If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

SAT (Fall 2023)

Course: SAT (Fall 2023) > Unit 6

Lesson 6: Additional Topics in Math: lessons by skill
Test prep
SAT (Fall 2023)
About the SAT Math Test
Additional Topics in Math: lessons by skill
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Congruence and similarity | Lesson

Google Classroom

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

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.
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!

try: find angle measures in an intersection of three lines
Based on the figure above, what is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
What is the value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
What is the value of c ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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 ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
What is the value of b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
What is the value of c ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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 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:
XY=k(AB)YZ=k(BC)XZ=k(AC)XYAB=YZBC=XZAC=k\begin{aligned} \maroonD{XY}&=k(\purpleD{AB}) \\\\ \maroonD{YZ} &=k(\purpleD{BC}) \\\\ \maroonD{XZ} &=k(\purpleD{AC}) \\\\ \dfrac{\maroonD{XY}}{\purpleD{AB}}&=\dfrac{\maroonD{YZ}}{\purpleD{BC}}=\dfrac{\maroonD{XZ}}{\purpleD{AC}}=k \end{aligned}
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!

try: use similarity to find side length
In the figure above, triangles A, B, D and B, C, D are similar. The length of start overline, B, D, end overline is 6, and the length of start overline, C, D, end overline is 12.
Based on the figure, start overline, C, D, end overline is the
of triangle B, C, D, and start overline, B, D, end overline is both the
of triangle B, C, D and the
of triangle A, B, D.
What is the length of start overline, A, D, end overline ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Your turn!

practice: find an angle measure
Intersecting lines p, q, and r are shown above. What is the value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

practice: find an angle measure
In the figure above, lines ell and m are parallel, y, equals, 30, and z, equals, 45. What is the value of x ?
Choose 1 answer:

practice: find a side length
In the figure above, segments A, B and D, E are parallel, and segments A, D and B, E intersect at C. What is the length of segment B, E ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want to join the conversation?

Log in
  • blobby green style avatar for user Noah T
    Posted a year ago. Direct link to Noah T's post “I dont like math. Help”
    I dont like math. Help
    (21 votes)
    Default Khan Academy avatar avatar for user
  • eggleston blue style avatar for user dagmaweamanuel
    Posted 3 months ago. Direct link to dagmaweamanuel's post “This in my opinion, is th...”
    This in my opinion, is the worst topic to go through compared to the rest.
    (7 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user redemptionottah06
    Posted 2 years ago. Direct link to redemptionottah06's post “So in the first video can...”
    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)
    Default Khan Academy avatar avatar for user
    • orange juice squid orange style avatar for user Evan
      Posted 2 years ago. Direct link to Evan's post “If the problem stated tha...”
      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 timestamp
      1:08
      in 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)
  • leaf blue style avatar for user PizzaMan21
    Posted 4 months ago. Direct link to PizzaMan21's post “I noticed in the explanat...”
    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)
    Default Khan Academy avatar avatar for user