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High School Geometry: Homework Help Resource13 chapters | 142 lessons

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Lesson Transcript

Instructor:
*Betty Bundly*

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

In this lesson we will learn about trigonometry and geometry. The SAS theorem ties together those two subjects into a formula which allows us to find the area of a triangle.

Triangles may seem to be just another geometric shape, with three sides and three angles. However, they are special enough to have an entire subject devoted to themâ€”trigonometry. What makes triangles so interesting is that certain sides and certain angles relative to those sides have a special relationship. One of these relationships is found in the SAS theorem and the SAS formula (which uses the principles of the theorem to calculate the area of a triangle).

The **SAS theorem** states that two triangles are equal if two sides and the angle between those two sides are equal. The angle between the two sides is also called the **included angle**. In this diagram, if angle *C* = angle *X*, and side *a* = side *z* and side *b* = side *y*, then by the SAS theorem, these two triangles would be equal. SAS stands for side angle side.

In the diagram, notice that the angles are represented with capital letters and the sides of the triangle are represented with lowercase letters. If SAS is true for two triangles, this means that everything else about these triangles is equalâ€”all the other angles, all the other sides, and the area of the triangles.

This theorem from geometry led to a formula in trigonometry, called the SAS area formula. Using the **SAS area formula**, you can find the area of a triangle if you know the length of two sides of a triangle and included angle. Specifically, if the sides of the included angle are symbolized as *b* and *c*, and the included angle is called *A*, then:

area of triangle *ABC* = (*b*c**sin *A*) / 2

It's important that the angle you use for the area calculation lies between the two sides you use for the calculation. In fact, using this idea for any triangle, there are three ways to calculate the area.

Why? Looking at triangle *ABC*, we see there are three angles and each one is included between two different sides: angle *C* is the included angle for sides *a* and *b*; angle *B* is the included angle for sides *a* and *c*; angle *A* is the included angle for sides *b* and *c*. Any set of this trio of parts could be used to calculate the area, if you know the value of those three parts.

Before we move forward with SAS triangles, let's review a little trigonometry. A common formula from geometry is the formula for the area of a triangle. This formula says that *area* = *b*h* / 2, where *b* is a side of the triangle called the base, and *h* is the height of the triangle, where the height is always at 90 degrees to the base. Using SAS and this area formula, we will see why the SAS area formula works.

In a right triangle, the side opposite the right angle is called the **hypotenuse**. If you take the length of side of the triangle opposite a particular angle divided by the length of the hypotenuse, this ratio is called the **sine** of that angle (abbreviated sin).

Notice that *h* in our triangle is the height of the right triangle and it is also the side opposite the angle *A*. Therefore, using the hypotenuse, we can make the sine ratio:

sin *A* = height / hypotenuse or sin *A* = *h / c*

But, this *h* is the same *h* in the formula *area = b*h* / 2. What would the geometry formula for the area of a triangle look like if we used this *h* instead? The following steps show how to get this answer.

- Using algebra,
*h = c**sin*A* - Substitute this expression into area =
*b*h*/ 2: area =*b* c**(sin*A*) / 2

How does SAS fit into this picture? Notice that according to our diagram and from the new formula for area, the angle *A* is the included angle for the sides *b* and *c*.

You may wonder, why do we need another area formula? There are actually even more formulas for the area of a triangle than the two given here. But, one advantage of the SAS area formula is that you do not need to know the height of the triangle. You only need to know the length of two sides and the measure of the included angle. This will be helpful if, for example, the triangle you are measuring is not a right triangle, and the height is not given.

**SAS theorem** states that two triangles are equal if two sides and the angle between those two sides are equal. While the geometry formula for the area of a triangle is often used, the SAS theorem is used with trigonometry to provide an alternate method to calculate the area of a triangle.

Using the **SAS area formula**, you can find the area of a triangle if you know the length of two sides of a triangle and included angle. Specifically, if the sides of the included angle are symbolized as *b* and *c*, and the included angle is called *A*, then the area of triangle *ABC* equals (*b*c**sin *A*) / 2.

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High School Geometry: Homework Help Resource13 chapters | 142 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Mathematical Proof: Definition & Examples 3:41
- Perpendicular Slope: Definition & Examples
- Side-Angle-Side (SAS) Triangle: Definition, Theorem & Formula 4:57
- Triangle Inequality: Theorem & Proofs
- Two-Column Proof in Geometry: Definition & Examples 5:48
- What is a Vector in Math? - Definition & Examples 6:27
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