Proving Triangles Similar
Standard
G.SRT.5 Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Goals for this section:
Students will learn how to prove similarity in two triangles using the AA~, SAS~, and SSS~ postulates. Previous knowledge of geometric reasoning such as parallel lines and transversals will be recalled in this lesson.
What is similarity?
Two polygons are similar (~) if corresponding angles are congruent and if the lengths of the corresponding sides are proportional.
- In other words, two polygons have the same shape but not necessarily the same size.
Proving Similarity
Angle-Angle Similarity Theorem (AA~)
If two angles of one triangle are congruent to two angles of another triangle then the triangles are similar.
Side-Angle-Side Similarity Theorem (SAS~)
If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar.
Side-Side-Side Similarity Theorem (SSS~)
If the corresponding sides of two triangles are proportional, then the triangles are similar.
Examples:
1. Prove ΔABC ~ ΔQRS.
Since we are not given angles, we cannot use AA~ or SAS~. We need to use the SSS~ postulate.
Tip: Identify our corresponding segments. We can determine this from what we're supposed to prove; ΔABC ~ ΔQRS.
Corresponding parts:
So now create ratios since we know what parts are corresponding. Are all the ratios equal? If yes, they're similar by the SSS~ postulate. If not, then they're not similar.
2. Prove ΔEFT ~ ΔHIG.
Hint: Remember your properties concerning angles, parallel lines, and transversals!
- Vertical Angles, Corresponding Angles, Supplementary Angles, Alternate Interior Angles, Alternate Exterior Angles, Same-Side Interior Angles.
Proof:
3. Which condition is sufficient to show that ΔABC ~ ΔQPR? Select all that apply.
a. RP = 4.5
b. m∠Q = 63
c. m∠P = 81
d. m∠R = 81
Hint: Use the theorems that we know, AA~, SAS~, and SSS~. What values would satisfy the theorems? Remember to find which angles/segments correspond with each other!
Corresponding parts:
What would we need to satisfy AA~?
What would we need to satisfy SAS~?
What would we need to satisfy SSS~?
To refresh your memory on triangle congruence by SSS and SAS, click on the link to take you back to Math 1.