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### Course: Digital SAT Math > Unit 13

Lesson 2: Congruence, similarity, and angle relationships: advanced# Congruence, similarity, and angle relationships | Lesson

A guide to congruence, similarity, and angle relationships on the digital SAT

## What are congruence, similarity, and angle relationship problems?

Congruence, similarity, and angle relationship problems ask us to solve for an unknown value using , (shown below), and intersections of lines.

**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.

To solve for unknown angle measures, we need to know the following information:

- 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 ${{x}^{\circ}}+{{y}^{\circ}}+{{z}^{\circ}}={180}^{\circ}$ , 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 ${ABC}$ and ${XYZ}$ 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 ${ABC}$ and ${XYZ}$ , the following is true:

Let's try applying the properties of similar triangles. In the figure below, $\stackrel{\u2015}{BD}$ is parallel to $\stackrel{\u2015}{AE}$ . If $BC=10$ , $BD=14$ , and $AE=21$ , what is the length of $\stackrel{\u2015}{AC}$ ?

### Try it!

## Your turn!

## Want to join the conversation?

- Highlight of Khan SAT Prep :the comments(206 votes)
- u got this! believe in urself whoever is reading this!!(162 votes)
- Thank you so much. It actually made me feel better >3(34 votes)

- just keep reading and try to understand the text and concept, you'll get through this, happy learning folks.(158 votes)
- As long as the line is straight there is nothing to worry about.(134 votes)
- i really like that fact that there is a lot of kind and supportive people in the comments :)

Don't give up! You've got this !!(58 votes) - May we all get reward of our hard work . Ameen(25 votes)
- May we all get 1600 by our hard work. Amen.(32 votes)

- how come the questions are the same from foundation to advanced?(16 votes)
- but from practice test i took literally none were advanced(16 votes)

- how do you know the proportional relationship, that's what I am really struggling with right now.(7 votes)
- You know how triangles have a shorter leg, longer leg and a hypotenuse right? Just make sure you are using the right leg when comparing the triangles. Just because they look to be on the same side doesn't make them similar.(10 votes)

- can someone please explain the short and long leg question? what is the short and long leg?(5 votes)
- heyy, i'm not a pro but i'll try my best. So in a right triangle there is the hypotenuse and two other lines that form the triangle. From those two lines the longer one is called the 'long leg' and the shorter one i called the 'short leg'. Hope it helps)(8 votes)