Similarity Chapter 8

Similar Polygons I can find corresponding lengths, perimeters, and areas in similar polygons.

Similar Polygons Vocabulary (page 223 in Student Journal) similar figures (~): figures with the same shape, but different size

Similar Polygons Core Concepts (page 223 and 224 in Student Journal) Corresponding Parts of Similar Polygons corresponding angles are congruent corresponding sides are proportional (the ratio of any 2 corresponding sides is equal to the scale factor) Perimeters of Similar Polygons Theorem (Theorem 8.1) If 2 polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

Similar Polygons Areas of Similar Polygons Theorem (Theorem 8.2) If 2 polygons are similar, then the ratio of their areas is equal to the square of the ratios of their corresponding side lengths.

Similar Polygons Examples (page 225 in Student Journal) #2) The polygons are similar. Find the value of x.

Similar Polygons Solution #2) x = 10

Similar Polygons ABCDE ~ KLMNP #3) Find the scale factor from KLMNP to ABCDE. #7) Find the ratio of the perimeters from ABCDE to KLMNP. #8) Find the ratio of the areas from ABCDE to KLMNP.

Similar Polygons Solutions #3) 3/2 #7) 2/3 #8) 4/9

Similar Polygons Additional Example (space on pages 223 and 224 in Student Journal) a) Determine if GNMH is similar to MLKH. Explain.

Similar Polygons Solution a) yes, all corresponding angles are congruent and all corresponding side lengths are proportional with a scale factor of 2.5

Proving Triangle Similarity by AA I can use the Angle-Angle Similarity Theorem.

Proving Triangle Similarity by AA Core Concepts (page 228 in Student Journal) Angle-Angle Similarity Theorem (AA~) (Theorem 8.3) If 2 angles in 1 triangle are congruent to 2 angles in another triangle, then the triangles are similar.

Proving Triangle Similarity by AA Examples (pages 229 and 230 in Student Journal) Determine if the triangles are similar. If so, write a similarity statement. If not, explain. #1)

Proving Triangle Similarity by AA Solution #1) yes, by AA~ triangle ABC is similar to triangle EFD

Proving Triangle Similarity by AA Show (prove) that the 2 triangles are similar. #3)

Proving Triangle Similarity by AA Solution #3) angle C is congruent to angle C (reflexive), angle E is congruent to angle BDC (corresponding angles), triangle ACE is similar to triangle BCD (AA~)

Proving Triangle Similarity by AA Additional Example (space on page 228 in Student Journal) a) A flag pole casts a shadow 45 feet long. At the same time a boy, who is 5’8” tall casts a shadow that is 51 inches long. How tall is the flag pole?

Proving Triangle Similarity by AA Solution a) 60 feet

Proving Triangle Similarity by SSS and SAS I can use the Side-Side-Side Similarity Theorem and the Side-Angle-Side Similarity Theorem.

Proving Triangle Similarity by SSS and SAS Core Concepts (pages 233 and 234 in Student Journal) Side-Side-Side Similarity Theorem (SSS~) (Theorem 8.4) If the corresponding sides of 2 triangles are proportional, then the triangles are similar. Side-Angle-Side Similarity Theorem (SAS~) (Theorem 8.5) If an angle in 1 triangle is congruent to an angle in another triangle and the sides that include the 2 angles are proportional, then the triangles are similar.

Proving Triangle Similarity by SSS and SAS Examples (pages 234 and 235 in Student Journal) #1) Determine whether triangle RST is similar to triangle ABC.

Proving Triangle Similarity by SSS and SAS Solution #1) no, the corresponding sides are not proportional

Proving Triangle Similarity by SSS and SAS #3) Find the value of x that makes triangle RST similar to triangle HGK.

Proving Triangle Similarity by SSS and SAS Solution #3) x = -3

Proving Triangle Similarity by SSS and SAS Additional Example (spaces on pages 233 and 234 in Student Journal) a) The drawing represents a scale drawing of a triangular roof truss. The actual lengths of the 2 upper sides are 18 feet and 40 feet, with an included angle of 110 degrees. Is the scale drawing similar to the actual truss?

Proving Triangle Similarity by SSS and SAS Solution a) no, the sides are not proportional

Proportionality Theorems I can use the Triangle Proportionality Theorem and its converse.

Proportions in Triangles Core Concepts (pages 238 and 239 in Student Journal) Triangle Proportionality Theorem (Theorem 8.6) If a line parallel to 1 side of a triangle intersects the other 2 sides, then it divides the 2 sides proportionally. Converse of the Triangle Proportionality Theorem (Theorem 8.7) If a line divides 2 sides of a triangle proportionally, then it is parallel to the 3rd side.

Proportions in Triangles Three Parallel Lines Theorem (Theorem 8.8) If 3 parallel lines intersect 2 transversals, then they divide the transversals proportionally. Triangle Angle Bisector Theorem (Theorem 8.9) If a ray bisects an angle of a triangle, then it divides the opposite side into 2 segments that are proportional to the other 2 sides of the triangle.

Proportions in Triangles Examples (pages 239 and 240 in Student Journal) #2) Find the length of segment AB.

Proportions in Triangles Solution #2) 14/9

Proportions in Triangles Determine if segment AB is parallel to segment XY. #4)

Proportions in Triangles Solution #4) The ratios of corresponding lengths are proportional, so by the Converse of the Triangle Proportionality Theorem the segments are parallel.

Proportions in Triangles #5) Use the diagram to complete the proportion.

Proportions in Triangles Solution #5) XZ

Proportions in Triangles #8) Find the value of x.

Proportions in Triangles Solution #8) x = 3