Geometry Reference
Triangle Similarity and Congruence
Congruent triangles match exactly. Similar triangles keep the same shape while corresponding side lengths scale by one constant factor.
Solved Examples
Jump directly to a worked example by proof condition or proportion task.
Fact Table
| Conclusion | Enough Information | What It Guarantees |
|---|---|---|
| Congruent triangles | SSS, SAS, ASA, AAS, or HL for right triangles. | All matching sides and angles are equal. |
| Similar triangles | AA, SSS proportionality, or SAS proportionality. | Matching angles are equal and sides have one scale factor. |
| Scale factor | $k=\frac{\text{image side}}{\text{original side}}$. | Lengths multiply by $k$; areas multiply by $k^2$. |
| Indirect measurement | Parallel rays or matching angles create similar triangles. | Use corresponding sides in the same order. |
Content Formulas
Similar Sides
$$\frac{a'}a=\frac{b'}b=\frac{c'}c=k$$
Similar Areas
$$\frac{A'}A=k^2$$
Correspondence
$$\triangle ABC\sim\triangle DEF\Rightarrow A\leftrightarrow D,\ B\leftrightarrow E,\ C\leftrightarrow F$$
Write corresponding vertices in matching order before creating a proportion. A correct calculation with mismatched sides gives the wrong answer.
Classic Examples
Test Similarity with Side Ratios
Are triangles with side lengths $3,4,5$ and $6,8,10$ similar?
Solution Steps
- Pair the shortest, middle, and longest sides.
- Compare each corresponding ratio.
- State the similarity conclusion and scale factor.
$$\frac63=2,\qquad\frac84=2$$$$\frac{10}{5}=2$$$$\boxed{\text{Yes; the triangles are similar with }k=2.}$$
Test Congruence with SAS
Two triangles each have sides $3$ and $4$ with an included angle of $60^\circ$. Are they congruent?
Solution Steps
- Identify the two matching side lengths.
- Confirm the given angle is included between them.
- Apply the SAS congruence criterion.
$$3=3,\qquad4=4$$$$\text{included angle }60^\circ=60^\circ$$$$\boxed{\text{The triangles are congruent by SAS.}}$$
Use a Scale Factor
A triangle with side length $5$ is enlarged by scale factor $3$. Find the new side length and area scale factor.
Solution Steps
- Multiply the original side by the scale factor.
- Square the scale factor for areas.
- State both scale results.
$$5(3)=15$$$$3^2=9$$$$\boxed{\text{new side }15,\quad\text{area scale factor }9}$$
Measure Height Indirectly
A $6$-foot stick casts a $4$-foot shadow. At the same time, a tree casts a $20$-foot shadow. Find the tree’s height.
Solution Steps
- Set equal the matching height-to-shadow ratios.
- Multiply across to isolate the tree height.
- State the measured height.
$$\frac64=\frac{h}{20}$$$$4h=120\Rightarrow h=30$$$$\boxed{30\text{ feet}}$$