Learn on PengiBig Ideas Math, Course 3Chapter 3: Angles and Triangles

Lesson 4: Using Similar Triangles

In this Grade 8 lesson from Big Ideas Math, Course 3 (Chapter 3: Angles and Triangles), students learn how to identify similar triangles using the Angle-Angle (AA) criterion, which states that two triangles are similar when two pairs of their angles are congruent. Students also apply indirect measurement, using proportional relationships in similar triangles to calculate distances or heights that cannot be measured directly, such as the height of a flagpole.

Section 1

The Angle-Angle (AA) Similarity Criterion

Property

To prove two triangles are similar (∼\sim), you do not need to check all their side lengths or all three angles. If you can prove that just two angles of one triangle are congruent (equal) to two angles of another triangle, the triangles are guaranteed to be similar. This is the AA Similarity Criterion.

Examples

  • Standard AA: △ABC\triangle ABC has angles of 45∘45^\circ and 60∘60^\circ. △DEF\triangle DEF has angles of 45∘45^\circ and 60∘60^\circ. Because two pairs match (45∘=45∘45^\circ=45^\circ and 60∘=60∘60^\circ=60^\circ), △ABC∼△DEF\triangle ABC \sim \triangle DEF.
  • The Hidden Match: △PQR\triangle PQR has angles of 30∘30^\circ and 80∘80^\circ. △XYZ\triangle XYZ has angles of 30∘30^\circ and 70∘70^\circ. Are they similar?
    • Find the missing angle in △PQR\triangle PQR: 180∘−(30∘+80∘)=70∘180^\circ - (30^\circ + 80^\circ) = 70^\circ.
    • Now we see △PQR\triangle PQR has a 30∘30^\circ and a 70∘70^\circ angle, matching △XYZ\triangle XYZ. Yes, they are similar!

Explanation

Why does AA work? Because of the Triangle Angle Sum Theorem. The three interior angles of any triangle must always add up exactly to 180∘180^\circ. Therefore, if two angles are already matched, the third angle has no choice but to be exactly the same! The AA criterion is the ultimate shortcut in geometry—it saves you from doing unnecessary measurements.

Section 2

Properties of Similar Triangles

Property

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio. For triangles, if △ABC\triangle ABC is similar to △XYZ\triangle XYZ, this property holds true.

Corresponding angles are equal:
m⁡∠A=m⁡∠X\operatorname{m}\angle A = \operatorname{m}\angle X
m⁡∠B=m⁡∠Y\operatorname{m}\angle B = \operatorname{m}\angle Y
m⁡∠C=m⁡∠Z\operatorname{m}\angle C = \operatorname{m}\angle Z

Corresponding sides are in the same ratio:

ax=by=cz\dfrac{a}{x} = \dfrac{b}{y} = \dfrac{c}{z}

Section 3

Indirect Measurement Using Similar Triangles

Property

Similar triangles can be used to find unknown measurements indirectly. When objects and their shadows are measured at the same time, they form similar right triangles because the sun's rays are parallel. By setting up a proportion using corresponding sides of these similar triangles, we can solve for unknown heights or distances.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

The Angle-Angle (AA) Similarity Criterion

Property

To prove two triangles are similar (∼\sim), you do not need to check all their side lengths or all three angles. If you can prove that just two angles of one triangle are congruent (equal) to two angles of another triangle, the triangles are guaranteed to be similar. This is the AA Similarity Criterion.

Examples

  • Standard AA: △ABC\triangle ABC has angles of 45∘45^\circ and 60∘60^\circ. △DEF\triangle DEF has angles of 45∘45^\circ and 60∘60^\circ. Because two pairs match (45∘=45∘45^\circ=45^\circ and 60∘=60∘60^\circ=60^\circ), △ABC∼△DEF\triangle ABC \sim \triangle DEF.
  • The Hidden Match: △PQR\triangle PQR has angles of 30∘30^\circ and 80∘80^\circ. △XYZ\triangle XYZ has angles of 30∘30^\circ and 70∘70^\circ. Are they similar?
    • Find the missing angle in △PQR\triangle PQR: 180∘−(30∘+80∘)=70∘180^\circ - (30^\circ + 80^\circ) = 70^\circ.
    • Now we see △PQR\triangle PQR has a 30∘30^\circ and a 70∘70^\circ angle, matching △XYZ\triangle XYZ. Yes, they are similar!

Explanation

Why does AA work? Because of the Triangle Angle Sum Theorem. The three interior angles of any triangle must always add up exactly to 180∘180^\circ. Therefore, if two angles are already matched, the third angle has no choice but to be exactly the same! The AA criterion is the ultimate shortcut in geometry—it saves you from doing unnecessary measurements.

Section 2

Properties of Similar Triangles

Property

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio. For triangles, if △ABC\triangle ABC is similar to △XYZ\triangle XYZ, this property holds true.

Corresponding angles are equal:
m⁡∠A=m⁡∠X\operatorname{m}\angle A = \operatorname{m}\angle X
m⁡∠B=m⁡∠Y\operatorname{m}\angle B = \operatorname{m}\angle Y
m⁡∠C=m⁡∠Z\operatorname{m}\angle C = \operatorname{m}\angle Z

Corresponding sides are in the same ratio:

ax=by=cz\dfrac{a}{x} = \dfrac{b}{y} = \dfrac{c}{z}

Section 3

Indirect Measurement Using Similar Triangles

Property

Similar triangles can be used to find unknown measurements indirectly. When objects and their shadows are measured at the same time, they form similar right triangles because the sun's rays are parallel. By setting up a proportion using corresponding sides of these similar triangles, we can solve for unknown heights or distances.

Examples