Question

which postulate or theorem proves two triangles are congruent if two angles and the included side are equal?
a. sas
b. asa
c. sss
d. aas
in triangle △abc, ab = 9cm, bc = 12cm, and ac = 15cm. in triangle △def, de = 18cm, ef = 24cm. what is the length of df if the triangles are similar?
a. 40 cm
b. 20 cm
c. 30 cm
d. 35 cm

Explanation:

Step1: Recall congruence postulates

SAS is side - angle - side (two sides and the included angle equal), ASA is angle - side - angle (two angles and the included side equal), SSS is side - side - side (all three sides equal), and AAS is angle - angle - side (two angles and a non - included side equal). So for two angles and the included side equal, the postulate is ASA.

Step2: Find the similarity ratio

For similar triangles $\triangle ABC$ and $\triangle DEF$, find the ratio of corresponding sides. $\frac{DE}{AB}=\frac{18}{9} = 2$, $\frac{EF}{BC}=\frac{24}{12}=2$.

Step3: Calculate the length of $DF$

Since the triangles are similar and the ratio of corresponding sides is 2, and $AC = 15$ cm, then $DF=2\times AC$. So $DF = 2\times15=30$ cm.

Answer:

1. B. ASA
2. C. 30 cm