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## Digital SAT Math

### Course: Digital SAT Math>Unit 5

Lesson 2: Congruence, similarity, and angle relationships: foundations

# 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

Khan Academy video wrapper
Worked example: Triangle angles (intersecting lines)See video transcript

### 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!

try: find angle measures in an intersection of three lines
Based on the figure above, what is the value of $a$ ?
What is the value of $b$ ?
What is the value of $c$ ?

try: find angle measures of parallel lines and transversals
The figure above shows two horizontal lines and two intersecting transversals.
What is the value of $a$ ?
What is the value of $b$ ?
What is the value of $c$ ?

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

### Solving similar triangles

Khan Academy video wrapper
Solving similar trianglesSee video transcript

### 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:
$\begin{array}{rl}XY& =k\left(AB\right)\\ \\ YZ& =k\left(BC\right)\\ \\ XZ& =k\left(AC\right)\\ \\ \frac{XY}{AB}& =\frac{YZ}{BC}=\frac{XZ}{AC}=k\end{array}$
Let's try applying the properties of similar triangles. In the figure below, $\stackrel{―}{BD}$ is parallel to $\stackrel{―}{AE}$. If $BC=10$, $BD=14$, and $AE=21$, what is the length of $\stackrel{―}{AC}$ ?

### Try it!

try: use similarity to find side length
In the figure above, triangles $ABD$ and $BCD$ are similar. The length of $\stackrel{―}{BD}$ is $6$, and the length of $\stackrel{―}{CD}$ is $12$.
Based on the figure, $\stackrel{―}{CD}$ is the
of triangle $BCD$, and $\stackrel{―}{BD}$ is both the
of triangle $BCD$ and the
of triangle $ABD$.
What is the length of $\stackrel{―}{AD}$ ?

practice: find an angle measure
Intersecting lines $p$, $q$, and $r$ are shown above. What is the value of $x$ ?

practice: find an angle measure
In the figure above, lines $\ell$ and $m$ are parallel, $y=30$, and $z=45$. What is the value of $x$ ?

practice: find a side length
In the figure above, segments $AB$ and $DE$ are parallel, and segments $AD$ and $BE$ intersect at $C$. What is the length of segment $BE$ ?

## Want to join the conversation?

• Whatever is it, don't give up. You can do anything! Trust me.
• made my day
• just keep reading and try to understand the text and concept, you'll get through this, happy learning folks.
• As long as the line is straight there is nothing to worry about.
• u got this! believe in urself whoever is reading this!!
• Thank you so much. It actually made me feel better >3
• Highlight of Khan SAT Prep :the comments
• how come the questions are the same from foundation to advanced?
• but from practice test i took literally none were advanced
• how do you know the proportional relationship, that's what I am really struggling with right now.