Question 1
The angles 52 degrees match up. So if the sides are in proportion, then the triangles are similar.
Notice how 8/12 = 18/27 is a true equation. So the sides are in proportion to one another.
Therefore, the triangles <u>are similar</u>. I used the <u>SAS similarity theorem</u>.
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Question 2
Let's find the missing angle of the triangle on the right. For now, make that missing angle to be x.
Rule: For any triangle, the three angles always add up to 180
x+70+32 = 180
x+102 = 180
x = 180-102
x = 78
The missing angle of the triangle on the left is 78 degrees
Repeat for the other triangle as well
y+78+32 = 180
y+110 = 180
y = 180-110
y = 70
Both triangles have the angles 32, 70, 78 in whatever order you wish.
Since the angles match up like this, <u>the triangles are similar</u> because of the <u>AA Similarity theorem</u>
Technically, we only need 2 pairs of congruent angles at minimum. But having 3 congruent pairs doesn't hurt either.
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Question 3
We have a pair of congruent 54 degree angles shown. There's another pair of angles at the very top that overlap (aka shared angles).
Since we have two pairs of congruent angles, <u>the two triangles are similar</u>.
This time we use the <u>AA Similarity theorem</u>
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Question 4
Let's divide the longest side of the bottom triangle by the longest side of the triangle on top
48/32 = 1.5
Now let's divide the shortest sides of each triangle in the same order
36/24 = 1.5
Repeat for the middle-most sides
45/30 = 1.5
We get the same linear scale factor 1.5 each time. Therefore, <u>the triangles are similar</u> due to the <u>SSS Similarity theorem</u>
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Question 5
Any angle is equal to itself due to the <u>reflexive property</u>. So that will go for reason number 2.
Reason 3 is <u>SAS similarity</u> because we have the sides in proportion (statement 1) and the angles are congruent (statement 2).
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