The labeling in the figure shown here is the convention we...

Name: Problem 6, Chapter 7: Precalculus: Graphs and Models
Uploaded: 2020-06-25T02:19:22-08:00
Duration: 7 min 2 s
Channel: Ahmad Reda
Description: The labeling in the figure shown here is the convention we will follow in this exercise set. Your answers to some problems may differ slightly from those in the book, depending on the order in which you solve for the sides and angles of a given triangle. (FIGURE CANNOT COPY) See Problem 5. Is there an SSA congruence theorem? An SSS congruence theorem? Explain.

The labeling in the figure shown here is the convention we will follow in this exercise set. Your answers to some problems may differ slightly from those in the book, depending on the order in which you solve for the sides and angles of a given triangle. (FIGURE CANNOT COPY) See Problem 5. Is there an SSA congruence theorem? An SSS congruence theorem? Explain.

The labeling in the figure shown here is the convention we will follow in this exercise set. Your answers to some problems may differ slightly from those in the book, depending on the order in which you solve for the sides and angles of a given triangle.
(FIGURE CANNOT COPY)
See Problem 5. Is there an SSA congruence theorem? An SSS congruence theorem? Explain.

Key Concepts

Triangle Congruence Theorems

Triangle congruence theorems are criteria used to determine whether two triangles are congruent, meaning all corresponding sides and angles are equal. Commonly accepted theorems include Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). These criteria guarantee that a triangle is uniquely determined by the given pieces of information, ensuring that if two triangles satisfy one of these conditions, they are congruent.

SSA and the Ambiguous Case

The SSA condition, which specifies two sides and a non-included angle, does not guarantee triangle congruence because it can lead to the ambiguous case. This means that for a given set of two sides and an angle not between them, there may be two distinct triangles, one unique triangle, or no triangle at all, depending on the relative lengths of the sides and the given angle. Therefore, SSA is not considered a valid congruence theorem in general.

SSS Congruence Theorem

The SSS congruence theorem states that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. This is a reliable method for determining congruence because the lengths of all sides uniquely determine the shape and size of the triangle.

Transcript

00:01 We have two types of triangles.

00:03 First type is ssa triangle.

00:09 The second type is s s s triangle.

00:17 The first type means that we have two sides and other angle not the including angle which means we have side a and side c and any of the angle gamma or alpha.

00:33 For example, we have alpha, but we don't have beta.

00:40 This is the type of ssa triangle.

00:44 The second type sss means that we know all the measurement of the three sides, a, b and c.

00:53 The question is, we want to figure out if these two types satisfy congruance theorem.

01:04 To find any type or to figure out that any type, satisfy congruance theorem or not first we check the determinancy of this triangle which means if we can get unique values of all measurement of the triangle then the triangle is deterministic and it satisfies congruance theorem let's apply on the first type for example ssa we can give an example we have a and alpha can we get unique values of b gamma and beta this is the question we have to answer let's find out how can we use law of cosines or law of signs we can't use law of signs here we can't use low of cosines we can use law of signs we can use law of signs let's try to find for example the angle we can get the angle gamma by using these two fractions we can say that gamma equals sine inverse of c divided by a multiplied by sine alpha we know here c a and alpha and then we can get gamma but note that that sine inverse can give us two angles can give us an acute angle and can give us an obvious angle...

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