#Geometry Study Guide: Congruence, Similarity, and Angles 📐

Hey there! Let's get you prepped for the SAT Math section with a supercharged review of congruence, similarity, and angle relationships. This is your go-to guide for tonight—let's make every minute count!

#Foundational Concepts: Why These Topics Matter

These concepts are the bedrock of geometry. They're not just about memorizing rules; they're about understanding how shapes interact and how to solve real-world problems. You'll see these ideas pop up in various forms on the SAT, so mastering them is key. Let's dive in!

#Congruent Figures: Exact Copies

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• What it is: Congruent figures are identical in both size and shape. Think of them as perfect copies of each other.
• Symbol: We use the symbol ≅ to denote congruence (e.g., ΔABC ≅ ΔXYZ).
• Transformations: Congruent figures can be mapped onto each other using rigid transformations (translations, rotations, and reflections) without any change in size or shape.

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These are the shortcuts to proving that two figures are congruent:

• SSS (Side-Side-Side): If all three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.
• AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.

#Properties of Congruent Figures

• Corresponding Sides: Have equal lengths.
• Corresponding Angles: Have equal measures.
• Rigid Transformations: Congruent figures remain identical in size and shape under rigid transformations.

#Similar Figures: Scaled Versions

#Defining Similarity

• What it is: Similar figures have the same shape but can be different sizes. They are essentially scaled versions of each other.
• Symbol: We use the symbol ~ to denote similarity (e.g., ΔABC ~ ΔXYZ).
• Scale Factor (k): The ratio of the lengths of corresponding sides in similar figures. This is your key to scaling up or down!

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Here's how to prove that two figures are similar:

• AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
• SSS (Side-Side-Side): If the ratios of the lengths of corresponding sides of two triangles are equal, then the triangles are similar.
• SAS (Side-Angle-Side): If the ratios of the lengths of two corresponding sides are equal and the included angles are equal, then the triangles are similar.

#Applying Similarity

• Calculating Dimensions: Use the scale factor to find missing side lengths. If a side in Figure A is 'x' and the corresponding side in Figure B is 'y', then x = k * y.
• Area Relationships: If the scale factor is 'k', the ratio of areas is k². (Area of Figure A = k² × Area of Figure B).
• Volume Relationships (3D Figures): If the scale factor is 'k', the ratio of volumes is k³. (Volume of Figure A = k³ × Volume of Figure B).
• Solving Problems: Set up proportions using corresponding sides and the scale factor to solve for unknowns.

#Angle Relationships: Lines and Triangles

#Intersecting Lines

• Four Angles: Two intersecting lines create four angles.
• Vertical Angles: Angles opposite each other are congruent (equal).
• Triangle Sum: The sum of the angles in any triangle is always 180°.
• Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

#Parallel Lines and Transversals

• Corresponding Angles: When a transversal intersects parallel lines, corresponding angles are congruent.
• Alternate Interior Angles: These are also congruent when a transversal intersects parallel lines.
• Alternate Exterior Angles: These are congruent as well.
• Same-Side Interior Angles: These angles are supplementary (add up to 180°).
• Same-Side Exterior Angles: Also supplementary.

#Right Triangles

• Hypotenuse: The side opposite the right angle.
• Legs: The two sides that form the right angle.
• Pythagorean Theorem: $$a^2 + b^2 = c^2$$ (where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse).

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Okay, here's what to really focus on for the exam:

• Congruence vs. Similarity: Know the difference and when to apply each concept.
• Angle Relationships: Master the rules for parallel lines and transversals, and the properties of triangles.
• Scale Factor: Understand how to use it to find missing lengths, areas, and volumes.
• Pythagorean Theorem: Essential for right triangle problems.

#Last-Minute Tips

• Time Management: Don't get bogged down on one question. If you're stuck, move on and come back later.
• Common Pitfalls: Watch out for misinterpreting diagrams or mixing up congruence and similarity.
• Strategy: Read the question carefully, draw diagrams if needed, and use your formulas wisely.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. In the figure below, lines m and n are parallel. If angle 1 measures 110 degrees, what is the measure of angle 7?

   (A) 70 degrees (B) 110 degrees (C) 20 degrees (D) 90 degrees

   

2. Triangle ABC is similar to triangle DEF. If AB = 6, DE = 9, and BC = 8, what is the length of EF?

   (A) 10 (B) 12 (C) 14 (D) 16

3. Two triangles are congruent if:

   (A) They have the same shape (B) They have the same size (C) They have the same size and shape (D) They have corresponding angles equal

#Free Response Question

Triangle ABC has vertices A(1, 2), B(4, 6), and C(1, 6). Triangle DEF has vertices D(5, 2), E(11, 10), and F(5, 10).

(a) Show that triangle ABC is a right triangle. [2 points]

(b) Show that triangle DEF is a right triangle. [2 points]

(c) Determine if triangle ABC is similar to triangle DEF. Justify your answer. [3 points]

(d) If the two triangles are similar, find the scale factor. [2 points]

Scoring Breakdown:

(a) Calculate the slopes of AB and AC. If the slopes are negative reciprocals, then the lines are perpendicular, and the triangle is a right triangle.
* Slope of AB = (6-2) / (4-1) = 4/3 * Slope of AC = (6-2) / (1-1) = undefined * Slope of BC = (6-6) / (1-4) = 0 * Since AB and BC are perpendicular, triangle ABC is a right triangle. (1 point for correct slopes, 1 point for correct conclusion)

(b) Calculate the slopes of DE and DF. If the slopes are negative reciprocals, then the lines are perpendicular, and the triangle is a right triangle. * Slope of DE = (10-2) / (11-5) = 8/6 = 4/3 * Slope of DF = (10-2) / (5-5) = undefined * Slope of EF = (10-10) / (5-11) = 0 * Since DE and EF are perpendicular, triangle DEF is a right triangle. (1 point for correct slopes, 1 point for correct conclusion)

(c) Compare the ratios of corresponding sides. If the ratios are equal, then the triangles are similar. * Length of AB = √((4-1)² + (6-2)²) = √(9+16) = √25 = 5 * Length of AC = √((1-1)² + (6-2)²) = √16 = 4 * Length of BC = √((4-1)² + (6-6)²) = √9 = 3 * Length of DE = √((11-5)² + (10-2)²) = √(36+64) = √100 = 10 * Length of DF = √((5-5)² + (10-2)²) = √64 = 8 * Length of EF = √((11-5)² + (10-10)²) = √36 = 6 * Ratio of AB to DE = 5/10 = 1/2 * Ratio of AC to DF = 4/8 = 1/2 * Ratio of BC to EF = 3/6 = 1/2 * Since all ratios are equal, triangle ABC is similar to triangle DEF. (1 point for correct side lengths, 1 point for correct ratios, 1 point for correct conclusion)

(d) The scale factor is the ratio of corresponding sides. From part (c), the scale factor is 1/2. (1 point for correct scale factor, 1 point for correct conclusion)

You've got this! Go into the exam with confidence, knowing you're well-prepared. Good luck! 🚀