Parallel Lines and Proportional Parts Similar Triangles Parallel Lines and Proportional Parts

Angle-Angle (AA) Similariry If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. No, they are not similar; the angles are not congruent

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Yes; △LPQ ≅ △LJK

Side-Side-Side (SSS) Similarity If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

Side-Angle-Side (SAS) Similarity If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Yes; △JKL ∼ △QPM; SSS, ratio 4:3

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Examples Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Yes; △WTZ ∼ △WYX; SAS, ratio 1:2

Examples Find each measure. QP and MP

Examples Find each measure. QP and MP 5/(5+x) = 6/(9 3/5) 48 = 30+6x QP = 3; MP = 8

Examples Find each measure. WR and RT

Examples Find each measure. WR and RT (x+6)/8 = (2x+6)/10 WR = 8, RT = 10

Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths.

Converse of Triangle Proportionality Theorem If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.

Examples If PS = 12.5, SR = 5, and PT = 15, find TQ.

Examples If PS = 12.5, SR = 5, and PT = 15, find TQ. 12.5/5 = 15/x x = 6; TQ = 6

Examples In △DEF, DG is half the length of GF, EH = 6, and HF = 10. Is DE ∥ GH?

Examples In △DEF, DG is half the length of GF, EH = 6, and HF = 10. Is DE ∥ GH? No; GF:DG = 2:1; HF:EH = 5:3

Midsegment A midsegment of a triangle is a segment with endpoints that are the midpoints of two sides of the triangle. Every triangle has three midsegments.

Triangle Midsegment Theorem A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side.

Examples Find the measure. DE

Examples Find the measure. DE 15/2 = 7.5

Examples Find the measure. DB

Examples Find the measure. DB FE = ½ AB DB = ½ AB FE = DB = 9.2

Examples Find the measure. m∠FED

Examples Find the measure. m∠FED FE ∥ AB ∠FED ≅ ∠EDB; Alt. Int. ∠

Proportional Parts of Parallel Lines If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Congruent Parts of Parallel Lines If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Examples Find x.

Examples Find x. 7x-2 = 4x+3 3x = 5 x = 5/3

Examples Find x.

Examples Find x. 2x+1 = 3x-5 6 = x