Integrated Math 2 / Unit 6: Similarity
6.7 Similarity Proofs
Notes
A proof lists statements with reasons. To prove triangles similar, show congruent angles or proportional sides.
- Useful angle reasons: vertical angles are congruent, alternate interior angles are congruent when lines are parallel, corresponding angles are congruent when lines are parallel, and a shared angle is congruent to itself (reflexive).
- Once two pairs of angles are congruent, conclude by AA.
- Then use "corresponding sides of similar triangles are proportional" to find lengths.
- Example: segments $AB \parallel CD$ cross at $E$. $\angle AEB \cong \angle CED$ (vertical angles) and $\angle BAE \cong \angle DCE$ (alternate interior angles), so $\triangle ABE \sim \triangle CDE$ by AA.
Practice
Lines $AD$ and $BC$ cross at $E$. Why is $\angle AEB \cong \angle DEC$?
Angles opposite each other at an intersection are vertical angles, which are congruent.
In a proof, two pairs of corresponding angles in two triangles are shown congruent. Which reason finishes the proof that the triangles are similar?
Two congruent angle pairs is the AA similarity criterion.
$DE \parallel BC$ in triangle $ABC$ with $D$ on $AB$ and $E$ on $AC$. Which reason shows $\angle ADE \cong \angle ABC$?
$DE$ and $BC$ are parallel and cut by transversal $AB$, so corresponding angles are congruent.
Ready for more? Take the Unit 6 test.