Question

are these triangles similar?
h
57°
f
33°
g
r
q
57°
37°
s
yes
no

Explanation:

Step1: Find missing angle in △HFG

The sum of angles in a triangle is $180^\circ$.
$\angle F = 180^\circ - 57^\circ - 33^\circ = 90^\circ$

Step2: Find missing angle in △RQS

The sum of angles in a triangle is $180^\circ$.
$\angle R = 180^\circ - 57^\circ - 37^\circ = 86^\circ$

Step3: Compare angle sets

△HFG angles: $57^\circ, 33^\circ, 90^\circ$
△RQS angles: $57^\circ, 37^\circ, 86^\circ$
No two pairs of corresponding angles are equal, so the triangles do not have matching angle sets required for similarity.
Wait correction: Recheck Step 2: $\angle R = 180 - 57 - 37 = 86$. Step 1: $\angle F = 90$. The only common angle is $57^\circ$, but the other angles do not match. Wait no, original answer was wrong. Correct answer is no.

Corrected Answer:

no

Step1: Calculate ∠F in △HFG

Sum of angles = $180^\circ$.
$\angle F = 180^\circ - 57^\circ - 33^\circ = 90^\circ$

Step2: Calculate ∠R in △RQS

Sum of angles = $180^\circ$.
$\angle R = 180^\circ - 57^\circ - 37^\circ = 86^\circ$

Step3: Compare all angle pairs

△HFG: $57^\circ, 33^\circ, 90^\circ$
△RQS: $57^\circ, 37^\circ, 86^\circ$
Only one pair of angles is equal; similarity requires two pairs of congruent corresponding angles (AA similarity criterion), which is not met.

Answer:

yes