Math  /  Geometry

Question(Geometry, Unit 3, Lesson 10) Which of these statements is true? Select the correct choice.
A To know whether 2 triangles are similar, it is enough to know the measure of 1 angle.
B To know whether 2 triangles are similar, it is enough to know the length of 1 side.
C To know whether 2 triangles are similar, it is enough to know the measure of 2 angles in each triangle.
D To know whether 2 triangles are similar, it is enough to know the measure of 2 sides in each triangle.

Studdy Solution

STEP 1

What is this asking? Which facts guarantee that two triangles are similar? Watch out! Don't mix up congruence and similarity!
Similar triangles have the same *shape*, but not necessarily the same *size*.

STEP 2

1. Angle-Angle Similarity
2. Side-Side-Side Similarity
3. Side-Angle-Side Similarity
4. Ruling out the wrong answers

STEP 3

Hey everyone!
Let's talk about similar triangles!
Two triangles are similar if their corresponding angles are equal, and their corresponding sides are proportional.
If we know two angles of one triangle are equal to two angles of another triangle, then the third angles *must* also be equal (because the angles in a triangle add up to 180180^\circ).
This is the **Angle-Angle (AA)** similarity criterion.

STEP 4

Another way to know if two triangles are similar is if all three pairs of corresponding sides are in the same **proportion**.
For example, if one triangle has sides of length 33, 44, and 55, and another has sides of length 66, 88, and 1010, they're similar!
That's because 63=84=105=2\frac{6}{3} = \frac{8}{4} = \frac{10}{5} = \mathbf{2}.
They all have the same **scaling factor**!
This is the **Side-Side-Side (SSS)** similarity criterion.

STEP 5

One more important rule: **Side-Angle-Side (SAS)** similarity.
If two pairs of corresponding sides are in the same proportion *and* the angles between those sides are equal, then the triangles are similar.
Imagine triangles with sides 33 and 44 with a 3030^\circ angle between them, and another with sides 66 and 88 with a 3030^\circ angle between them.
Similar triangles!

STEP 6

Option A says one angle is enough.
Nope! Imagine a tiny equilateral triangle and a huge equilateral triangle.
They both have 6060^\circ angles, but they're not the same size, so just one angle isn't enough.

STEP 7

Option B says one side is enough.
Definitely not!
Think of a long, skinny triangle and a short, fat triangle.
They could both have a side of length 55, but be totally different shapes.

STEP 8

Option D says two sides are enough.
Not quite!
We need information about the angle between those sides too, or the triangles could still be different shapes.

STEP 9

Option C says two angles are enough.
Yes! Remember our **Angle-Angle** similarity?
If two angles match, the third *has* to match too!

STEP 10

The correct answer is C.
Knowing two angles in each triangle is enough to prove similarity.

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