7-3 Additional Practice Proving Triangles Similar

7+3 Additional Practice Proving Triangles Similar: Mastering Similarity Theorems

Proving triangle similarity is a cornerstone of geometry, opening doors to solving complex problems involving proportions, lengths, and areas. While mastering the three primary similarity postulates – AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side) – is crucial, deepening your understanding requires extensive practice. This article provides seven core examples followed by three challenging bonus problems, designed to solidify your grasp of triangle similarity theorems and equip you with the skills to tackle any problem thrown your way.

Understanding the Three Similarity Postulates: A Quick Recap

Before diving into the practice problems, let's briefly review the three postulates that form the bedrock of triangle similarity proofs:

• AA (Angle-Angle Similarity Postulate): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Remember, if two angles are congruent, the third angle must also be congruent due to the Angle Sum Theorem (the sum of angles in a triangle is 180°).

• SAS (Side-Angle-Side Similarity Postulate): If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. The key here is the included angle – the angle formed by the two proportional sides.

• SSS (Side-Side-Side Similarity Postulate): If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. All three sides must be in the same proportion.

7 Core Practice Problems: Proving Triangle Similarity

Let's tackle seven problems illustrating the application of these postulates. Each problem will detail the steps required for a successful proof.

Problem 1: AA Similarity

Given: In ΔABC and ΔDEF, ∠A ≅ ∠D and ∠B ≅ ∠E.

Prove: ΔABC ~ ΔDEF (ΔABC is similar to ΔDEF)

Solution: Since ∠A ≅ ∠D and ∠B ≅ ∠E, by the AA Similarity Postulate, ΔABC ~ ΔDEF.

Problem 2: SAS Similarity

Given: In ΔABC and ΔXYZ, AB/XY = BC/YZ = 2/3, and ∠B ≅ ∠Y.

Prove: ΔABC ~ ΔXYZ

Solution: We are given that two pairs of sides are proportional (AB/XY = BC/YZ = 2/3) and the included angles ∠B and ∠Y are congruent. Therefore, by the SAS Similarity Postulate, ΔABC ~ ΔXYZ.

Problem 3: SSS Similarity

Given: In ΔPQR and ΔSTU, PQ/ST = QR/TU = RP/US = 1/2.

Prove: ΔPQR ~ ΔSTU

Solution: All three pairs of corresponding sides are proportional (PQ/ST = QR/TU = RP/US = 1/2). Therefore, by the SSS Similarity Postulate, ΔPQR ~ ΔSTU.

Problem 4: Combining AA and Proportions

Given: In ΔLMN and ΔOPQ, ∠L ≅ ∠O and LM/OP = LN/OQ = 1.5.

Prove: ΔLMN ~ ΔOPQ

Solution: We're given that ∠L ≅ ∠O. Additionally, we have the proportion LM/OP = LN/OQ. While this isn't a full SSS or SAS, because we already have one pair of congruent angles, we can use the AA postulate indirectly. If we can prove another pair of angles congruent (or the third pair by the angle sum theorem), we have similarity. In this case, since the sides LM and LN are proportional to OP and OQ, the triangles are similar by SAS (or you could argue that the triangles are similar by AA if the proportionality implies the angle at M is equal to the angle at P).

Problem 5: Using Auxiliary Lines

Given: Two parallel lines intersected by two transversals. The transversals create segments AB and CD on one parallel line and segments EF and GH on the other parallel line.

Prove: ΔABE ~ ΔCDG

Solution: Since the lines are parallel, we can use the properties of parallel lines and transversals to show that ∠ABE ≅ ∠CDG (corresponding angles) and ∠AEB ≅ ∠CGD (corresponding angles). By AA similarity, ΔABE ~ ΔCDG.

Problem 6: Triangles within Triangles

Given: ΔABC with a line segment DE parallel to BC, where D is on AB and E is on AC.

Prove: ΔADE ~ ΔABC

Solution: Because DE || BC, ∠ADE ≅ ∠ABC and ∠AED ≅ ∠ACB (corresponding angles). By AA similarity, ΔADE ~ ΔABC.

Problem 7: Right Triangles and Altitudes

Given: A right-angled triangle with an altitude drawn to the hypotenuse.

Prove: The two smaller triangles formed are similar to each other and to the original triangle.

Solution: This involves multiple similarity proofs. Let the right-angled triangle be ABC, with the altitude AD drawn to the hypotenuse BC. By considering the angles in each of the three triangles (ABC, ADB, ADC), you can apply AA similarity to show that ΔADB ~ ΔABC ~ ΔADC. This is a classic example illustrating how altitudes in right triangles create similar triangles.

3 Bonus Challenge Problems: Stepping Up Your Game

These problems require a deeper understanding and application of similarity postulates and theorems.

Bonus Problem 1: Indirect Proof

Given: In ΔABC, AB/AC = BC/AB. Prove that ∠BAC = 2∠ABC.

Solution: This problem requires an indirect proof approach. Assume that ∠BAC ≠ 2∠ABC. Using the Law of Sines and the given proportion, manipulate the equations to demonstrate a contradiction, thus proving the initial assumption incorrect and confirming that ∠BAC = 2∠ABC.

Bonus Problem 2: Using Similar Triangles to Find Lengths

Given: A triangle with sides of length 6, 8, and 10. A smaller similar triangle is formed within the larger triangle, with its longest side having length 4. Find the lengths of the other two sides of the smaller triangle.

Solution: Since the triangles are similar, the ratios of corresponding sides are equal. Set up proportions using the given information to solve for the lengths of the other two sides of the smaller triangle.

Bonus Problem 3: Area and Similarity

Given: Two similar triangles. The ratio of their corresponding sides is 2:3. What is the ratio of their areas?

Solution: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of the areas is (2/3)² = 4/9.

Conclusion: Mastering Triangle Similarity Through Practice

Proving triangle similarity is a skill honed through consistent practice. By working through these examples, you've built a strong foundation in understanding and applying the AA, SAS, and SSS postulates. Remember that complex problems often require a combination of these postulates and clever use of geometric properties. Don't hesitate to revisit these problems, and continue seeking out more challenging examples to further refine your skills in this crucial area of geometry. The more you practice, the more intuitive and confident you will become in tackling any triangle similarity problem.