Similarity and Congruence
Determine Similar Figures
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Question 1 of 3
1. Question
Are the two triangles similar?Hint
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Help VideoSimilar triangles have corresponding side lengths which are proportional (equal). This rule is called SideSideSide `(SSS)` Similarity.First, let’s label the values of each side lengthFor triangle `ABC`:`AB=6, ``BC=12, ``AC=9`For triangle `DEF`:`DE=4, ``EF=8, ``DF=6`To prove that `ABC` is similar to `DEF` the following equation must be true$$\frac{\color\green{AB}}{\color\green{DE}}=\frac{\color{#004ec4}{BC}}{\color{#004ec4}{EF}}=\frac{\color{#e85e00}{AC}}{\color{#e85e00}{DF}}$$ Definition of Similarity $$\frac{\color\green{6}}{\color\green{4}}=\frac{\color{#004ec4}{12}}{\color{#004ec4}{8}}=\frac{\color{#e85e00}{9}}{\color{#e85e00}{6}}$$ Plug in the side lengths $$\frac{\color\green{3}}{\color\green{2}}=\frac{\color{#004ec4}{3}}{\color{#004ec4}{2}}=\frac{\color{#e85e00}{3}}{\color{#e85e00}{2}}$$ Simplify each fraction Since corresponding side lengths are proportional, `ABC` is similar to `DEF`Yes, `ABC` is similar to `DEF` 
Question 2 of 3
2. Question
Are the two triangles similar?Hint
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Help VideoSimilar triangles have corresponding angles which are proportional (equal). This rule is called AngleAngleAngle `(A A A)` Similarity.First, take note of equal angles from the diagram.To prove that `ABC` is similar to `DEF`, check for equal angles.`angle A` `=` `angle D` as shown by the dot `angle B` `=` `angle E` as shown by the purple mark `angle C` `=` `angle F` Angle Sum of Triangle Since all angles are equal, `ABC` and `DEF` are equiangular and therefore similar.Yes, `ABC` is similar to `DEF` 
Question 3 of 3
3. Question
Are the two triangles similar?Hint
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Help VideoIf two triangles have two corresponding side lengths and an included angle to be proportional, then the triangles are similar. This rule is called SideAngleSide `(SAS)` Similarity.First, lets label the values of each side lengthFor triangle `ABC`:`AB=8, ``AC=17.6`For triangle `DEF`:`DE=5, ``DF=11``angle A = angle D` as shown by the dotsTo prove that `ABC` is similar to `DEF` the following equation must be true$$\frac{\color\green{AB}}{\color\green{DE}}=\frac{\color{#e85e00}{AC}}{\color{#e85e00}{DF}}$$ Definition of Similarity $$\frac{\color\green{8}}{\color\green{5}}=\frac{\color{#e85e00}{17.6}}{\color{#e85e00}{11}}$$ Plug in the side lengths $$\frac{\color\green{8}}{\color\green{5}}=\frac{\color{#e85e00}{8}}{\color{#e85e00}{5}}$$ Simplify each fraction Since corresponding side lengths are proportional, and included angles are equal, `ABC` is similar to `DEF`Yes, `ABC` is similar to `DEF`