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GEOMETRY MODULE 2 LESSON 14-17 SIMILAR TRIANGLES Similar Triangles: Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent. Notation: ~ Similarity of triangles areβ¦ ο· Reflexive: A triangle is similar to itself. βπ΄π΅πΆ ~ βπ΄π΅πΆ ο· Symmetric: If βπ΄π΅πΆ ~ βπ·πΈπΉ, then βπ·πΈπΉ~βπ΄π΅πΆ. ο· Transitive: If βπ΄π΅πΆ ~ βπ·πΈπΉ and βπ·πΈπΉ~βπΊπ»πΌ, then βπ΄π΅πΆ~βπΊπ»πΌ. CRITERION FOR TWO TRIANGLES TO BE SIMILAR ο· Angle-Angle (AA) ο· Side-Side-Side (SSS) ο· Side-Angle-Side (SAS) AA Criterion If the two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. β π΄ β β π· and β π΅ β β πΈ, so βπ΄π΅πΆ ~ βπ·πΈπΉ. SSS Criterion If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. π΄π΅ π·πΈ π΅πΆ π΄πΆ = πΈπΉ = π·πΉ , so βπ΄π΅πΆ ~ βπ·πΈπΉ. SAS Criterion MOD2 L14-17 1 If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. π΄π΅ π·πΈ π΄πΆ = π·πΉ and β π΄ β β π·, so βπ΄π΅πΆ ~ βπ·πΈπΉ. PRACTICE Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and side-length measures. 1. x y The missing angles are calculated using Sum of the Angles in a Triangle= 180°. We find that there are indeed two pairs of equal corresponding angles. By AA criterion, the triangles are similar. To find the missing sides requires a bit more work. First set up the proportions. 6 7 π₯ = = 8 28 π¦ 3 π₯ 0.75 = π¦ Since x is opposite the smallest angle, we need to choose a value for x that is smaller than 6 and 7. 5 Let π₯ = 5. Therefore, π¦ = 0.75 = MOD2 L14-17 20 3 2 2. 4 3.14 π₯ = = 12 π¦ 16.5 1 3.14 1 π₯ = πππ = 3 π¦ 3 16.5 π¦ = 3(3.14) = 9.42 π₯= 16.5 3 = 5.5 Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 1. A. B. π΄π΅ π΄πΆ π΅πΆ = β² β²= β² β² π΄π΅ π΄πΆ π΅β²πΆβ² 2.82 5.4 7 = = 1.41 2.7 3.5 2 = 2 = 2 βπ΄π΅πΆ~βπ΄β² π΅ β² πΆ β² ππ¦ πππ πΆπππ‘πππππ π΄π΅ π΄πΆ π΅πΆ = = πΈπΉ π·πΉ π·πΈβ² 1 5.83 6.4 = = 0.68 2.13 2.42 1.47 β 2.74 β 2.64 The triangles are not similar. MOD2 L14-17 3 2. A. π΄π΅ πΆπ΅ = π·πΈ πΈπΉ 3 3.13 = 1.5 1.565 2 = 2 βπ΄π΅πΆ~βπ·πΈπΉ ππ¦ ππ΄π πΆπππ‘πππππ B. π΄πΈ π΅πΈ = π·πΈ πΈπΆ 2.24 3.16 = 4.48 6.32 .5 = .5 βπ΄πΈπ΅~βπ·πΈπΆ ππ¦ ππ΄π πΆπππ‘πππππ A Brian is photographing the Washington Monument and wonders how tall the monument is. Brian places his 5ft. camera tripod approximately 100yd. from the base of the monument. Lying on the ground, he visually aligns the top of his tripod with the monument and marks his location on the ground approximately 2ft. 9in from the center of his tripod. Use Brianβs measurements to approximate the height B of the Washington Monument. Currently, the given values are presented in three different units. We must convert to the same unit. We will convert to feet. 100π¦π = 300ππ‘ and 2ππ‘ 9ππ = 2.75ππ‘ E D π΄πΈ πΈπΆ = π΅π· πΆπ· π΄πΈ 302.75 = 5 2.75 π΄πΈ = (302.75)(5) = 550.5 ππ‘ 2.75 MOD2 L14-17 4 C ON YOUR OWN 1. Catarinaβs boat has come untied and floated A away on the lake. She is standing atop a cliff that is 35ft above the water in a lake. If she stands 10ft. from the edge of the cliff, she C B can visually align the top of the cliff with the water at the back of her boat. Her eye level is 5 ½ ft. above the ground. Approximately how far out from the cliff is Catarinaβs boat? D E π΄πΈ π΅πΆ = πΆπ· π·πΈ 5.5 10 = 35 π·πΈ π΄πΈ = 350 = 63.6 ππ‘ 5.5 MOD2 L14-17 5 Μ Μ Μ Μ β₯ ππ Μ Μ Μ Μ Μ and ππ Μ Μ Μ Μ Μ β₯ ππ Μ Μ Μ Μ . Given the diagram to the right, ππ Show that βπππ~βπππ. STEP JUSTIFICATION Μ Μ Μ Μ β₯ ππ Μ Μ Μ Μ Μ and ππ Μ Μ Μ Μ Μ β₯ ππ Μ Μ Μ Μ ππ Given β πππ β β πππ Right angles = 90° so angles equal to each other β π β β π Both triangles share β π , so Reflexive βπππ~βπππ AA Criterion Are the triangles shown similar? Explain. If the triangles are similar, write the similarity statement. Yes, the triangles are similar. βπ΄π΅πΆ~βπ΄π·πΈ by AA because β π΄π·πΈ β β π΄π΅πΆ, and both triangles share β π΄. MOD2 L14-17 6