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G.3(B)·Geometry·Logical Reasoning & ProofSupporting

Similar-Triangle Reasoning

Draw valid conclusions from angle relationships in similar triangles.

Students draw valid conclusions from angle information, proving triangles similar by AA and distinguishing similarity from congruence.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice45 minutes

Learn the concept

The third angle follows

The interior angles of every triangle sum to 180°. If two pairs of angles are congruent, then the remaining angles must also be congruent.

AA similarity

When two angles of one triangle are congruent to two angles of another, the triangles are similar. Corresponding sides are proportional, but they need not be equal.

Write a valid conclusion

List corresponding vertices in order, such as △ABC ~ △DEF when ∠A ≅ ∠D and ∠B ≅ ∠E. Then AB/DE = BC/EF = AC/DF. Congruence requires additional information about side lengths.

Worked example

In △PQR and △XYZ, ∠P ≅ ∠X = 40° and ∠Q ≅ ∠Y = 75°. What conclusions are valid?

  1. The third angles each measure 180° − 40° − 75° = 65°, so ∠R ≅ ∠Z.
  2. Two pairs of congruent angles prove △PQR ~ △XYZ by AA.
  3. The corresponding sides are proportional: PQ/XY = QR/YZ = PR/XZ.
  4. No side lengths are given, so congruence cannot be concluded.

△PQR ~ △XYZ, with ∠R ≅ ∠Z and proportional corresponding sides

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

7th Grade STAAR Practice Attributes of Similarity (7.5A - #8)

Five Minute Math · Aaron Daffern, Dallas ISD

Identify the attributes of similar figures: congruent corresponding angles and proportional sides.

Watch on YouTube
Texas public school video

Geometry: Deductive Reasoning in Quadrilaterals

UT IPSI · Kid2Kid / Texas Education Agency

Model deductive reasoning that justifies conclusions from given geometric properties.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. ∠A ≅ ∠D and ∠B ≅ ∠E. If AB = 6, DE = 9, and BC = 8, find EF.

    Show answer and reasoning

    EF = 12

    The triangles are similar by AA. The scale factor from △ABC to △DEF is 9/6 = 1.5, so EF = 8(1.5) = 12.

  2. 2. Can two pairs of congruent angles prove that △ABC ≅ △DEF?

    Show answer and reasoning

    No

    The triangles must have the same shape but may have different sizes. Angles alone prove similarity, not congruence.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 8 · 1 point

For triangles ABCABC and DEFDEF, ∠A≅∠D\angle A \cong \angle D and ∠B≅∠E\angle B \cong \angle E. Based on this information, which statement is a reasonable conclusion?

  1. ∠C≅∠D\angle C \cong \angle D because they are corresponding angles of congruent triangles.
  2. CA‾≅FD‾\overline{CA} \cong \overline{FD} because they are corresponding parts of congruent triangles.
  3. ∠C≅∠F\angle C \cong \angle F because they are corresponding angles of similar triangles.
  4. AB‾≅DE‾\overline{AB} \cong \overline{DE} because they are corresponding parts of similar triangles.
Show STAAR answer and explanation

∠C≅∠F\angle C \cong \angle F because they are corresponding angles of similar triangles.

Two pairs of corresponding congruent angles establish triangle similarity by AA. The third angles therefore correspond and are congruent: ∠C≅∠F\angle C\cong\angle F.

∠C≅∠D\angle C \cong \angle D because they are corresponding angles of congruent triangles.: C corresponds to F, not D, and AA establishes similarity rather than proving congruence.

CA‾≅FD‾\overline{CA} \cong \overline{FD} because they are corresponding parts of congruent triangles.: Corresponding sides of similar triangles are proportional, not necessarily congruent.

AB‾≅DE‾\overline{AB} \cong \overline{DE} because they are corresponding parts of similar triangles.: AB and DE correspond, but similarity alone does not make their lengths equal.

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Learning objectives

By the end of the lesson, students can:

  • Use the triangle angle sum to show that the third pair of angles is congruent.
  • Conclude similarity by AA and write a correct correspondence statement.
  • Explain why two pairs of congruent angles do not prove congruence.

Instructional sequence

  1. 1

    Engage

    5 min

    Show two triangles with matching angles but different sizes. Ask which conclusions are certain.

  2. 2

    Direct instruction

    10 min

    Use the 180° angle sum to justify the third angle and introduce AA similarity.

  3. 3

    Sort conclusions

    12 min

    Sort statements as justified or not justified by given angle information.

  4. 4

    Proportional sides

    12 min

    Use the correspondence to write proportions and find missing side lengths.

  5. 5

    Closure

    6 min

    Students write a justified conclusion for the exit ticket.

Misconceptions to address

““Two pairs of congruent angles prove the triangles are congruent.””

Teacher move: Show triangles with the same angles but different sizes. Angles alone fix the shape, not the size, so they prove similarity.

““The order of the letters does not matter.””

Teacher move: In a similarity statement, corresponding vertices must be listed in the same order so proportions match.