Similar Triangles Proofs Worksheet With Answers – Easy Guide
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In this article, we will explore the concept of similar triangles, which is an essential topic in geometry. Similar triangles play a significant role in understanding proportional relationships and solving various real-life problems. This guide will provide you with an easy-to-follow worksheet filled with proofs and solutions to reinforce your understanding of this topic. 📐
What are Similar Triangles? 🤔
Similar triangles are triangles that have the same shape but may differ in size. This means that their corresponding angles are equal, and their corresponding sides are in proportion. If two triangles are similar, we can express this as:
- Triangle ABC ∼ Triangle DEF (read as "Triangle ABC is similar to Triangle DEF").
Key Properties of Similar Triangles
- Angle-Angle (AA) Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- Side-Side-Side (SSS) Criterion: If the corresponding sides of two triangles are in proportion, then the triangles are similar.
- Side-Angle-Side (SAS) Criterion: If one angle of a triangle is equal to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.
The Importance of Similar Triangles in Geometry 📏
Understanding similar triangles is crucial for several reasons:
- Problem Solving: They can be used to find unknown lengths and angles in various geometric shapes.
- Real-Life Applications: Similar triangles are used in various fields such as architecture, engineering, and even art.
- Foundation for Trigonometry: Many trigonometric concepts rely on the properties of similar triangles.
Similar Triangles Proofs Worksheet 📝
To solidify your understanding of similar triangles, we have prepared a worksheet that includes several proofs. Below are some exercises to help you practice:
Exercise 1: Prove Similarity using the AA Criterion
Given: Triangle ABC and Triangle DEF where ∠A = ∠D = 50° and ∠B = ∠E = 70°.
Proof:
- Since ∠A = ∠D and ∠B = ∠E, we can conclude that ∠C = ∠F (because the sum of angles in a triangle is 180°).
- Thus, by the AA criterion, Triangle ABC ∼ Triangle DEF.
Exercise 2: Prove Similarity using the SSS Criterion
Given: Triangle GHI and Triangle JKL where GH = 4 cm, HI = 6 cm, and GI = 5 cm. Also, JK = 8 cm, KL = 12 cm, and JL = 10 cm.
Proof:
-
Find the ratio of corresponding sides:
- GH : JK = 4 : 8 = 1 : 2
- HI : KL = 6 : 12 = 1 : 2
- GI : JL = 5 : 10 = 1 : 2
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All ratios are equal; thus, by the SSS criterion, Triangle GHI ∼ Triangle JKL.
Exercise 3: Prove Similarity using the SAS Criterion
Given: Triangle MNO and Triangle PQR where ∠M = ∠P = 40°, MN = 6 cm, and PQ = 9 cm. NO = 4 cm and QR = x cm.
Proof:
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First, use the ratio of the sides that include the equal angles:
- MN : PQ = 6 : 9 = 2 : 3
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According to the SAS criterion, if the sides MN and PQ are proportional, then the triangles are similar.
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If NO : QR = 4 : x, to maintain similarity, we know 2/3 must hold:
- Set up the proportion: 4/x = 2/3
- Cross-multiply: 2x = 12 → x = 6 cm
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Therefore, Triangle MNO ∼ Triangle PQR.
Answer Key for the Worksheet 📋
| Exercise | Proof Type | Conclusion |
|---|---|---|
| 1 | AA | Triangle ABC ∼ Triangle DEF |
| 2 | SSS | Triangle GHI ∼ Triangle JKL |
| 3 | SAS | Triangle MNO ∼ Triangle PQR |
Important Notes to Remember 🔑
- Always ensure that the corresponding angles are equal when determining similarity using the AA criterion.
- When using the SSS criterion, all ratios of corresponding sides must be equal.
- For the SAS criterion, verify that one pair of angles is equal, and the sides adjacent to those angles maintain a consistent ratio.
Conclusion
Understanding similar triangles and their properties is vital in the study of geometry. With the exercises provided in this worksheet and the solutions offered, you can reinforce your knowledge and prepare yourself for more advanced mathematical concepts. Use the above proofs as a guide, and remember to practice regularly. Happy studying! 📚