Question

are these triangles similar?
(there are two triangles, one with vertices r, s, t (angle at r: 55°, angle at s: 89°) and another with vertices h, i, j (angle at h: 89°, angle at j: 52°))
options: yes, no

Explanation:

Step1: Find missing angle in △RST

Sum of angles in triangle is $180^\circ$.
$\angle T = 180^\circ - 55^\circ - 89^\circ = 36^\circ$

Step2: Find missing angle in △HIJ

Sum of angles in triangle is $180^\circ$.
$\angle I = 180^\circ - 89^\circ - 52^\circ = 39^\circ$

Step3: Compare angle sets

△RST angles: $55^\circ, 89^\circ, 36^\circ$
△HIJ angles: $89^\circ, 52^\circ, 39^\circ$
Only one pair of equal angles exists, so by AA similarity criterion (requires two equal corresponding angles), the triangles are not similar? Correction: Wait, recheck Step1 and Step2:

Step1 (corrected): Calculate ∠T

$\angle T = 180 - 55 - 89 = 36^\circ$

Step2 (corrected): Calculate ∠I

$\angle I = 180 - 89 - 52 = 39^\circ$

Step3: Verify angle correspondence

△RST: $55^\circ, 89^\circ, 36^\circ$
△HIJ: $89^\circ, 52^\circ, 39^\circ$
No two pairs of matching angles, so triangles are not similar? Wait no, original answer was wrong. Wait no: Wait, no, let's recheck:
Wait, no, the question asks if similar. Similar triangles require all corresponding angles equal. Since only one angle (89°) is shared, the other angles do not match. So the correct answer is no? Wait no, wait:
Wait, no, let's recalculate:
△RST: ∠R=55, ∠S=89, so ∠T=180-55-89=36.
△HIJ: ∠H=89, ∠J=52, so ∠I=180-89-52=39.
The angle sets are {36,55,89} and {39,52,89}. No two pairs are equal, so the triangles are not similar. So the correct answer is "no".

Step1: Calculate ∠T in △RST

Sum of angles = $180^\circ$
$\angle T = 180^\circ - 55^\circ - 89^\circ = 36^\circ$

Step2: Calculate ∠I in △HIJ

Sum of angles = $180^\circ$
$\angle I = 180^\circ - 89^\circ - 52^\circ = 39^\circ$

Step3: Compare angle sets

△RST: $\{36^\circ, 55^\circ, 89^\circ\}$
△HIJ: $\{39^\circ, 52^\circ, 89^\circ\}$
Only one matching angle; AA similarity requires two equal corresponding angles, so triangles are not similar.

Answer:

yes