The Gemmorah in Moed Katan 17 says:
א"ר יוחנן... אם דומה הרב למלאך ה' יבקשו תורה מפיו
ואם לאו אל יבקשו תורה מפיוRabbi Yoḥanan said as follows: ... If the teacher is similar to an angel [malakh] of the Lord, then seek Torah from his mouth, etc
How does (should) an angel of Hashem look/behave that a Rabbi must resemble him?
The prophet Zachariya (3,7), according to the Targums explanation, refers to angels as "those that stand by".
'Thus saith the LORD of hosts: If thou wilt walk in My ways, and if thou wilt keep My charge, and wilt also judge My house, and wilt also keep My courts, then I will give thee free access among these that stand by.
The accepted reason that angels are called "those that stand by" is because they are created at a specific spiritual level and have no way to climb higher, as apposed to us humans who's life mission is to reach higher and higher.
Rabbi Yoḥanan is teaching us that one should only learn from a Rabbi who is similar to an angel in this aspect, that he is willing to halt his own self-growth in order to allow his students to grow.
I heard this explanation from Rabbi Yehoshua Heshil Eichenstein Rosh Yeshiva of Yad Aharon some 20 odd years ago.
In his diary entry for August 6th, 1754 R. Hayyim Joseph David Azulai describes meeting R. Jacob Joshua Falk:
ותכף הלכתי לקבל פני הרב בעל פני יהושע הנז' ומראהו כמראה מלאך האלדים
I went immediately 'to receive the countenance' of the Rabbi, the author of P'nei Yehoshua aforem. 'And his appearance is as the appearance of an angel of the L-rd'!
(Cymerman translation)
Thus, to answer your question, an approximation of what an angel looks like is what R. Falk looked like. So if a rabbi looks like R. Falk, he should be okay.
If I have understood your question correctly, you are asking how it is possible for a Rebbi to be on the level of an angel? Or more precisely, how can they resemble what is essentially a celestially-rooted being when humans are inherently ‘earthly’ in form?
Rav Michel Yehuda Lefkowitz zt"l noted that the answer is that just as a מלאך has no personal aspirations but rather is driven purely to satisfy the will of Hashem, so too a Rebbi must be focused entirely on fuelling the success of his students.1
Thus, a Rebbi must be consistent with his appointed title. This means that he must not be deficient in the traits necessary for a מחנך. If a teacher is lacking in his level of observance, character or learning it will serve to hamper his personal example. Failure to be a true role model will abjectly impact on his students. The fact that we liken our Rebbeim to angels is a testament to the high regard with which we view them. They serve to mould and fashion our children into 'עובדי ה, and for this alone, we must endeavour to ensure that our children receive the correct level of teaching. Indeed, Rabbi Moshe Shternbach שליט"א notes that one must be quick to ascertain that their children have a רבי who is a ירא שמים and this means seeing him with your own eyes to verify if he will provide a good example.2
Likewise, Rabbi Yaakov Neiman זצ"ל in his דרכי מוסר also explains how important it is for a teacher to be a ירא שמים. When an educator has this מדה his influence is all the more impacting on his תלמידים. For example, when a teacher relates the miraculous experiences of יציאת מצרים, and his absolute 'אמונת ה is evident, it enters into the heart of his students and leaves a lasting impression. This is the whole understanding behind a Rebbi resembling a malach as he notes:
לכן אמרו חז"ל אם הרב דומה למלאך ד' תורה בקשו מפיו ואם לא אל יבקשו תורה מפיו כי הכל תלוי ברב שמלמד תורה לכן מצאנו בחז"ל שאמר רב לברי' כשתשב ללמוד תורה מרבך ראה בפניו שנאמר והי' עיניך רואות את מוריך כי ראיית פני הרב מביאה יראת שמים
Therefore Chazal says if a Rav resembles an angel of Hashem, seek Torah from his mouth and if not do not seek Torah from his mouth because it all depends on the Rav who teaches Torah. Therefore we find in Chazal that Rav said to his son when you sit to learn Torah from your Rebbi make sure to see his face. As it says "And your eyes shall see your teacher" because seeing the face of one's Rav leads one to become fearful of heaven.3
Conversely, when a teacher says something that lacks conviction, his students will see this lackadaisical approach and will not absorb the message. For this reason, R’ Neiman writes elsewhere that it is crucial that a Rebbi teaches with a mindset that is completely לשמה.4
It is only when an act is performed לשמה that Hashem can allow His שכינה to settle. This is because when a person is of pure heart and mind; his resulting actions truly reflect his intentions. This purity consequently provides a foothold for the Divine presence to cling to. This means that teachers have a tremendous responsibility as when they avoid acting in a לשמה manner their students may not learn properly. This in turn, can result in disinterested students which can ח"ו leave them open to foreign and evil influence. Thus the aim is for them to strive to be 'angelic' by providing a solid example and maintaining a strong 'יראת ה which filters down to their students.
1 As brought down by HaRav Michel Yehuda Lefkowitz זצ"ל in קונטרוס יסודות החינוך מרבותינו גדולי ישראל שליט"א: דברים שנאמרו בכינוס מאות מחנכים ומנהלים רוחניים, כסלו תשע"א, 'הרב הדומה למלאך שכל כוונתו תועלת התלמיד', עמוד 5 - He writes there;
דכמו שמלאך אין לו שום כוונה לתועלת עצמו וכל התכלית שלו זה רק לקיים רצון השי"ת, כך גם הרב המלמד את תורת ד' צריך שיהיה כל כוונתו לתועלת הצלחת התלמיד
2 Refer to ספר אורחות הבית, חלק ראשון, פרק ששה עשר, אות ז', עמוד קכב. He writes:
"והיו עיניך רואות את מוריך היאך הוא מתנהג באורחות חייו וכשהוא מתפלל שיהא לדוגמא ומופת לתלמידיו"
“And your eyes themselves should see your teacher, how he acts in his daily life and when he davens he should be an example and source of wonder to his students”.
In a similar vein, the חפץ חיים adds:
"מאד מאד יתחזק האדם לראות שבנו ישיג חינוך טהור בידיעת התורה והמצות וכמו שנהגו אבותינו מלפנים בדרך החינוך ללמד את בניהם חומש ורש"י שיהיו עי"ז יהודי נאמן לד' ותורתו"
“It is highly important for a person to see that his child receive a pure chinuch with Torah knowledge and Mitzvos, and like our fathers before them were accustomed in the ways of chinuch to teach their children Chumash and Rashi, and through this, a Jew will believe in Hashem and His Torah”. As written in בית ישראל, פרק ט
3 'דרכי מוסר, פרשת יתרו, 'והגדת לבנך (p.121). He begins by asking why it was that יתרו was so excited by משה’s report of the miracles that had occurred in the build up to and during יציאת מצרים, when he had already heard the same account prior? R’ Neiman answers,"אינו דומה שומע מפי הרב לשומע מפי התלמיד תלוי מי "המשמיע – “One cannot compare hearing something from the mouth of a Rov to hearing it from the mouth of the student, as it is dependent on the speaker.” This means that it was only when יתרו heard the miraculous happenings of מצרים from משה that it left a lasting impression.
An answer may lie with interpretations of the phrase רבנן...דמצייני כמלאכי השרת רבנן found in Nedarim 20a.
Furthermore, see Maharal, Nesivos Olam, Nesiv Haemes 3:3:
כי אין התורה בארץ רק שהאדם שיש בו התורה נחשב בשמים מצד התורה שהיא מן העליונים, ומפני כך אמרו בכל מקום (נדרים כ', ב') מאן מלאכי רבנן שבשביל התורה שיש עמהם אשר התורה היא מן השמים נחשבו רבנן מלאכים.
It is not that the Torah is here on earth, rather that one who is here on earth with Torah in him is as if he is elevated to the heavens, for the Torah is from the upper heavenly realms. This is why it is frequently stated ‘who are angles? Rabbis!’, since they have acquired Torah knowledge, which is from the heavens, they are considered like angles.
Thus, instead of figuring out if the teacher is similar to an angel, perhaps discern if an angel is similar to him — and through that one can certainly be successful.
This question was answered by the Lubavitcher Rebbe, you can watch the full content here with English subtitles, worthwhile to listen to the entire explanation: https://www.youtube.com/watch?v=JVFXZK6LDb8
Summary of the answer -
Torah instructs: "Provide yourself with a mentor." But how does one know who to choose as a spiritual guide? The Talmud Tractate Berachos teaches: "A mentor must be like an angel of G-d". In his code of Jewish law, Maimonides offers a clue to the conduct of an angel: He describes the future world, "when there will be no envy, rivalry, war or hatred," and concludes: "At that time the righteous will resemble the Ministering Angels." So to resemble an angel of G-d means, to be free of the corruptions of one's body. The Talmud Tractate Yevamos also offers insight: "There are three identifying signs of a Jew: He is modest, compassionate, and giving." Observe whether the potential mentor has these selfless qualities. If he does, one may confidently turn to him for spiritual direction and life-guidance.
