Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "SAS similarity"

(Created page with "One of the main 3 main theorems for proving the similarity between <math>2</math> triangles. Similarity (specifically for triangles here) means all the angles in these triangl...")
 
 
Line 1: Line 1:
−
One of the main 3 main theorems for proving the similarity between <math>2</math> triangles. Similarity (specifically for triangles here) means all the angles in these triangles are equal and all sides are proportional to each other. SAS similarity is a similarity theorem stating that if there are two proportional sides and an angle in between it is equal, then they are similar by SAS similarity.
+
One of the main <math>3</math> main theorems for proving the similarity between <math>2</math> triangles. Similarity (specifically for triangles here) means all the angles in these triangles are equal and all sides are proportional to each other. SAS similarity is a similarity theorem stating that if there are two proportional sides and an angle in between it is equal, then they are similar by SAS similarity.
  
 
Example Problem: There are two isosceles triangles, <math>\triangle ABC</math> and <math>\triangle DEF</math>.  
 
Example Problem: There are two isosceles triangles, <math>\triangle ABC</math> and <math>\triangle DEF</math>.  
−
<math>m\angle B = m\angle E</math> = 32^\circ<math>, and </math>\overline{AB} = 9<math>, </math>\overline{DE} = 3<math>, </math>\overline{BC} = 2<math>, </math>\overline{EF} = 6<math>. What are the measures of </math>m\angle BAC, m\angle BCA, m\angle EDF<math> and </math>m\angle EFD<math> (All of these are base angles for their respective triangles)?
+
<math>m\angle B = m\angle E= 32^\circ</math>, and <math>\overline{AB} = 9</math>, <math>\overline{DE} = 3</math>, <math>\overline{BC} = 2</math>, <math>\overline{EF} = 6</math>. What are the measures of <math>m\angle BAC, m\angle BCA, m\angle EDF</math> and <math>m\angle EFD</math> (All of these are base angles for their respective triangles)?
  
−
Example Solution: Since sides </math>\overline{AB}<math> and </math>\overline{DE}<math> with their common ratio being </math>3<math>, and </math>\overline{EF}<math> and </math>\overline{BC}<math> also having a common ratio of </math>3<math>, and both of them sharing a common angle </math>\angle B<math> and </math>\angle E<math>, </math>\triangle ABC<math> and </math>\triangle DEF<math> are similar by SAS Similarity. Now, if </math>2<math> triangles are similar, all their angles are equal, which means that </math>m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD<math> since they are all base angles of similar triangles. If one of the non-base angles is </math>32^\circ<math>, it means that </math>\frac{(180-32)}{2}<math>, which is equal to </math>\frac{148}{2}<math>, which is </math>74^\circ<math>. So </math>m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD = 74^\circ$.
+
Example Solution: Since sides <math>\overline{AB}</math> and <math>\overline{DE}</math> with their common ratio being <math>3</math>, and <math>\overline{EF}</math> and <math>\overline{BC}</math> also having a common ratio of <math>3</math>, and both of them sharing a common angle <math>\angle B</math> and <math>\angle E</math>, <math>\triangle ABC</math> and <math>\triangle DEF</math> are similar by SAS Similarity.  
 +
 
 +
Now, if <math>2</math> triangles are similar, all their angles are equal, which means that <math>m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD</math> since they are all base angles of similar triangles. If one of the non-base angles is <math>32^\circ</math>, it means that <math>\frac{(180-32)}{2}</math>, which is equal to <math>\frac{148}{2}</math>, which is <math>74^\circ</math>. So <math>m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD = 74^\circ</math>.

Latest revision as of 23:40, 7 June 2021

One of the main $3$ main theorems for proving the similarity between $2$ triangles. Similarity (specifically for triangles here) means all the angles in these triangles are equal and all sides are proportional to each other. SAS similarity is a similarity theorem stating that if there are two proportional sides and an angle in between it is equal, then they are similar by SAS similarity.

Example Problem: There are two isosceles triangles, $\triangle ABC$ and $\triangle DEF$. $m\angle B = m\angle E= 32^\circ$, and $\overline{AB} = 9$, $\overline{DE} = 3$, $\overline{BC} = 2$, $\overline{EF} = 6$. What are the measures of $m\angle BAC, m\angle BCA, m\angle EDF$ and $m\angle EFD$ (All of these are base angles for their respective triangles)?

Example Solution: Since sides $\overline{AB}$ and $\overline{DE}$ with their common ratio being $3$, and $\overline{EF}$ and $\overline{BC}$ also having a common ratio of $3$, and both of them sharing a common angle $\angle B$ and $\angle E$, $\triangle ABC$ and $\triangle DEF$ are similar by SAS Similarity.

Now, if $2$ triangles are similar, all their angles are equal, which means that $m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD$ since they are all base angles of similar triangles. If one of the non-base angles is $32^\circ$, it means that $\frac{(180-32)}{2}$, which is equal to $\frac{148}{2}$, which is $74^\circ$. So $m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD = 74^\circ$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=SAS_similarity&oldid=155560"