Exploring Similarity in Triangles

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Mrs. Casias Math explains the concept of similar triangles, focusing on the angle-angle criterion. The video breaks down how to determine if triangles are similar by ensuring two corresponding angles are congruent, which automatically makes the third angle the same. Through various examples, the criteria for similar triangles are explored, showing calculations to ascertain missing angles and confirm similarity or difference in the triangles.

- Determining similar triangles by checking congruent angles! 🎨
- Calculation practices for missing angles to determine triangle similarity. 📏
- The angle-angle criterion makes identifying similar triangles straightforward! 🔍

- Similar triangles have congruent angles but differ in size! 🤔
- Two angles being equal in two triangles means the third one is automatically the same, confirming similarity! 🔄
- Even without exact measures, understanding vertical angles and shared angles helps determine similarity. 🎯

Mrs. Casias delves into the fascinating world of geometry, specifically targeting similar triangles and how to spot them. By understanding that similar triangles have two angles that are congruent, students can quickly determine similarity even when sizes differ. The lesson simplifies the process, emphasizing the angle-angle criterion.

The video provides multiple examples of pairs of triangles, where viewers are guided through finding missing angles and comparing them to known angles. This approach teaches students to confidently identify whether triangles are similar or not using basic arithmetic operations and logical deductions.

Beyond exact measures, the lesson explores the significance of shared angles, such as vertical and common angles in overlaps, making it easy for students to apply these concepts to both theoretical and practical problems. Mrs. Casias's explanations ensure that students gain a solid understanding of similar triangles and the principles governing them.

**00:00 - 00:30**hey guys today we are going to be looking at angles in similar triangles we're going to answer the question what are similar triangles and how do we determine if triangles are similar or not so similar triangles have congruent angles but are not the same size so like right here we have the same angle measures but they're not the same sized triangle so in a triangle if two of the**00:30 - 01:00**corresponding angles are congruent like b and b prime and a and a prime then that means the triangles are similar and that automatically makes the third angle the same in both triangles so we're gonna go through these sets of triangles and determine if they're similar or not by using this if they have two of the same angles that automatically makes the third one the same so let's look at these triangles right here they both have an angle measure of 67 so we have one set of corresponding**01:00 - 01:30**angles that are the same now i need to find the third angle in this triangle right here and if the third angle is 48 then that means this one's automatically 65 so they are similar and remember we find the missing interior angle by seeing 67 plus 65 plus that missing angle setting it equal to 180**01:30 - 02:00**so let's find x i'm just going to use my calculator for this 65 plus 67 is 132. so 132 plus that missing angle is equal to 180 i'm gonna subtract 132 and that will tell me what that missing angle is 180 minus 132 is 48 degrees so 67 and 67 were the same**02:00 - 02:30**48 and 48 are the same and that means this one also has to be 65 degrees so these triangles are similar because we had two angles that we proved to the same which means that the third one is also the same okay let's look at number two these are right triangles so they both have a 90 degree angle now let's see if we can find another set of angles**02:30 - 03:00**that are the same so i'm going to find this missing angle right here i'm going to do 90 plus 54 plus x equals 180 and 90 plus 54 is 144 so i just subtract 144 from 80 to figure out that missing angle measure so 180 minus 144**03:00 - 03:30**is 36. so this angle right here is 36 so i do have two sets of angles that are the same and that automatically makes the third angle the same as well they're both 54. so these triangles are similar okay let's look at number three so i do have one set of angle measures that are the same**03:30 - 04:00**now i'm going to find this missing angle here and see if it's equal to 60 if not then it won't be similar so 80 plus 30 plus x equals 180 that will help me find the missing angle measure by using this equation so now i'm going to combine 80 plus 30 that's 110 and 110 plus x is 180 i'm going to subtract 110**04:00 - 04:30**and i get that x is 70. so this angle right here is 70. um on this triangle i have 80 60 and why don't we find out what that third one would be see if any of them are the same 80 plus 60 plus x equals 180 so that be 140 plus x equals 180 i'd subtract 140**04:30 - 05:00**and i get x equals 40. so these angles are 80 60 and 40. and the angles in this triangle are 80 70 and 30. so we don't have corresponding congruent angles so these triangles are not similar**05:00 - 05:30**and i knew that once this angle was not 60 because then there was no way that this one could have been 30 but i just wanted to prove that to you guys by solving for that angle and finding it was 40. okay let's look at this one so i do have two angles that are the same 75 now let's find this angle here and see if it is 50 if it is that means that they are similar triangles so 75 plus 55**05:30 - 06:00**plus x equals 180 and 75 plus 55 is 130 so 130 plus x equals 180 i'm gonna subtract 130 so x equals 50 because 130 or 180 minus 130 is 50. so i do have two sets of corresponding angles that automatically**06:00 - 06:30**makes the third one the same as well so these triangles are similar okay number five this one i can automatically tell that it is not going to be similar triangles because similar triangles have to have two angles that are the same so these might be the same**06:30 - 07:00**but even if they are my other two sets of angles are different so this one is not similar because there will not be two sets of congruent**07:00 - 07:30**angles okay number six so 61 i have that in both of my triangles they didn't give me another set of angles in either the triangles but whenever we talked about parallel lines cut by transversal we learned about vertical angles so these angles right here are vertical angles they are opposite angles formed by intersecting lines so we know that those angles are congruent**07:30 - 08:00**and we have two sets of congruent angles so that automatically makes the third angle the same as well so these are similar triangles because there are two sets of congruent angles even though we don't**08:00 - 08:30**know what these exact angle measures are we know that they are congruent because they are vertical angles okay let's look at number seven i have this bigger triangle and i have this smaller triangle**08:30 - 09:00**and they share this angle right here this 90 degree angle so they both have one angle that's the same already that right angle it's 90 degrees now i need to find one of those angles and see if it's the same in the other triangle so i'm going to do the smaller triangle i'm going to find that angle and see if it als is also 30 and is congruent to that second angle so 90 plus 60 plus x equals 180 90 plus 60 is 150**09:00 - 09:30**so my new equation is 150 plus x equals 180 and now i'm going to subtract 150 and x equals 30. so this angle is 30 i have another set of congruent angles i had the 90 degree angle and these are**09:30 - 10:00**both 30 so since two of them are the same these are similar triangles all right let's look at number eight so i have this large outside triangle and i have this smaller inside triangle up here and i already see one angle that's the same on both of them they both share this angle of 85 degrees so that**10:00 - 10:30**angle is the same i just need to find one other angle pair that's the same so i'm going to find this missing angle right here and see if it is also 42 so let's add the three angles of that pink triangle together 85 plus 53 plus x i'm going to set it equal to 180 and 85 plus 53 is 138 so my new equation is 138 plus x equals**10:30 - 11:00**180 i'm going to subtract 138 and that will tell me what that missing angle is so 180 minus 138 is 42. so i have two sets of angles that are the same they share this one of 85 and then this one's 42. that automatically makes these over here 53 so these triangles**11:00 - 11:30**all have these same angles so they are similar