6-3/6-4: Proving Triangles Similar Geometry Chapter 6 6-3/6-4: Proving Triangles Similar

Warm-Up For the given polygons, a.) Determine whether the polygons are similar b.) If the polygons are similar, write a similarity statement and find the scale factor. 𝑨 𝟐𝟏 𝑩 𝑪 𝑫 𝟏𝟓 𝟐𝟒 𝟏𝟐 𝑾 𝑿 𝒀 𝒁 𝟐𝟖 𝟐𝟎 𝟑𝟐 𝟏𝟔

Warm-Up In the diagram, ∆𝑫𝑬𝑭 ~ ∆𝑴𝑵𝑷. Find the value of x. 𝑴 𝑵 𝑷 𝟐𝟒 𝟏𝟖 𝟑𝟎 𝑫 𝑬 𝑭 𝟔 𝟖 𝒙

Proving Triangles Similar Objective: Students will be able to use theorems and postulates to prove that triangles are similar. Agenda Postulates/Theorems for Similarity Practice

Proving Similar Polygons To identify that two polygons are similar, then you must identify that:

Definition: Similar Polygons To identify that two polygons are similar, then you must identify that: 1.) Corresponding angles are congruent

Definition: Similar Polygons To identify that two polygons are similar, then you must identify that: 1.) Corresponding angles are congruent 2.) Corresponding sides are in proportion (i.e. Their side lengths have the same ratio.)

Definition: Similar Polygons To identify that two polygons are similar, then you must identify that: 1.) Corresponding angles are congruent 2.) Corresponding sides are in proportion (i.e. Their side lengths have the same ratio.) *For Triangles, however, we can prove that they are similar more quickly with the following:

A Postulate for Similar Triangles Postulate 22 – AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 𝒀 𝑿 𝒁 If <𝑿 ≅ <𝑱 And <𝒀≅ <𝑲 𝑲 𝑱 𝑳 Then ∆𝑱𝑲𝑳 ~ ∆𝑿𝒀𝒁

Theorems for Similar Triangles Theorem 6.2 – SSS Similarity Theorem: If the sides of two triangles are in proportion, then the triangles are similar. If 𝑴𝑵 𝑻𝑼 = 𝑵𝑷 𝑼𝑽 = 𝑴𝑷 𝑻𝑽 𝑻 𝑼 𝑽 𝑴 𝑵 𝑷 Then ∆𝑴𝑵𝑷 ~ ∆𝑻𝑼𝑽

Theorems for Similar Triangles Theorem 6.3 – SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. 𝑫 𝑬 𝑭 If <𝑨≅ <𝑫 𝑨𝑩 𝑫𝑬 = 𝑨𝑪 𝑫𝑭 𝑨 𝑩 𝑪 Then ∆𝑨𝑩𝑪 ~ ∆𝑫𝑬𝑭

Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 6 9 12 𝑪 𝑫 𝑬

Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬

Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬 𝑲𝑵 𝑫𝑬 = 𝟐𝟏 𝟗 = 𝟕 𝟑

Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 𝑴𝑲 𝑪𝑫 = 𝟏𝟒 𝟔 = 𝟕 𝟑 6 9 12 𝑪 𝑫 𝑬 𝑲𝑵 𝑫𝑬 = 𝟐𝟏 𝟗 = 𝟕 𝟑 𝑴𝑵 𝑪𝑬 = 𝟐𝟖 𝟏𝟐 = 𝟕 𝟑

Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 14 21 28 𝑴 𝑲 𝑵 6 9 12 𝑪 𝑫 𝑬 𝑴𝑲 𝑪𝑫 = 𝑲𝑵 𝑫𝑬 = 𝑴𝑵 𝑪𝑬 = 𝟕 𝟑 ∆𝑴𝑲𝑵 ~ ∆𝑪𝑫𝑬 By SSS Similarity

Example 2 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫 𝟐𝟔° 𝑯 𝑮 𝑲 𝟔𝟒°

Example 2 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. <𝑫 ≅ <𝑮 (Both Right Angles) 𝑬 𝑪 𝑫 𝟐𝟔° 𝑯 𝑮 𝑲 𝟔𝟒°

Example 2 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫 𝟐𝟔° <𝑫 ≅ <𝑮 (Both Right Angles) <𝑪≅ <𝑲 (Why?) <𝑬≅ <𝑯 (Why?) 𝑯 𝑮 𝑲 𝟔𝟒°

Example 2 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑬 𝑪 𝑫 𝟐𝟔° ∆𝑫𝑬𝑪 ~ ∆𝑮𝑯𝑲 By AA Similarity 𝑯 𝑮 𝑲 𝟔𝟒°

Example 3 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪 16 24 32 𝑴 𝑵 𝑷

Example 3 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪 𝑨𝑪 𝑵𝑷 = 𝟐𝟎 𝟑𝟐 = 𝟓 𝟖 16 24 32 𝑴 𝑵 𝑷 𝑩𝑪 𝑴𝑷 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑 𝑨𝑩 𝑴𝑵 = 𝟏𝟎 𝟏𝟔 = 𝟓 𝟖

Example 3 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 10 20 16 𝑨 𝑩 𝑪 𝑨𝑪 𝑵𝑷 = 𝟐𝟎 𝟑𝟐 = 𝟓 𝟖 16 24 32 𝑴 𝑵 𝑷 𝑩𝑪 𝑴𝑷 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑 𝑨𝑩 𝑴𝑵 = 𝟏𝟎 𝟏𝟔 = 𝟓 𝟖 The Triangles are NOT Similar

Example 4 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾

Example 4 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾 𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑

Example 4 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾 𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑 <𝑷𝑼𝑹 ≅ <𝑾𝑼𝑻 (Why?)

Example 4 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑼 𝑷 𝑻 𝑹 𝑾 ∆𝑷𝑼𝑹 ~ ∆𝑾𝑼𝑻 By SAS Similarity

Example 5 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁

Example 5 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑷𝑼 𝑾𝑼 = 𝟏𝟐 𝟗 = 𝟒 𝟑 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁 𝑹𝑼 𝑻𝑼 = 𝟖 𝟔 = 𝟒 𝟑 <𝑷𝑼𝑹 ≅ <𝑾𝑼𝑻 However…

Example 5 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 8 9 6 𝑿 𝑾 𝒀 𝑻 𝒁 The congruent angles do not fall between the proportional sides. Therefore, the Triangles are NOT Similar.

Example 6 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭

Example 6 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭 <𝑨𝑪𝑩 ≅ <𝑭𝑪𝑫 (Why?) <𝑨 ≅ <𝑭 (Why?)

Example 6 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨 𝑩 𝑪 𝑫 𝑭 ∆𝑨𝑪𝑩 ~ ∆𝑭𝑪𝑫 By AA Similarity

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱 𝑸 𝑹 𝑺

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱 𝑸 𝑹 𝑺 <𝑰 ≅ <𝑺 (Both 𝟕𝟎° Angles) <𝑱≅ <𝑹 (Why?) <𝑯≅ <𝑸 (Why?)

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑯 𝑰 𝑱 <𝑰 ≅ <𝑺 (Both 𝟕𝟎° Angles) <𝑱≅ <𝑹 <𝑯≅ <𝑸 𝑸 𝑹 𝑺 ∆𝑯𝑰𝑱 ~ ∆𝑸𝑺𝑹 By AA Similarity

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬 𝟓𝟔° 𝑸 𝑹 𝑷 𝟑𝟑°

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬 𝟓𝟔° Only 1 pair of angles are congruent. Not enough for AA. 𝑸 𝑹 𝑷 𝟑𝟑° The Triangles are NOT Similar

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑭 𝑮 𝑬 𝟓𝟔° <𝑬≅ <𝑹 (Both 𝟗𝟎° Angles) 𝒎<𝑭=𝟓𝟔°; 𝒎<𝑮=𝟑𝟒° 𝒎<𝑸=𝟓𝟕°; 𝒎<𝑷=𝟑𝟑° 𝑸 𝑹 𝑷 𝟑𝟑°

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮

Final Practice 𝑮𝑰 𝑯𝑰 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑 𝑮𝑯 𝑯𝑲 = 𝟐𝟎 𝟑𝟎 = 𝟐 𝟑 Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮 𝑮𝑰 𝑯𝑰 = 𝟏𝟔 𝟐𝟒 = 𝟐 𝟑 𝑮𝑯 𝑯𝑲 = 𝟐𝟎 𝟑𝟎 = 𝟐 𝟑 𝑯𝑰 𝑰𝑲 = 𝟐𝟒 𝟑𝟔 = 𝟐 𝟑

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 16 24 30 36 20 𝑯 𝑰 𝑲 𝑮 ∆𝑯𝑰𝑮 ~ ∆𝑰𝑲𝑯 By SSS Similarity

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 12 32° 𝑨 𝑪 𝑫 18 125° 22 23° 𝑶 𝑵 𝑷 33

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. 𝑨𝑪 𝑶𝑵 = 𝟏𝟐 𝟐𝟐 = 𝟔 𝟏𝟏 12 32° 𝑨 𝑪 𝑫 18 125° 22 23° 𝑶 𝑵 𝑷 33 𝑪𝑫 𝑵𝑷 = 𝟏𝟖 𝟑𝟑 = 𝟔 𝟏𝟏 <𝑪 ≅ <𝑵 (Why?)

Final Practice Determine whether the triangles are similar. If they are, write a similarity statement. Justify your choice with a postulate or theorem. ∆𝑨𝑪𝑫 ~ ∆𝑶𝑵𝑷 By SAS Similarity 12 32° 𝑨 𝑪 𝑫 18 125° 22 23° 𝑶 𝑵 𝑷 33