Or

Tutorial

**Theorem 57** - If 2 triangles have the 3 angles of one = respectively to the 3 angles of the other, then the triangles are ~.

**Corollary 57-1** - If 2 angles of one triangle are = respectively to 2 angles of another, then the triangles are ~.

**Corollary 57 -2** - 2 right triangles are similar if an acute angle of one is = to an acute angle of the other.

**Theorem 58** - If 2 triangles have 2 pairs of sides proportional and the included angles = respectively, then the 2 triangles are ~. (s.a.s.)

**Corollary 58-1** - If the legs of one triangle are proportional to the legs of another, the triangles are similar. (l.l.)

**Theorem 59** - If 2 triangles have their sides respectively proportional, then the triangles are ~. (s.s.s.)

*Read through the proofs for T's and c's in your text before moving on to the exercises.

Source: ABEKA Plane Geometry 2006

Prove that 2 isosceles triangles are similar if any angle of one equals the corresponding angle of the other.

* case 1:* equal vertex angles

Given: Isosceles triangle ABC, isosceles triangle DEF

AC=BC; DF=EF; <C=<F

Prove: triangle ABC ~ triangle DEF

Draw 2 isosc. triangles, different sizes for your diagrams.

statements for case 1 proof:

2. <A = <D

3. triangle ABC ~ triangle DEF

case 2: equal base angles

statements for case 2 proof:

2. <A = <B; <D = <E

3. <B = <E

4. triangle ABC ~ triangle DEF

Source: ABEKA Plane Geometry 2006

case 1:

2. isosc. triangles with = vertex angles have = base angles also.

3. a.a.

case 2:

2. in any isosc. triangle, the angles opposite the = sides are =.

3. quantities that are = to = quantities are =.

4. a.a.

Source: ABEKA Plane Geometry 2006

If AC = 9, AE = 3, AB = 12, AD = 4, prove: triangle ABC ~ triangle ADE

*diagram in text

given: triangle ABC; DE; AC = 9; AE = 3; AB = 12; AD = 4

prove: triangle ABC ~ triangle ADE

statements for proof:

2. 12:4 = 9:3

3. AB:AD = AC:AE

4. <A = <A

5. triangle ABC ~ triangle ADE

Source: ABEKA Plane Geometry 2006

2. definition of proportion

3. substitution

4. identity

5. S.A.S.

Source: ABEKA Plane Geometry 2006

Prove that if a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle

*look at what the problem tells you, and draw the needed diagram. Remember, the line is drawn parallel to the base of the triangle, this line will be the transversal to the sides of the triangle. If 2 parallel lines are cut by a transversal, what do you know about the corresponding angles?

given: triangle ABC; DE parallel AB

prove: triangle DEC ~ triangle ABC

statements for proof:

2. <CDE = <A; <CED = <B

3. triangle ABC ~ triangle DEC

Source: ABEKA Plane Geometry 2006

2. if 2 parallel lines are cut by a transversal, then the corresponding angles are =.

3. a.a.

Source: ABEKA Plane Geometry 2006

In the triangle ABC, AE is the altitude on BC and CD is the altitude on AB. Prove: triangle CBD ~ triangle AEB

given: triangle ABC; AE altitude to BC; CD altitude to AB

*draw your diagram

statements for proof:

2. <B = <B

3. AE perpendicular to BC; CD perpendicular to AB

4. <AEB, <CDB are right angles

5. triangle CDB and triangle AEB are right triangles

6. triangle CDB ~ triangle AEB

Source: ABEKA Plane Geometry 2006