Have sides on the transversal going in the same direction
3. Have other sides going in parallel directions

In this figure, ∠2 and ∠6 are corresponding angles. So are ∠3 and ∠7, ∠1 and ∠5, and ∠4 and ∠8.

Corresponding Angle Theorem

The Corresponding Angle Theorem states:

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

This means corresponding angles have the same measure.

For example, if m∠2 = 30°, then m∠6 = 30° as well.

Example Problems

1. If l1 || l2 and m∠6 = 70°, find m∠8.

• ∠6 and ∠8 are corresponding angles, so they are congruent.
• Therefore, m∠8 = 70°.

2. In the same figure, if m∠8 = 70°, find m∠3.

• ∠3 and ∠8 are vertical angles, so they are congruent.
• Therefore, m∠3 = 70°.

3. Again, if m∠8 = 70°, find m∠7.

• ∠7 and ∠8 form a linear pair, so they are supplementary.
• Supplementary angles add up to 180°.
• So, m∠7 + 70° = 180°
• Solve for m∠7: m∠7 = 180° – 70° = 110°

Remember:

• Corresponding angles are on the same side of the transversal and “match” in position
• Corresponding angles are congruent (they have the same measure)
• You can use the Corresponding Angle Theorem to solve problems involving parallel lines and transversals

2. Alternate Interior Angles

When a transversal intersects two parallel lines, it creates pairs of alternate interior angles. Alternate interior angles:

1. Are between the parallel lines (in the “interior”)
2. Are on opposite sides of the transversal

In this figure, ∠3 and ∠6 are alternate interior angles. So are ∠4 and ∠5.

The “S” Pattern

Alternate interior angles form an “S” or “Z” pattern across the transversal.

This can help you identify them quickly.

Alternate Interior Angle Theorem

The Alternate Interior Angle Theorem states:

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

This means alternate interior angles have the same measure.

For example, if m∠3 = 50°, then m∠6 = 50° as well.

Example Problems

1. If l1 || l2 and m∠4 = 80°, find m∠5.

• ∠4 and ∠5 are alternate interior angles, so they are congruent.
• Therefore, m∠5 = 80°.

2. In the same figure, if m∠3 = 60°, find m∠6.

• ∠3 and ∠6 are alternate interior angles, so they are congruent.
• Therefore, m∠6 = 60°.

Remember:

• Alternate interior angles are between the parallel lines and on opposite sides of the transversal
• They form an “S” or “Z” pattern
• Alternate interior angles are congruent (they have the same measure)
• You can use the Alternate Interior Angle Theorem to solve problems involving parallel lines and transversals

3. Alternate Exterior Angles

When a transversal intersects two parallel lines, it creates pairs of alternate exterior angles. Alternate exterior angles:

1. Are outside the parallel lines (in the “exterior”)
2. Are on opposite sides of the transversal

In this figure, ∠1 and ∠8 are alternate exterior angles. So are ∠2 and ∠7.

Alternate Exterior Angle Theorem

The Alternate Exterior Angle Theorem states:

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

This means alternate exterior angles have the same measure.

For example, if m∠1 = 50°, then m∠8 = 50° as well.

Example Problem

If l1 || l2 and m∠1 = 80°, find m∠2, m∠3, m∠6, and m∠8.

1. To find m∠2:

• ∠1 and ∠2 form a linear pair, so they are supplementary.
• Supplementary angles add up to 180°.
• So, 80° + m∠2 = 180°
• Solve for m∠2: m∠2 = 180° – 80° = 100°

2. To find m∠3:

• ∠1 and ∠3 are corresponding angles, so they are congruent.
• Therefore, m∠3 = 80°.

3. To find m∠6, you can use either ∠1 or ∠2:

• Using ∠1: ∠1 and ∠6 are vertical angles, so they are congruent. Therefore, m∠6 = 80°.
• Using ∠2: ∠2 and ∠6 form a linear pair, so they are supplementary. So, m∠6 = 180° – m∠2 = 180° – 100° = 80°.

4. To find m∠8:

• ∠1 and ∠8 are alternate exterior angles, so they are congruent.
• Therefore, m∠8 = 80°.

Remember:

• Alternate exterior angles are outside the parallel lines and on opposite sides of the transversal
• Alternate exterior angles are congruent (they have the same measure)
• You can use the Alternate Exterior Angle Theorem to solve problems involving parallel lines and transversals

Interior Angles of a Polygon

As we saw earlier, a triangle has three interior angles. Let’s call them ∠A, ∠B, and ∠C.

Triangle Sum Theorem

The Triangle Sum Theorem states:

The sum of the measures of the interior angles of any triangle is always 180°.

In symbols:

m∠A + m∠B + m∠C = 180°

This is true for all triangles, regardless of who draws them or what they look like.

Example Problem

If m∠ABC = 3x + 15, m∠ACB = x + 20, and m∠BAC = x, find the value of x.

1. We know the sum of the measures of the interior angles is 180°.
2. So, (3x + 15) + (x + 20) + x = 180°
3. Simplify: 5x + 35 = 180°
4. Subtract 35 from both sides: 5x = 145
5. Divide both sides by 5: x = 29

Therefore, the value of x is 29.

Remember:

• Interior angles are the angles inside a polygon
• In a triangle, the sum of the measures of the interior angles is always 180°
• You can use the Triangle Sum Theorem to set up an equation and solve for unknown values

The General Formula for Polygon Angles

In a polygon with n sides, there are also n interior angles. For example:

• A triangle (3 sides) has 3 interior angles
• A square (4 sides) has 4 interior angles
• A pentagon (5 sides) has 5 interior angles

We can find the sum of the interior angles of any polygon using this formula:

Sum of interior angles = 180(n – 2)

where n is the number of sides of the polygon.

The formula is based on the idea that any polygon can be divided into triangles by drawing diagonals from one vertex. The number of triangles is always 2 less than the number of sides (n – 2).

Since the sum of the angles in each triangle is 180°, multiplying 180° by the number of triangles (n – 2) gives the sum of all the interior angles.

Example Problem

Find the sum of the interior angles of a dodecagon (12-sided polygon).

1. In this case, n = 12.
2. Plug n into the formula: Sum of interior angles = 180(12 – 2)
3. Simplify: Sum of interior angles = 180(10) = 1800°

Therefore, the sum of the interior angles of a dodecagon is 1800°.

Remember:

• The number of interior angles in a polygon equals the number of sides
• You can find the sum of the interior angles of any polygon using the formula 180(n – 2), where n is the number of sides
• The formula is based on dividing the polygon into triangles, each with an angle sum of 180°

Measure a Regular Polygon’s Interior Angles

A regular polygon is a polygon where all sides are equal in length and all interior angles are equal in measure. Examples include equilateral triangles, squares, and regular pentagons.

For a regular polygon with n sides, the measure of each interior angle can be calculated using this formula:

Measure of each interior angle = (180(n – 2)) ÷ n

where n is the number of sides of the polygon.

1. The formula (180(n – 2)) gives the sum of all interior angles in the polygon.
2. Dividing this sum by n gives the measure of each individual interior angle, since they are all equal in a regular polygon.

Example 1: Equilateral Triangle

An equilateral triangle has 3 sides (n = 3). Using the formula:

Measure of each interior angle = (180(3 – 2)) ÷ 3 = 180 ÷ 3 = 60°

So each interior angle of an equilateral triangle measures 60°.

Example 2: Regular Decagon

A regular decagon has 10 sides (n = 10). Using the formula:

Measure of each interior angle = (180(10 – 2)) ÷ 10 = 1440 ÷ 10 = 144°

So each interior angle of a regular decagon measures 144°.

Remember:

• This formula only works for regular polygons, where all sides and angles are equal
• First calculate the sum of all interior angles using (180(n – 2))
• Then divide by n to find the measure of each individual angle

Perpendicularity

Two lines are perpendicular if they intersect at a right angle (90°). In other words, when two lines cross and form four 90° angles, they are perpendicular.

Perpendicular lines form a “T shape” or an inverted “T shape.” This is an easy way to spot them.

The symbol ⊥ is used to show that two lines are perpendicular. For example, if lines l1 and l2 are perpendicular, we write: l1 ⊥ l2.

When two lines are perpendicular:

1. They form four right angles, each measuring 90°.
2. The angles are linear pairs, meaning they share a common side and their other sides form a straight line.

Example Problem

Lines l1 and l2 are perpendicular. If angle 1 measures x°, and angle 2 measures (x + 40)°, find the value of x.

1. Since the angles are linear pairs, they are supplementary. Supplementary angles add up to 180°.
2. So, x + (x + 40) = 180°
3. Simplify: 2x + 40 = 180°
4. Subtract 40 from both sides: 2x = 140°
5. Divide both sides by 2: x = 70°

Therefore, the value of x is 70°.

Remember:

• Perpendicular lines intersect at a 90° angle
• They form a “T shape” or inverted “T shape”
• The symbol ⊥ means perpendicular
• Angles formed by perpendicular lines are right angles and linear pairs