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+Phew. I spent two days finishing Unit 3, Congruence, from the High
+School Geometry course on Khan Academy. I went through a lot of geometry
+proofs. I can definitely feel my progress slowing down a little, but it
+was worth the time.
+
+When two triangles have the same lengths for all corresponding sides and
+the same measures for all corresponding angles, we can say the two
+triangles must be congruent. However, we do not necessarily need all of
+those facts to prove congruence. There are several shortcuts: SSS, SAS,
+ASA, AAS, and HL.
+
+SSS stands for side-side-side. When three pairs of corresponding sides
+are congruent, the two triangles are congruent.
+
+Proof: For triangles ABC and DEF, suppose we are given that segment AB
+is congruent to segment DE, segment BC is congruent to segment EF, and
+segment AC is congruent to segment DF. We map segment AB onto segment DE
+through a rigid transformation. Rigid transformations preserve distances
+and angle measures. Then draw a circle centered at D with radius DF, and
+another circle centered at E with radius EF. The image of point C must
+lie at one of the intersections of the two circles. If the image point
+C' lands on F, we are done. Otherwise, it lies on the opposite side of
+segment DE, and reflecting the triangle across DE maps C' onto F.
+
+SAS stands for side-angle-side. If two corresponding sides are
+congruent, and the included angles between them are congruent, then the
+two triangles are congruent.
+
+Proof: For triangles ABC and DEF, suppose we are given that segment AB
+is congruent to segment DE, segment BC is congruent to segment EF, and
+angle B is congruent to angle E. We map segment AB onto segment DE
+through a rigid transformation. Rigid transformations preserve distances
+and angle measures. Then draw a circle centered at E with radius EF. The
+image point C' lies at the intersection of the circle and the ray
+forming angle E with segment DE. If C' lands on F, we are done.
+Otherwise, it lies on the opposite side of segment DE, and reflecting
+the triangle across DE maps C' onto F.
+
+ASA stands for angle-side-angle. When two pairs of corresponding angles
+are congruent, and the included side between them is congruent, then the
+two triangles are congruent.
+
+Proof: For triangles ABC and DEF, suppose we are given that segment BC
+is congruent to segment EF, angle B is congruent to angle E, and angle C
+is congruent to angle F. We map segment BC onto segment EF through a
+rigid transformation. Rigid transformations preserve distances and angle
+measures. The two angles determine two rays, and their intersection
+determines the image point A'. If A' lands on D, we are done. Otherwise,
+it lies on the opposite side of segment EF, and reflecting the triangle
+across EF maps A' onto D.
+
+AAS stands for angle-angle-side. When two pairs of corresponding angles
+are congruent, and a non-included side is congruent, then the two
+triangles are congruent.
+
+Proof: The interior angles of a triangle sum to 180°. Since two angles
+are already known, we can calculate the third angle. Then we can apply
+the ASA postulate to prove the triangles are congruent.
+
+HL stands for hypotenuse-leg. If the hypotenuses and one pair of
+corresponding legs of two right triangles are congruent, then the
+triangles are congruent.
+
+Proof: This is a special case of right triangles. Using the Pythagorean
+Theorem, we can calculate the missing side length. Once we know all
+three side lengths, we can apply the SSS postulate.
+
+There is also an SSA case—side-side-angle—which is not a valid triangle
+congruence postulate.
+
+Suppose we are given that segment AB is congruent to segment DE, segment
+AC is congruent to segment DF, and angle C is congruent to angle F. We
+map segment AC onto segment DF through a rigid transformation. Rigid
+transformations preserve distances and angle measures. Then draw a
+circle centered at D with radius DE. The image point B' may lie at one
+of two intersections between the circle and the ray forming angle F with
+segment DF. In some cases, both positions satisfy the given conditions
+but form different triangles. Therefore, SSA is ambiguous and does not
+guarantee triangle congruence.
+
+The unit also covers the reflexive, symmetric, and transitive properties
+of congruence.
+
+Reflexive property: A relation is reflexive if every object is related
+to itself. For example, if triangles ABC and ADC share side AC, then
+segment AC is congruent to itself by the reflexive property.
+
+Symmetric property: If A is related to B, then B is also related to A.
+For example, if segment AB is congruent to segment BC, then segment BC
+is also congruent to AB by the symmetric property.
+
+Transitive property: If A is related to B, and B is related to C, then A
+is related to C. For example, if angle A is congruent to angle B, and
+angle B is congruent to angle C, then angle A is congruent to angle C by
+the transitive property.
+
+In the evening, I spent some time watching the keynote of Google I/O
+2026. Google introduced a new family of Gemini models—Gemini 3.5 Flash,
+which is highly capable and outperforms Gemini 3.1 Pro. It even
+outperforms Claude Opus 4.7 and GPT 5.5 in some benchmarks. The crazy
+thing is that its output speed is incredibly fast, reaching nearly 280
+tokens per second. The Pro version will be released later next month.
+
+Google also lowered the price of the highest-tier Gemini subscription
+from $250 to $200. One thing that stops me from using Gemini is that I
+cannot opt out of AI traning unless I gave up my chat history. I feel
+uncomfortable about that, almost as if I am being force into it.
+
+During the keynote, Google—an AI-first company—also demonstrated how
+deeply they are integrating Gemini 3.5 Flash into their products and
+across the entire ecosystem: Google Search, Shopping, YouTube, Gmail,
+and more. In the near future, Google Search will leverage Antigravity to
+generate interactive demos from scratch to help people understand new
+ideas or concepts. That was probably the feature I liked the most.
+
+I feel deeply touched by how AI may completely change people's work and
+daily lives. Software may become dispoable and one-time use. At the end
+of the keynote, Google introduced a new version of Google Glasses. I
+guess I probably will not be able to experience them, but I can already
+think of one use case that would suit me very well right now: whenever I
+need help with a math problem, I would not need to take out my phone and
+snap a photo anymore. I could simply talk to Gemini directly through the
+glasses—it sees what I see.
+
+Oh, and there was also another keynote dedicated specifically to
+developers. I guess I need to set aside some extra time to watch that as
+well.