Sum and Difference of Angles Practice Problems & Solutions

Master angle relationships with practice problems on corresponding angles, alternate angles, adjacent angles, and parallel lines. Step-by-step solutions included.

📚Master Angle Relationships Through Targeted Practice
  • Identify and calculate corresponding angles between parallel lines
  • Solve for unknown angles using alternate angle properties
  • Apply adjacent angle relationships to find missing measurements
  • Work with vertically opposite angles in intersection problems
  • Calculate angles in triangles using parallel line theorems
  • Distinguish between acute, obtuse, right, and straight angles in complex figures

Understanding Types of Angles

Complete explanation with examples

What is an angle?

Definition: Angles are created at the intersection between two lines. As seen in the following illustration

Angles are created at the intersection between two lines

The angle in the illustration is called AB AB . We could also call it angle ∢ABC \sphericalangle ABC . The important thing is that the middle letter is the one at the intersection of the lines.

For example, in this case:

We could also call it angle ABC

The angle is ∢BCD \sphericalangle BCD or ∢DCB \sphericalangle DCB . Both notations are correct for the same angle.

We usually mark the angle with an arc as follows:

The angle is BCD

The marked angle is ∡ABC ∡ABC . Sometimes we will denote angles using Greek letters, for example:

α α or β β

Before the name of the angle, we should note the angle symbol, like this:

∡ ∡

Together it looks like this:

∡CBA ∡CBA or ∡α ∡α

Next, we will delve into the size of angles, the different types, and those that are created when a line intersects two parallel lines.

Detailed explanation

Practice Types of Angles

Test your knowledge with 27 quizzes

Find the size of angle \( \alpha \).

27.727.727.7AAABBBCCC41

Examples with solutions for Types of Angles

Step-by-step solutions included
Exercise #1

What type of angle is α \alpha ?

αα

Step-by-Step Solution

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

Answer:

Straight

Exercise #2

What is the size of the missing angle?

80

Step-by-Step Solution

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is 180∘180^\circ. Given that one angle is 80∘80^\circ, we can calculate the missing angle using the following steps:

  • Step 1: Recognize that the given angle α=80∘\alpha = 80^\circ and the missing angle β\beta form a straight line.
  • Step 2: Use the angle sum property for a straight line: α+β=180∘ \alpha + \beta = 180^\circ
  • Step 3: Substitute the known value: 80∘+β=180∘ 80^\circ + \beta = 180^\circ
  • Step 4: Solve for the missing angle β\beta: β=180∘−80∘=100∘ \beta = 180^\circ - 80^\circ = 100^\circ

Therefore, the size of the missing angle is 100∘100^\circ.

Answer:

100°

Video Solution
Exercise #3

Indicates which angle is greater

Step-by-Step Solution

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Answer:

Video Solution
Exercise #4

Indicates which angle is greater

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer:

Video Solution
Exercise #5

Which angle is greater?

Step-by-Step Solution

The angle in diagram (a) is more acute, meaning it is smaller:

Conversely, the angle in diagram (b) is more obtuse, making it larger.

Answer:

Video Solution

Frequently Asked Questions

Everything you need to know about Types of Angles

What are corresponding angles and how do I identify them?

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Corresponding angles are formed when a transversal cuts two parallel lines, positioned on the same side of the transversal and at the same level relative to each parallel line. They are always equal in measure. Look for angles that occupy the same relative position at each intersection point.

How do I solve problems involving alternate angles?

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Alternate angles are equal when formed by parallel lines and a transversal, but they're on opposite sides of the transversal and at different levels. To solve: 1) Identify the parallel lines, 2) Locate the transversal, 3) Find angles on opposite sides that aren't at the same level, 4) Set them equal and solve.

What's the difference between adjacent angles and vertically opposite angles?

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Adjacent angles share a common vertex and side, and their measures add up to 180° when they form a straight line. Vertically opposite angles are formed by two intersecting lines and face each other across the intersection point - they are always equal in measure.

How do I calculate missing angles in triangles with parallel lines?

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Use these steps: 1) Identify corresponding or alternate angles created by parallel lines, 2) Transfer known angle measures using these relationships, 3) Apply the triangle angle sum theorem (angles total 180°), 4) Set up an equation and solve for the unknown angle.

What are the key angle relationships I need to memorize?

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Essential relationships include: • Corresponding angles are equal (parallel lines) • Alternate angles are equal (parallel lines) • Adjacent angles sum to 180° on a straight line • Vertically opposite angles are equal • Triangle angles sum to 180° • Right angles measure exactly 90°

How do I know when to use angle properties with parallel lines?

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Look for these indicators in problems: parallel line symbols (∥), statements mentioning 'parallel lines,' diagrams showing lines that don't intersect, or problems asking about 'corresponding' or 'alternate' angles. These signal that you should apply parallel line angle theorems.

What's the easiest way to remember acute, right, obtuse, and straight angles?

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Use this memory system: Acute angles are 'cute and small' (less than 90°), Right angles form perfect corners (exactly 90°), Obtuse angles are 'obese and big' (between 90° and 180°), Straight angles form straight lines (exactly 180°). Visual practice helps reinforce these concepts.

How do I solve complex angle problems step by step?

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Follow this systematic approach: 1) Read the problem carefully and identify given information, 2) Draw or label the diagram clearly, 3) Identify which angle relationships apply, 4) Write equations based on these relationships, 5) Solve algebraically, 6) Check your answer makes geometric sense.

More Types of Angles Questions

Click on any question to see the complete solution with step-by-step explanations

Sum and Difference of Angles

Triangle Angle Analysis: Can 30°, 60°, and 90° Form a Valid Triangle? Triangle Angle Verification: Do 56°, 89°, and 17° Form a Valid Triangle? Triangle Angle Problem: Can 90, 115, and 35 Degrees Form a Triangle? Triangle Classification: Analyzing Angles 53°, 117°, and 21° Calculate Angle ABC in Parallelogram with Given Angles 40° and 44°

Continue Your Math Journey

Master Types of Angles first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Right angleAcute AnglesObtuse AnglePlane angleTypes of angles (right, acute, obtuse, straight)Angle NotationAngle Bisector

Topics Learned in Later Sections

Sides, Vertices, and AnglesSum and Difference of AnglesSum of Angles in a PolygonThe Sum of the Interior Angles of a TriangleExterior angles of a triangle

Practice by Question Type

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