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Similar Figures

Study Guide

Key Definition

Two figures are similar if they have the same shape, their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal.

Important Notes

  • Similar figures have corresponding angles that are congruent.
  • The scale factor is the ratio of any two corresponding lengths.
  • $\sim$ means 'is similar to'.
  • When $ABC \sim DEF$, $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$.
  • Ratios of corresponding sides are equal: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}$.

Mathematical Notation

$\sim$ means 'is similar to'$\cong$ means 'is congruent to'$\angle$ denotes an angle$\frac{a}{b}$ represents a fraction$\sqrt{x}$ represents a square rootRemember to use proper notation when solving problems

Why It Works

The concept of similarity helps us understand proportional relationships in geometry, allowing us to apply known measurements to unknown ones using scale factors.

Remember

In similar figures, all corresponding angles are equal, and all corresponding side lengths are proportional.

Quick Reference

Corresponding Angles:$\angle A \cong \angle D$Corresponding Sides:$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}$

Understanding Similar Figures

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

Similar figures have the same shape with proportional sides.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Which of the following ratios could represent the scale factor between two congruent figures?

A$1 : 2 : 3$B$2 : 3 : 5$C$3 : 4 : 5$D$1 : 1 : 1$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You have a photo that you want to enlarge. The original dimensions are 4 inches by 6 inches. If the enlarged photo is 8 inches wide, what will be its height?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Prove that two triangles with two pairs of congruent angles are similar.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Given $\triangle ABC \sim \triangle DEF$, if $AB = 6$, $BC = 8$, and $DE = 9$, find $EF$.

A$12$B$15$C$18$D$10$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

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