Question

what does the \side - angle - side\ similarity criterion verify? a. equal angles and one side b. equal sides and one angle c. all sides proportional d. proportional sides and one angle which of the following represents a correct ratio if triangles are similar by sas? a. ab + de = ac + df b. ab de = ac df c. ab - de = ac - df d. ab/de = ac/df if ∠a = ∠d = 80°, ab = 16 cm, ac = 24 cm, and de = 8 cm, what is the value of df if △abc ~ △def? a. 12 cm b. 14 cm c. 18 cm d. 10 cm

Explanation:

Step1: Recall SAS similarity criterion

The Side - Angle - Side (SAS) similarity criterion states that if two sides of one triangle are proportional to two corresponding sides of another triangle and the included angles are equal, the two triangles are similar.

Step2: Answer the first question

For the question "What does the 'Side - Angle - Side' similarity criterion verify?", the answer is that it verifies proportional sides and one (included) angle. So the answer is d. Proportional sides and one angle.

Step3: Recall ratio for similar triangles by SAS

For similar triangles by SAS, the ratio of corresponding sides is equal. That is, if $\triangle ABC\sim\triangle DEF$ by SAS, then $\frac{AB}{DE}=\frac{AC}{DF}$. So for the question "Which of the following represents a correct ratio if triangles are similar by SAS?", the answer is d. $AB/DE = AC/DF$.

Step4: Solve for the length of $DF$

Given $\angle A=\angle D = 80^{\circ}$, $\triangle ABC\sim\triangle DEF$ by SAS. We know that $\frac{AB}{DE}=\frac{AC}{DF}$. Substitute $AB = 16$ cm, $AC = 24$ cm, and $DE = 8$ cm into the proportion $\frac{16}{8}=\frac{24}{DF}$. Cross - multiply gives $16\times DF=24\times8$. Then $16DF = 192$, and $DF=\frac{192}{16}=12$ cm. So the answer for the third question is a. 12 cm.

Answer:

1. d. Proportional sides and one angle
2. d. $AB/DE = AC/DF$
3. a. 12 cm