8-3 Similar Triangles notes

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent Triangles8-3 Similar TrianglesEmphasize the importance of labeling ∆’s with sides in same “position” (in name).Warm-up1. Do the triangles ABC and DEF have the same size and shape? YesBE1245°111245°1160° 75°60° 75°AC DF122. Do triangles MNP and RST have the same size? NoThe same shape? Yes3. Name the triangles in this figure. AEB and ADC4. Find the unknown angle measure.X = 42.5, ∠A = 85°, ∠B = 42.5°, ∠C = 52.5°Today we will:1. Apply the properties of similar triangles.2. Prove that triangles are similar.Next class we will:1. Continue with 8-38-3 Similar Triangles

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent Triangles8-3 Similar TrianglesCongruent Triangles - Triangles in which corresponding parts, sides andangles, are equal in measure.Example∆ABC ≅ ∆DEF; AB = DE; BC = EF; AC = DFm∠A = m∠D; m∠B = m∠E; m∠C = m∠FSimilar Triangles - Triangles in which corresponding angles are equal inmeasure and corresponding sides are in proportion.The ratio of their measures are all equal.Example∆UVW ~ ∆XYZm∠U = m∠X; m∠V = m∠Y; m∠W = m∠ZUV VW UW= =XY YZ XZSide note: ≅ means congruent and ~ means similar.Triangle similarity postulateIf two angles of one triangle are equal in measure to two angles of anothertriangle, then the two triangles are similar. AA (Angle Angle) similarity.Example∆ABC ~ ∆DEF∆ABC ~ ∆EBD8-3 Similar Triangles

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesExample 1: For each pair of triangles, tell whether the triangles are similar.a. b. c.Solution 1In order for 2 triangles to be similar, we need two pairs of angles to be equal, AAsimilarity.a. No, because we only have one pair of angles that are equal.b. Yes, ∆MON ~ ∆QOP because m∠N = m∠P and m∠MON = m∠QOP(vertical angles)c. Yes, ∆RST ~ ∆UVW by Triangle Sum Theorem. m∠T = 75° so ∠T =∠W and ∠R = ∠U.Example 2∆JKL and ∆PQR are similar. Complete and solve each proportion.a. 10 x10 y= b. =20 ?20 ?Solution 2a. In the similarity, JLcorresponds to PR and JKcorresponds to PQ.PR PQ=JL JK10 x=20 3020x= 300x = 15J3020K25LPx10▲JKL~ ▲PQRQRyb. In the similarity, QR corresponds to KL and JL corresponds to PR.PR QR=JL KL10 y=20 2520y= 250y = 12.58-3 Similar Triangles

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesOverlapping similar trianglesWe can separate these ∆s to better see the corresponding parts.XXPQYXZYZPQOverlapping Similar Triangles TheoremIf a line is drawn from a point on one side of a triangle parallel to anotherside, then it forms a triangle similar to the original triangle.Example 3Find the measure of VW.Solution 3Since VW || YZ, ∆VWX ~ ∆ZYX.VW/ZY = XW/XYVW 22=21 2828VW= 462VW = 16.5ZV21X22W6YThe measure of VW is 16.5 units.8-3 Similar Triangles

Mrs. Aitken’s Integrated 2Unit 8 Similar and Congruent TrianglesExample 4: Find the measure of AD.Solution 4Use a proportion because DE II BC so ∆ABC ~ ∆ADE.AD AE=AB ACx 6=12 2424x= 72x = 3So, AD = 3Example 5Prove the Overlapping Similar Triangles TheoremGiven: ∆ABCDE ll BCProve: ∆ADE ~ ∆ABCSolution 5Statement1. ∆ABC; DE II BC 1. Given2. m∠ADE = m∠ABCm∠AED = m∠ACBJustification2. If 2 parallel lines cut by a transversal,then corresponding angles are equal.(Can be used for parallel lines and ∆s.)3. ∆ADE ~ ∆ABC 3. AA Similarity (Triangle SimilarityPostulate.8-3 Similar Triangles