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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 6

Lesson 6: Additional Topics in Math: lessons by skill- Volume word problems | Lesson
- Right triangle word problems | Lesson
- Congruence and similarity | Lesson
- Right triangle trigonometry | Lesson
- Angles, arc lengths, and trig functions | Lesson
- Circle theorems | Lesson
- Circle equations | Lesson
- Complex numbers | Lesson

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# Congruence and similarity | Lesson

## What are congruence and similarity problems, and how frequently do they appear on the test?

Congruence and similarity problems ask us to use and (shown below) to solve problems.

On your official SAT, you'll likely see

**1 to 2 questions**about congruence and similarity.**You can learn anything. Let's do this!**

## What are some common ways the SAT combines angle relationships?

### Finding angles in triangles

### Triangles and other angle relationships

On the SAT, we're expected to find unknown angle measures when only a few are given. More often than not, triangles are involved.

At the beginning of each SAT math section, the following information is provided as reference:

- The sum of the measures in degrees of the angles of a triangle is 180.

#### Triangles, vertical angles, and supplementary angles

One common type of figure on the SAT is a triangle formed by three intersecting lines, as shown below.

We know that start color #7854ab, x, degrees, end color #7854ab, plus, start color #ca337c, y, degrees, end color #ca337c, plus, start color #208170, z, degrees, end color #208170, equals, 180, degrees, but we also know how the angles outside the triangle relate to the inside angles based on the properties of and .

#### Triangles and parallel lines

Another common type of figure shows constructed using parallel lines.

Two similar triangles can be constructed from two parallel lines and two intersecting transversals, as shown below.

**Note:**Since the two triangles have different orientations, be careful when identifying the corresponding sides! In two similar triangles, the longest side in one corresponds to the longest side in the other and so on.

Two similar triangles can also be constructed by drawing a line inside a triangle that's parallel to one of the sides. In the example shown below, the line inside the triangle is parallel to the base of the triangle and divides the larger triangle into a similar smaller triangle and a quadrilateral.

### Try it!

## How do I use similarity to find side lengths?

### Solving similar triangles

### Setting up proportional relationships using similarity

Similar triangles have the same shape, but aren't necessarily the same size. In the figure below, triangles start color #7854ab, A, B, C, end color #7854ab and start color #ca337c, X, Y, Z, end color #ca337c are similar: they have the same angle measures, but not the same side lengths.

The corresponding side lengths of similar triangles are related by a constant ratio, which we can call k. For similar triangles start color #7854ab, A, B, C, end color #7854ab and start color #ca337c, X, Y, Z, end color #ca337c, the following is true:

Let's try applying the properties of similar triangles. In the figure below, start overline, B, D, end overline is parallel to start overline, A, E, end overline. If B, C, equals, 10, B, D, equals, 14, and A, E, equals, 21, what is the length of start overline, A, C, end overline ?

### Try it!

## Your turn!

## Want to join the conversation?

- I dont like math. Help(21 votes)
- I can't help you with that. I'm sorry.

I get that math is a hard thing to like, but can you at least try?(2 votes)

- This in my opinion, is the worst topic to go through compared to the rest.(7 votes)
- So in the first video can another way of solving it be;

180-121=59 instead of going through the whole process. I just want to know if it's going to work for other problems like it.(0 votes)- If the problem stated that the two diagonal lines were parallel, then you'd be able to assume angle x was equal to the angle Sal solved for at timestamp1:08in the video. Then you'd jest be able to do
*180 - 121 = 59 = x*. Since the problem didn't state that the lines were parallel, you wouldn't be able to assume that. For this problem, the lines ended up being parallel, so that logic did work out. Though lines that look close to parallel may not actually be parallel in all problems, so it's safer to do all the math. However, you could first try and prove the lines were parallel, and then you'd be able to follow that logic.

If you were pressed for time on the real SAT when you got that problem, you could just use your gut instinct and do*180 - 121 = 59 = x*. Though if you have the time, it's good to do the math out.

Have a great day! (:(6 votes)

- I noticed in the explanation of the third "Try It!" question, it doesn't tell you the length of the leg for the last dropdown option. It just says: "and the leg of triangle ABD."

It is obviously "longer leg" but still, it would have been nice if they included that in the explanation.(2 votes)