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

Course: Praxis Core Math > Unit 1

Lesson 5: Geometry
Test prep
Praxis Core Math
Lessons
Geometry
© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Congruence and similarity | Lesson

Google Classroom

What do congruent and similar mean?

Congruent triangles have both the same shape and the same size. In the figure below, triangles ABC and DEF are congruent; they have the same angle measures and the same side lengths.
Similar triangles have the same shape, but not necessarily the same size. In the figure below, triangles ABC and XYZ are similar, but not congruent; they have the same angle measures, but not the same side lengths.
Note: If two objects are congruent, then they are also similar.

What skills are tested?

  • Determining whether two triangles are congruent
  • Determining whether two triangles are similar
  • Using similarity to find a missing side length

What are the triangle congruence criteria?

Two triangles are congruent if they meet one of the following criteria.
: All three pairs of corresponding sides are equal.
: Two pairs of corresponding sides and the corresponding angles between them are equal.
: Two pairs of corresponding angles and the corresponding sides between them are equal.
: Two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are equal.
: The pair of
and another pair of corresponding sides are equal in two right triangles.

What are the triangle similarity criteria?

Two triangles are similar if they meet one of the following criteria.
: Two pairs of corresponding angles are equal.
: Three pairs of corresponding sides are proportional.
: Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

Finding missing side lengths in similar triangles

The SSS similarity criterion allows us to calculate missing side lengths in similar triangles. For similar triangles ABC and XYZ shown below:
XY=k(AB)YZ=k(BC)XZ=k(AC)XYAB=YZBC=XZAC=k
To calculate a missing side length, we:
  1. Write a proportional relationship using two pairs of corresponding sides.
  2. Plug in known side lengths. We need to know 3 of the 4 side lengths to solve for the missing side length.
  3. Solve for the missing side length.

Your turn!

TRY: PROPERTIES OF CONGRUENT TRIANGLES
Triangles ABC and DEF are shown above. If the two triangles are congruent, which of the following statements must be true?
Choose 1 answer:

TRY: CONGRUENCE CRITERIA
For triangles LMN and PQR, LM=PQ, and ∠M has the same measure as ∠Q. Which of the following statements, if true, is sufficient to show that the two triangles are congruent?
Choose all answers that apply:

TRY: IDENTIFYING SIMILAR TRIANGLES
Triangle ABC is shown above. Which of the following triangles are similar to Triangle ABC ?
Choose all answers that apply:

TRY: CALCULATING MISSING SIDE LENGTH
Triangles RST and XYZ are similar triangles. What is the length of RT ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Things to remember

Congruent triangles have the same corresponding angle measures and side lengths. The triangle congruence criteria are:
  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Angle-Angle-Side)
  • HL (Hypotenuse-Leg, right triangle only)
Similar triangles have the same corresponding angle measures and proportional side lengths. The triangle similarity criteria are:
  • AA (Angle-Angle)
  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
If triangles ABC and XYZ are similar, then their corresponding side lengths have the same ratio:
XYAB=YZBC=XZAC=k

Want to join the conversation?

Log in