AA Similarity

Grade 8 · geometry · 100 practice problems · read aloud

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AA Similarity: Angle-Angle Similarity

🔍 What is AA Similarity?

AA Similarity is a shortcut to prove two triangles are similar. If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar. This is useful because similar triangles have proportional sides, which helps us find missing lengths without measuring directly!

🧩 Step-by-Step Guide

  1. Identify: Look at the two triangles. Find two pairs of angles that you know are equal.
  2. Check: Confirm the angle measures are congruent. Remember, the sum of all angles in a triangle is always 180°.
  3. State: Write your similarity statement using the ∼ symbol (e.g., ΔABC ∼ ΔXYZ).
  4. Use: If needed, set up proportions with corresponding sides to solve for a missing side length.

📐 Visual Examples

Example 1: Are the triangles similar?

Triangle 1: Angles 50° and 60°.
Triangle 2: Angles 50° and 60°.

Step 1: Both triangles have a 50° angle and a 60° angle.
Step 2: The third angle in both must be 70° (because 180 - 50 - 60 = 70).
Step 3: Yes! By AA Similarity, the triangles are similar.

Example 2: Find the missing side.

Given ΔABC ∼ ΔDEF, where AB=4, BC=6, and the corresponding side DE=10. Find EF.

Step 1: Set up a proportion with corresponding sides: AB/DE = BC/EF.
Step 2: Plug in the known values: 4/10 = 6/EF.
Step 3: Cross-multiply: 4 * EF = 60.
Step 4: Solve: EF = 15.

⚠️ Common Mistakes

Matching the wrong angles: Always match corresponding vertices in the correct order when writing your similarity statement (e.g., if ∠A ≅ ∠D and ∠B ≅ ∠E, then ΔABC ∼ ΔDEF).

Assuming it's AAA: You only need two pairs of congruent angles. The third pair is automatically congruent because the angles in a triangle always add up to 180°.

Incorrect proportions: When setting up proportions, make sure corresponding sides are matched correctly. The first triangle's sides go on top (or bottom) consistently.

💡 Tips & Tricks

  • Look for shared angles or parallel lines, which often create equal corresponding angles.
  • Remember the "A" team: You only need two A's (Angles) to prove similarity.
  • If you know one angle in each triangle is 90°, you only need to find one more pair of congruent acute angles to use AA.

🎯 Practice Suggestions

  • Draw your own triangles with given angle measures and test the AA rule.
  • Find real-world examples, like shadows or ramps, where similar triangles are used.
  • Practice writing clear similarity statements and setting up proportions from them.
  • Use online quizzes or flashcards to quickly identify which triangles are similar by AA.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 ∛(512) = ?

Hint: Think about what number multiplied by itself three times gives the value inside the cube root symbol.

Show the answer

Answer: 8

  1. Factor 512 into smaller numbers. 512 is even, so divide by 2: 512 ÷ 2 = 256 256 ÷ 2 = 128 128 ÷ 2 = 64 64 ÷ 2 = 32 32 ÷ 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1 So, 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 That is 2^9.
  2. Cube root means exponent divided by 3. So ∛(2^9) = 2^(9/3) = 2^3.
  3. Compute 2^3 = 8.
  4. Check: 8 × 8 × 8 = 64 × 8 = 512. Correct. Thus, the cube root of 512 is 8. ANSWER: 8

We are asked to find the cube root of 512. That means we want a number \( x \) such that \( x^3 = 512 \).

2 2x + 5 = 13

Hint: To solve linear equations, isolate the variable by performing inverse operations on both sides of the equation.

Show the answer

Answer: 4

  1. Subtract 5 from both sides of the equation. This is done to isolate the term with the variable (2x) on one side. 2x + 5 - 5 = 13 - 5 2x = 8
  2. Divide both sides by 2. This is done to solve for x. 2x / 2 = 8 / 2 x = 4 So the solution is x = 4.

We are solving the equation: 2x + 5 = 13

3 ∛(8 × 27) = ?

Hint: First find the product inside the cube root, then determine what number cubed gives that result.

Show the answer

Answer: 6

  1. Understand the problem We need to find the cube root of the product of 8 and 27. That is: cube root of (8 × 27).
  2. Multiply inside the cube root 8 × 27 = 216 So the problem becomes: cube root of 216.
  3. Find the cube root of 216 We need a number that, when multiplied by itself three times, gives 216. Let’s check possible numbers: - 5 × 5 × 5 = 125 (too small) - 6 × 6 × 6 = 36 × 6 = 216 (correct)
  4. Conclusion Since 6 × 6 × 6 = 216, the cube root of 216 is 6. Final answer: 6

4 √(64) + 3² = ?

Hint: First evaluate the square root of a perfect square, then calculate the exponent, and finally add the results together.

Show the answer

Answer: 17

  1. Identify the operations in the problem. The expression is: square root of 64 plus 3 squared. That is: √(64) + 3²
  2. Calculate the square root of 64. The square root of 64 is the number that, when multiplied by itself, gives 64. Since 8 × 8 = 64, we have √(64) = 8.
  3. Calculate 3 squared. 3 squared means 3 × 3. 3 × 3 = 9, so 3² = 9.
  4. Add the two results. From
  5. 8 From
  6. 9 8 + 9 = 17
  7. State the final answer. The result is 17.

5 √(64) + 3² - 5 = ?

Hint: Remember to follow the order of operations: evaluate exponents and roots before addition and subtraction. For example, in an expression like √(25) + 2², you would first calculate the square root and the exponent.

Show the answer

Answer: 12

  1. Evaluate the square root √(64) means the positive square root of 64. Since 8 × 8 = 64, we have √(64) = 8.
  2. Evaluate the exponent 3² means 3 × 3 = 9.
  3. Substitute back into the expression The expression becomes: 8 + 9 − 5.
  4. Perform addition and subtraction from left to right First, 8 + 9 = 17. Then, 17 − 5 = 12. Final Answer: 12

Let's solve step-by-step:

6 √(64) + 3² - 2³ = ?

Hint: Remember to evaluate each term separately before combining them. Start with the square root, then the exponents, and finally perform the addition and subtraction in order.

Show the answer

Answer: 9

  1. Evaluate the square root √(64) means the positive square root of 64. Since 8 × 8 = 64, we have √(64) = 8.
  2. Evaluate the exponent 3² 3² means 3 × 3 = 9.
  3. Evaluate the exponent 2³ 2³ means 2 × 2 × 2 = 8.
  4. Substitute back into the expression Original: √(64) + 3² − 2³ Substitute: 8 + 9 − 8
  5. Perform the addition and subtraction from left to right First: 8 + 9 = 17 Then: 17 − 8 = 9 Final answer: 9

Let's solve step-by-step.

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