Three Triangles - Three Unknown Angles

Category: Trigonometry, Plane Geometry

"Published in Suisun City, California, USA"

Refer to the figure. Show that α + β = γ, and find tan γ.

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Solution:

Referring to the given figure above

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Since the two small triangles are right triangles, then we can get the other angle of two triangles as follows

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The two vertices of a big triangle are vertical angles with the other two triangles. The sum of the interior angles of a big triangle is calculated as follows

Therefore,

Consider again the figure above

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We can get the tangent of α and β as follows

If

Take tangent on both sides of the equation, we have

Substitute the values of tan α and tan β to the above equation

Therefore,

Functions - Inverse Functions

Category: Algebra

"Published in Suisun City, California, USA"

Find the inverse function for

Solution:

Consider the given equation above

We know that y = f(x). Substitute y = f(x) as follows

Solve for x for the equation above

Replace x with y and y with x to the above equation, we have

Therefore,

Proving - Congruent Triangles, 2

Category: Plane Geometry

"Published in Newark, California, USA"

In the given figure, if MR ≅ PN; NR ≅ MP, prove that ∆NOR
≅ ∆MOP.

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Solution:

 Consider the given figure above

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Proof:

1. Statement: MR ≅ PN and NR ≅ MP.

    Reason: Given items.

2. Statement: PR ≅ PR

    Reason: Reflexive property of congruence.

3. Statement: ∆MRP ≅ ∆NPR

    Reason: Side Side Side (SSS) Postulate.

4. Statement: ∠MPR ≅ ∠NRP
                      ∠PMR ≅ ∠PNR
                      ∠PRM ≅ ∠RPN

    Reason: Since ∆MRP ≅ ∆NPR, then all interior angles of a triangle are congruent to all interior angles of other triangle. 

5. Statement: ∠PRM ≅ ∠4 and ∠RPN ≅ ∠3.

    Reason: Reflexive property of congruence.

6. Statement: ∠PRM ≅ ∠RPN ≅ ∠3 ≅ ∠4

    Reason: Transitive property of congruence.

7. Statement: ∆POR is an isosceles triangle.

    Reason: The two angles of an isosceles triangle are congruent. Hence, ∠3 ≅ ∠4 at the base.

8. Statement: OP ≅ OR

    Reason: Since ∆POR is an isosceles triangle, then the two sides of an isosceles triangle are congruent.

9. Statement: ∠MPR = ∠1 + ∠3 and ∠NRP = ∠2 + ∠4

    Reason: Addition property of angles.

10. Statement: ∠1 ≅ ∠2

      Reason: If ∠MPR ≅ ∠NRP and ∠3 ≅ ∠4, then it follows that ∠1 ≅ ∠2.

11. Statement: ∆MOP ≅ ∆NOR 

      Reason: Side Angle Side (SAS) Postulate.