Question

What information is necessary to prove two triangles are similar by the SAS similarity theorem?

Answer 1

SAS similarity theorem says that the two sides of one triangle are proportional to the other sides of the another triangle an one included corresponding angles are congruent.

Further explanation:

SAS similarity theorem says that the two sides of one triangle are proportional to the other sides of the another triangle an one included corresponding angles are congruent.

It can be better explained through an example.

Step 1:

Consider the two triangles $\Delta OMN{\text{ and }}\Delta SQR$ that the corresponding sides are written below.

$\begin{aligned}ON \to SR \hfill\\OM \to QS \hfill \\MN \to QR \hfill\\\end{aligned}$  

Step 2:

The length of $ON$ is 8 units and the length of $SR$ is 32 units.

Therefore, the ratio of $SR$ and $ON$ can be expressed as,

$\begin{aligned}\frac{{SR}}{{ON}} &= \frac{{32}}{8}\\\frac{{SR}}{{ON}} &= 4\\\end{aligned}$  

Step 3:

The length of $OM$ is 15 units and the length of $SQ$ is 60 units.

Therefore, the ratio of $OM$ and $SQ$ can be expressed as,

$\begin{aligned}\frac{{SQ}}{{OM}} &= \frac{{60}}{{15}}\\\frac{{SQ}}{{OM}} &= 4\\\end{aligned}$  

Step 4:

In the triangle ${\text{ }}\Delta SQR$ the angle $\angle QSR = 60^\circ$.and in $\Delta OMN$ the angle $\angle NOM = 60^\circ$.

These two angles $\angle QSR = 60^\circ$ and $\angle NOM = 60^\circ$ are corresponding congruent angles.

Step 5:

It can be seen that the ratio of corresponding sides of the triangle are proportionalas,

$\dfrac{{SR}}{{ON}} = \frac{{SQ}}{{OM}} = 4$  

Therefore, it satisfies the condition of SAS similarity theorem.

Thus, $\Delta OMN{\text{ and }}\Delta SQR$ are similar by SAS theorem.

Learn more:  

1. Learn more about SSS similarity theorem brainly.com/question/3067398
2. Learn more about the symmetry for a function brainly.com/question/1286775
3. Learn more about midpoint of the segment brainly.com/question/3269852

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Triangle similarity.

Keywords: Triangle, similarity, sides, proportional, ratio, length, theorem, diagram, corresponding sides, angles, sum, fraction

Answer 2

Answer:

Sample Response: You need to show that two sides of one triangle are proportional to two corresponding sides of another triangle, with the included corresponding angles being congruent.

Step-by-step explanation:

that is the sample response