UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 7: Proving Similarity Instruction

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Amanda Stanley
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1 Prerequisite Skills This lesson requires the use of the following skills: identifying similar triangles using similarity statements to find unknown lengths and measures of similar triangles using the distance formula to find lengths of sides of triangles working with and simplifying square roots using the Pythagorean Theorem Introduction Design, architecture, carpentry, surveillance, and many other fields rely on an understanding of the properties of similar triangles. Being able to determine if triangles are similar and understanding their properties can help you solve real-world problems. Key Concepts Similarity Similarity statements include Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS). These statements allow us to prove triangles are similar. Similar triangles have corresponding sides that are proportional. It is important to note that while both similarity and congruence statements include an SSS and an SAS statement, the statements do not mean the same thing. Similar triangles have corresponding sides that are proportional, whereas congruent triangles have corresponding sides that are of the same length. Triangle Theorems The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally. This theorem can be used to find the lengths of various sides or portions of sides of a triangle. It is also true that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. U1-436

2 The Triangle Angle Bisector Theorem states if one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle. The Pythagorean Theorem, written symbolically as a 2 + b 2 = c 2, is often used to find the lengths of the sides of a right triangle, which is a triangle that includes one 90 angle. Drawing the altitude, the segment from the right angle perpendicular to the line containing the opposite side, creates two smaller right triangles that are similar. Common Errors/Misconceptions misidentifying congruent angles because of the orientation of the triangles incorrectly creating proportions between corresponding sides assuming a line parallel to one side of a triangle bisects the remaining sides rather than creating proportional sides misidentifying the altitudes of triangles incorrectly simplifying expressions with square roots U1-437

3 Guided Practice Example 1 A meterstick casts a shadow 65 centimeters long. At the same time, a tree casts a shadow 2.6 meters long. How tall is the tree? 1. Draw a picture to understand the information. x 1 m 65 cm 2.6 m U1-438

4 2. Determine if the triangles are similar. The rays of the sun create the shadows, which are considered to be parallel. Right angles are formed between the ground and the meterstick as well as the ground and the tree. x 1 m 65 cm 2.6 m Two angles of the triangles are congruent; therefore, by Angle-Angle Similarity, the triangles are similar. U1-439

5 3. Solve the problem. Convert all measurements to the same units. x 1 m 0.65 m Similar triangles have proportional sides. 2.6 m Create a proportion to find the height of the tree. height of stick height of tree = length of stick sshadow Create a proportion. length of tree s shadow = Substitute known values. x 2.6 (1)(2.6) = (0.65)(x) Find the cross products. 2.6 = 0.65x Simplify. x = 4 Solve for x. The height of the tree is 4 meters. U1-440

6 Example 2 Finding the distance across a canyon can often be difficult. A drawing of similar triangles can be used to make this task easier. Use the diagram to determine AR, the distance across the canyon. R x A 180 m B 90 m C 75 m D 1. Interpret the given information. A person standing at point A can sight a rock across the canyon at point R. Point C is selected so that CA is perpendicular to AR, the distance across the canyon. Point D is selected so that CD is perpendicular to CA and can be easily measured. The point of intersection of RD and CA, point B, can then be found. U1-441

7 2. Determine if the triangles are similar. A and C both measure 90 and are congruent. RBA DBC By the Angle-Angle Similarity Statement, RBA DBC. 3. Solve the problem. Similar triangles have proportional sides. Create a proportion to find the distance across the canyon. AB AR = Create a proportion. BC CD 180 = x Substitute known values (180)(75) = (90)(x) Find the cross products. 13,500 = 90x Simplify. x = 150 Solve for x. The distance across the canyon is 150 meters. U1-442

8 Example 3 To find the distance across a pond, Rita climbs a 30-foot observation tower on the shore of the pond and locates points A and B so that AC is perpendicular to CB. She then finds the measure of DB to be 12 feet. What is the measure of AD, the distance across the pond? C 30 ft A x D 12 ft B 1. Determine if the triangles are similar. ABC is a right triangle with C the right angle. CD is the altitude of ABC, creating two similar triangles, ACD and CBD. ABC ACD CBD 2. Solve the problem. Similar triangles have proportional sides. Create a proportion to find the distance across the pond. BD CD = Create a proportion. CD AD = Substitute known values. 30 x (12)(x) = (30)(30) Find the cross products. 12x = 900 Simplify. x = 75 Solve for x. The distance across the pond is 75 feet. U1-443

9 Example 4 To estimate the height of an overhang, a surveyor positions herself so that her line of sight to the top of the overhang and her line of sight to the bottom form a right angle. What is the height of the overhang to the nearest tenth of a meter? B D 1.75 m 8.5 m C A 1. Determine if the triangles are similar. ABC is a right triangle with C the right angle. CD is the altitude of ABC, creating two similar triangles, ACD and CBD. ABC ACD CBD U1-444

2. Solve the problem. Similar triangles have proportional sides. Create a proportion to find the height of the overhang. AD CD = CD BD Create a proportion. 1.75 8.5 = 8.5 x Substitute known values.

10 2. Solve the problem. Similar triangles have proportional sides. Create a proportion to find the height of the overhang. AD CD = CD BD Create a proportion = 8.5 x Substitute known values. (1.75)(x) = (8.5)(8.5) Find the cross products. 1.75x = Simplify. x 41.3 Solve for x. The length of BD is approximately 41.3 meters; however, the measure of the overhang is represented by AB. Find the length of AB = The height of the overhang is approximately 43.1 meters. U1-445

UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 6: Defining and Applying Similarity Instruction

UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 6: Defining and Applying Similarity Instruction Prerequisite Skills This lesson requires the use of the following skills: understanding that the sum of the measures of the angles in a triangle is 180 identifying both corresponding and congruent parts